[{"Name":"Motion Under Gravitational Force and Central Force","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro","Duration":"17m 47s","ChapterTopicVideoID":9183,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"Hello. In this lesson,"},{"Start":"00:01.755 ","End":"00:07.230","Text":"we\u0027re going to be speaking about the central force and what its characteristics are."},{"Start":"00:07.230 ","End":"00:11.805","Text":"Our central force is going to be of this format,"},{"Start":"00:11.805 ","End":"00:15.000","Text":"so it\u0027s going to be some force in terms of r,"},{"Start":"00:15.000 ","End":"00:19.210","Text":"our radius, and the radial direction."},{"Start":"00:19.490 ","End":"00:26.445","Text":"Our f(r) can be any function that has the variable r in it."},{"Start":"00:26.445 ","End":"00:34.560","Text":"For instance, some constant divided by r or a constant divided by r^2 or e^r,"},{"Start":"00:34.560 ","End":"00:41.830","Text":"any function with a variable r. The direction of the force is given here by r-hat."},{"Start":"00:41.830 ","End":"00:44.780","Text":"What does that mean? That means that if we have over"},{"Start":"00:44.780 ","End":"00:47.600","Text":"here our axes and here is our origin."},{"Start":"00:47.600 ","End":"00:54.020","Text":"The force acting on the body has to always be in the direction of r-hat,"},{"Start":"00:54.020 ","End":"00:56.490","Text":"that\u0027s what\u0027s written over here."},{"Start":"00:57.070 ","End":"01:03.085","Text":"Our r-hat is pointing from the origin to our body."},{"Start":"01:03.085 ","End":"01:07.205","Text":"It can be outwards if our force is positive."},{"Start":"01:07.205 ","End":"01:10.433","Text":"If for instance, our force is negative,"},{"Start":"01:10.433 ","End":"01:11.780","Text":"here there\u0027s a negative,"},{"Start":"01:11.780 ","End":"01:19.900","Text":"then our r vector or our F will be pointing inwards towards the origin."},{"Start":"01:19.900 ","End":"01:22.100","Text":"If the body is over here,"},{"Start":"01:22.100 ","End":"01:26.869","Text":"then our force is going to point either in this direction or in that direction,"},{"Start":"01:26.869 ","End":"01:28.835","Text":"in a straight line towards the origin,"},{"Start":"01:28.835 ","End":"01:30.650","Text":"depending on the sign of the force."},{"Start":"01:30.650 ","End":"01:32.705","Text":"If our body is over here,"},{"Start":"01:32.705 ","End":"01:36.710","Text":"then it\u0027s going to point outwards in a straight line from the origin out,"},{"Start":"01:36.710 ","End":"01:40.430","Text":"and if it\u0027s a negative then a straight line into the origin,"},{"Start":"01:40.430 ","End":"01:44.380","Text":"and similarly, if the body is any, which plays."},{"Start":"01:44.380 ","End":"01:49.992","Text":"Now examples for central force are gravity,"},{"Start":"01:49.992 ","End":"01:52.980","Text":"our g and also our electric force,"},{"Start":"01:52.980 ","End":"01:58.950","Text":"F_E For instance, with our sun,"},{"Start":"01:58.950 ","End":"02:01.295","Text":"so if our sun is over here,"},{"Start":"02:01.295 ","End":"02:03.590","Text":"so it\u0027s located at the origin,"},{"Start":"02:03.590 ","End":"02:09.665","Text":"and then our planet is always going to be pulled into the center of our sun."},{"Start":"02:09.665 ","End":"02:12.560","Text":"The gravitational pull from the sun."},{"Start":"02:12.560 ","End":"02:16.015","Text":"It doesn\u0027t matter where our planet or other planets or"},{"Start":"02:16.015 ","End":"02:19.910","Text":"other stars are in relation to our sun,"},{"Start":"02:19.910 ","End":"02:24.115","Text":"but the force is always going to be pulling them to the center."},{"Start":"02:24.115 ","End":"02:27.980","Text":"When we\u0027re dealing with our electrostatic forces,"},{"Start":"02:27.980 ","End":"02:32.855","Text":"so if we imagine that this is some charge and these are other charges."},{"Start":"02:32.855 ","End":"02:35.960","Text":"Always, depending on the sign of the charges,"},{"Start":"02:35.960 ","End":"02:40.195","Text":"if they\u0027re both positive or negative or 1 is positive and 1 Is negative,"},{"Start":"02:40.195 ","End":"02:43.880","Text":"so our central force is always going to be either"},{"Start":"02:43.880 ","End":"02:48.590","Text":"pushing or pulling into the center of the charge."},{"Start":"02:48.590 ","End":"02:53.010","Text":"Each one will be applying the force to the other one."},{"Start":"02:53.060 ","End":"03:00.340","Text":"They\u0027ll either be attractive or they\u0027ll repel one another."},{"Start":"03:00.470 ","End":"03:05.845","Text":"Now we have a bit of a basic understanding in what central force is,"},{"Start":"03:05.845 ","End":"03:10.780","Text":"these types of calculations come up all the way through physics, okay,"},{"Start":"03:10.780 ","End":"03:15.250","Text":"so from mechanics to your electricity courses and so on and so forth."},{"Start":"03:15.250 ","End":"03:19.340","Text":"It\u0027s very important to understand this concept."},{"Start":"03:19.340 ","End":"03:23.710","Text":"Now let\u0027s speak of some of the characteristics of the central force."},{"Start":"03:23.710 ","End":"03:26.305","Text":"First I\u0027m going to list the characteristics,"},{"Start":"03:26.305 ","End":"03:28.645","Text":"and then afterwards we\u0027re going to prove"},{"Start":"03:28.645 ","End":"03:32.105","Text":"the characteristics and show where they come from."},{"Start":"03:32.105 ","End":"03:38.705","Text":"The first characteristic is that a central force is always a conservative field."},{"Start":"03:38.705 ","End":"03:41.915","Text":"That means that in questions involving central force,"},{"Start":"03:41.915 ","End":"03:45.650","Text":"there will always be conservation of energy."},{"Start":"03:45.650 ","End":"03:49.170","Text":"This is a very useful fact,"},{"Start":"03:49.170 ","End":"03:56.080","Text":"and you\u0027re going to use this fact a lot when you\u0027re solving questions."},{"Start":"03:56.080 ","End":"04:02.105","Text":"The next thing is that the potential energy of central force is dependent solely on"},{"Start":"04:02.105 ","End":"04:07.898","Text":"r. Our u for potential energy is going to be a function of only r,"},{"Start":"04:07.898 ","End":"04:09.985","Text":"no Theta and no z."},{"Start":"04:09.985 ","End":"04:14.090","Text":"We can prove that a certain force is a central force if we see that"},{"Start":"04:14.090 ","End":"04:18.725","Text":"its potential energy is just in terms of r. Sometimes in a question,"},{"Start":"04:18.725 ","End":"04:23.365","Text":"you\u0027ll be asked to prove that a certain force is a central force."},{"Start":"04:23.365 ","End":"04:26.510","Text":"If you can show that the equation for the potential energy of"},{"Start":"04:26.510 ","End":"04:30.305","Text":"that force has only one variable, which is r,"},{"Start":"04:30.305 ","End":"04:35.621","Text":"so it depends solely on its distance from the origin,"},{"Start":"04:35.621 ","End":"04:40.255","Text":"then you have just proved that it is a central force."},{"Start":"04:40.255 ","End":"04:42.080","Text":"The next thing to note,"},{"Start":"04:42.080 ","End":"04:44.120","Text":"which is also super important,"},{"Start":"04:44.120 ","End":"04:49.890","Text":"is that the torque of a central force is always equal to 0."},{"Start":"04:50.030 ","End":"04:53.404","Text":"Why is that? If we look over here,"},{"Start":"04:53.404 ","End":"04:54.775","Text":"in order to find our torque,"},{"Start":"04:54.775 ","End":"04:59.230","Text":"we have to do r cross F. We can see that this is"},{"Start":"04:59.230 ","End":"05:04.160","Text":"our r and so we can see that it\u0027s in"},{"Start":"05:04.160 ","End":"05:06.860","Text":"a parallel line and it\u0027s either in the same direction or"},{"Start":"05:06.860 ","End":"05:09.611","Text":"the opposite direction to our force,"},{"Start":"05:09.611 ","End":"05:11.795","Text":"to our F, which means that they\u0027re parallel,"},{"Start":"05:11.795 ","End":"05:17.165","Text":"and r cross F when parallel is going to equal 0."},{"Start":"05:17.165 ","End":"05:21.275","Text":"Now we can say that if I torque is always equal to 0,"},{"Start":"05:21.275 ","End":"05:26.190","Text":"then we will always have conservation of angular momentum."},{"Start":"05:28.040 ","End":"05:32.715","Text":"Our angular momentum will always be a constant."},{"Start":"05:32.715 ","End":"05:38.365","Text":"Now this is only correct if there\u0027s no external forces acting on our body,"},{"Start":"05:38.365 ","End":"05:39.880","Text":"but in most of the questions,"},{"Start":"05:39.880 ","End":"05:41.185","Text":"especially at this stage,"},{"Start":"05:41.185 ","End":"05:46.905","Text":"we\u0027ll just have some system and there won\u0027t be external forces colliding or acting."},{"Start":"05:46.905 ","End":"05:49.285","Text":"Then this is ashore,"},{"Start":"05:49.285 ","End":"05:53.585","Text":"or a very good way to answer questions."},{"Start":"05:53.585 ","End":"05:58.695","Text":"Now let\u0027s write out the proofs for these characteristics."},{"Start":"05:58.695 ","End":"06:03.085","Text":"The first thing that we\u0027re going to do is look at this."},{"Start":"06:03.085 ","End":"06:05.785","Text":"A central force is always a conservative field,"},{"Start":"06:05.785 ","End":"06:08.890","Text":"and in that case, we have a conservation of energy."},{"Start":"06:08.890 ","End":"06:13.300","Text":"In order to prove that a force is a conservative field,"},{"Start":"06:13.300 ","End":"06:18.225","Text":"we have to do our rotor or a curl. What does that mean?"},{"Start":"06:18.225 ","End":"06:21.480","Text":"It\u0027s a Nabla or a Delta function,"},{"Start":"06:21.480 ","End":"06:25.265","Text":"crossed with our force,"},{"Start":"06:25.265 ","End":"06:27.065","Text":"and if it is equal to 0,"},{"Start":"06:27.065 ","End":"06:28.865","Text":"then it\u0027s a conservative force."},{"Start":"06:28.865 ","End":"06:31.224","Text":"This is what we have to check,"},{"Start":"06:31.224 ","End":"06:33.745","Text":"if I curl is equal to 0."},{"Start":"06:33.745 ","End":"06:41.440","Text":"Here we have the formula for our curl where A is some force vector."},{"Start":"06:41.450 ","End":"06:45.285","Text":"This is simply from your equation sheet."},{"Start":"06:45.285 ","End":"06:47.860","Text":"Our row, let\u0027s just write this,"},{"Start":"06:47.860 ","End":"06:55.505","Text":"so our A vector is equal to our force vector over here."},{"Start":"06:55.505 ","End":"06:59.840","Text":"Our row is equal to r_l,"},{"Start":"06:59.840 ","End":"07:04.645","Text":"our Phi is equal to a Theta."},{"Start":"07:04.645 ","End":"07:15.200","Text":"Now we can write that our force is equal to some F as a function of r in the r-direction."},{"Start":"07:15.200 ","End":"07:25.265","Text":"That means that our F_Theta component is going to be equal to 0,"},{"Start":"07:25.265 ","End":"07:30.667","Text":"and our F_z component is also going to be equal to 0."},{"Start":"07:30.667 ","End":"07:38.200","Text":"That means that our d of f(r) are divided"},{"Start":"07:38.200 ","End":"07:46.045","Text":"by d Theta is going to be equal to 0 because there\u0027s no Theta component,"},{"Start":"07:46.045 ","End":"07:50.710","Text":"and that means that our df is a function of r divided"},{"Start":"07:50.710 ","End":"07:56.770","Text":"by dz is also going to be equal to 0."},{"Start":"07:56.770 ","End":"08:00.850","Text":"With this information, let\u0027s look at our equation."},{"Start":"08:00.850 ","End":"08:04.945","Text":"I\u0027m going to cross out everything that has to do"},{"Start":"08:04.945 ","End":"08:10.855","Text":"with a component that isn\u0027t my r-component because they are equal to 0."},{"Start":"08:10.855 ","End":"08:13.600","Text":"Here I have the z-component,"},{"Start":"08:13.600 ","End":"08:14.980","Text":"so that\u0027s equal to 0,"},{"Start":"08:14.980 ","End":"08:17.275","Text":"Theta component is equal to 0."},{"Start":"08:17.275 ","End":"08:18.700","Text":"This is my r-component,"},{"Start":"08:18.700 ","End":"08:21.850","Text":"so I\u0027ll leave that, z-component is equal to 0."},{"Start":"08:21.850 ","End":"08:24.100","Text":"This is a Theta component,"},{"Start":"08:24.100 ","End":"08:27.910","Text":"it\u0027s equal to 0, and this is an r-component, so I\u0027ll leave that."},{"Start":"08:27.910 ","End":"08:30.475","Text":"Now we can see out of this entire equation,"},{"Start":"08:30.475 ","End":"08:34.040","Text":"we only have 2 components left."},{"Start":"08:34.290 ","End":"08:39.145","Text":"As we just saw, our force is dependent just on r,"},{"Start":"08:39.145 ","End":"08:40.945","Text":"it only has that variable."},{"Start":"08:40.945 ","End":"08:44.140","Text":"That means that when we take the derivative"},{"Start":"08:44.140 ","End":"08:47.598","Text":"of the Theta component or of the z-component,"},{"Start":"08:47.598 ","End":"08:53.015","Text":"it\u0027s going to be equal to 0 because we don\u0027t have a Theta or a z-component."},{"Start":"08:53.015 ","End":"08:56.080","Text":"Now, if we look at our 2-components,"},{"Start":"08:56.080 ","End":"08:59.605","Text":"here I\u0027m taking the derivative with respect to z,"},{"Start":"08:59.605 ","End":"09:02.110","Text":"and we just saw over here that that\u0027s going to be equal"},{"Start":"09:02.110 ","End":"09:04.870","Text":"to 0 because we don\u0027t have any z-components."},{"Start":"09:04.870 ","End":"09:10.075","Text":"Over here, I\u0027m taking the derivative with respect to my Theta."},{"Start":"09:10.075 ","End":"09:12.820","Text":"We just saw that because there\u0027s no Theta component,"},{"Start":"09:12.820 ","End":"09:18.115","Text":"the derivative with respect to Theta is also equal to 0."},{"Start":"09:18.115 ","End":"09:23.560","Text":"Now we\u0027ve gotten that this entire equation is going to be equal"},{"Start":"09:23.560 ","End":"09:29.665","Text":"to 0 when we\u0027re using this central force of this format."},{"Start":"09:29.665 ","End":"09:33.310","Text":"Now we saw that our curl is equal to 0,"},{"Start":"09:33.310 ","End":"09:38.480","Text":"and so therefore, this is a conservative force."},{"Start":"09:38.760 ","End":"09:43.600","Text":"Now let\u0027s look at number 3 because it\u0027s a bit shorter,"},{"Start":"09:43.600 ","End":"09:46.945","Text":"we\u0027ll come back to 2 in a second. Let\u0027s see."},{"Start":"09:46.945 ","End":"09:52.465","Text":"We know that our equation for our torque is equal to our r vector,"},{"Start":"09:52.465 ","End":"09:57.670","Text":"cross multiplication with our force vector."},{"Start":"09:57.670 ","End":"10:04.790","Text":"Our r vector is going from our origin until our body."},{"Start":"10:04.980 ","End":"10:09.640","Text":"In our case, if we\u0027re looking at our diagram,"},{"Start":"10:09.640 ","End":"10:13.510","Text":"this is going to be our r vector,"},{"Start":"10:13.510 ","End":"10:17.515","Text":"and its size is the length of the vector."},{"Start":"10:17.515 ","End":"10:21.595","Text":"Now when I\u0027m writing this out in polar coordinates,"},{"Start":"10:21.595 ","End":"10:26.740","Text":"so my r vector is going to be the size of my r vector,"},{"Start":"10:26.740 ","End":"10:31.540","Text":"which is r, and it\u0027s going in the radial direction."},{"Start":"10:31.540 ","End":"10:34.960","Text":"This really defines our r vector."},{"Start":"10:34.960 ","End":"10:38.380","Text":"Our r is the size and it\u0027s in the radial direction,"},{"Start":"10:38.380 ","End":"10:39.430","Text":"which is really what\u0027s happening."},{"Start":"10:39.430 ","End":"10:44.725","Text":"We\u0027re going from the origin in the radial direction to some body."},{"Start":"10:44.725 ","End":"10:49.540","Text":"We have our r vector cross multiplied with our F vector,"},{"Start":"10:49.540 ","End":"10:50.605","Text":"which as we know,"},{"Start":"10:50.605 ","End":"10:53.440","Text":"is our F as a function of r,"},{"Start":"10:53.440 ","End":"10:59.410","Text":"and this is also in the r-direction always when we\u0027re dealing with the central force."},{"Start":"10:59.410 ","End":"11:01.660","Text":"All of the constants,"},{"Start":"11:01.660 ","End":"11:04.150","Text":"we can take to the side,"},{"Start":"11:04.150 ","End":"11:08.905","Text":"so I\u0027ll have r multiplied by our F as a function of r. Then,"},{"Start":"11:08.905 ","End":"11:14.050","Text":"we\u0027re going to have our r-direction cross multiplied with our other r-direction."},{"Start":"11:14.050 ","End":"11:17.335","Text":"Now when we have our r-hat cross multiplied with our r-hat,"},{"Start":"11:17.335 ","End":"11:18.745","Text":"that\u0027s going to be equal 0,"},{"Start":"11:18.745 ","End":"11:22.150","Text":"0 times anything is equal to 0."},{"Start":"11:22.150 ","End":"11:25.840","Text":"That\u0027s how we get that our torque is equal to 0."},{"Start":"11:25.840 ","End":"11:28.810","Text":"The torque of a central force is always equal to 0."},{"Start":"11:28.810 ","End":"11:33.730","Text":"That means that we have conservation of angular momentum,"},{"Start":"11:33.730 ","End":"11:39.415","Text":"which means that our angular momentum is always going to be equal to some constant."},{"Start":"11:39.415 ","End":"11:42.160","Text":"That was our third characteristic."},{"Start":"11:42.160 ","End":"11:44.755","Text":"Now let\u0027s look at our second characteristic."},{"Start":"11:44.755 ","End":"11:48.070","Text":"Our second characteristic is that our potential energy of"},{"Start":"11:48.070 ","End":"11:53.590","Text":"a central force is always solely dependent on our r variable."},{"Start":"11:53.590 ","End":"11:57.220","Text":"The first thing we\u0027re going to do is we\u0027re going to be reminded between"},{"Start":"11:57.220 ","End":"12:01.420","Text":"the relationship of our force and potential energy."},{"Start":"12:01.420 ","End":"12:04.060","Text":"The equation is that our force,"},{"Start":"12:04.060 ","End":"12:05.695","Text":"which is a vector,"},{"Start":"12:05.695 ","End":"12:13.100","Text":"is equal to the negative gradient of our potential energy."},{"Start":"12:13.110 ","End":"12:20.785","Text":"My gradient function is some vector-type situation,"},{"Start":"12:20.785 ","End":"12:25.780","Text":"and I multiply it by some function which is not a vector."},{"Start":"12:25.780 ","End":"12:31.180","Text":"My force over here is a vector and it has a size and direction,"},{"Start":"12:31.180 ","End":"12:33.820","Text":"but my u over here is scalar."},{"Start":"12:33.820 ","End":"12:36.430","Text":"So it\u0027s a function and it has a size,"},{"Start":"12:36.430 ","End":"12:38.600","Text":"but it doesn\u0027t have direction."},{"Start":"12:38.600 ","End":"12:40.635","Text":"From the equation sheet,"},{"Start":"12:40.635 ","End":"12:44.865","Text":"we can see that our grad of F,"},{"Start":"12:44.865 ","End":"12:48.595","Text":"which is simply equal to this,"},{"Start":"12:48.595 ","End":"12:53.575","Text":"is equal to our df by dr,"},{"Start":"12:53.575 ","End":"13:01.360","Text":"here they write it as Rho in the r-direction plus 1 divided by r multiplied"},{"Start":"13:01.360 ","End":"13:10.975","Text":"by our df by d Theta plus over here,"},{"Start":"13:10.975 ","End":"13:14.330","Text":"our df by dz,"},{"Start":"13:15.150 ","End":"13:22.630","Text":"and this is in the Theta direction and this will be in the z-direction."},{"Start":"13:22.630 ","End":"13:29.875","Text":"Our function, our F is equal to our u, a potential energy."},{"Start":"13:29.875 ","End":"13:37.900","Text":"Again, our Phi is equal to our Theta and our Rho is equal to our r. Now,"},{"Start":"13:37.900 ","End":"13:40.870","Text":"if I take the grad of my u,"},{"Start":"13:40.870 ","End":"13:48.325","Text":"that\u0027s equal to my du by dr in the r-direction plus"},{"Start":"13:48.325 ","End":"13:56.005","Text":"1 over our du by d Theta in the Theta direction,"},{"Start":"13:56.005 ","End":"14:03.505","Text":"plus du by dz in the z-direction."},{"Start":"14:03.505 ","End":"14:07.330","Text":"Because we saw that our F is minus grad of u,"},{"Start":"14:07.330 ","End":"14:15.790","Text":"so this is going to be equal to our negative F. Then I can simply rewrite"},{"Start":"14:15.790 ","End":"14:25.300","Text":"this as being equal to my F as a function of r in the r-direction."},{"Start":"14:25.300 ","End":"14:29.650","Text":"This is what I meant to get and I\u0027m checking this."},{"Start":"14:29.650 ","End":"14:32.230","Text":"This is the answer I meant to get."},{"Start":"14:32.230 ","End":"14:41.515","Text":"My du by d Theta is going to be equal to 0 because in my F,"},{"Start":"14:41.515 ","End":"14:45.475","Text":"I don\u0027t have any Thetas or any z\u0027s."},{"Start":"14:45.475 ","End":"14:50.005","Text":"That\u0027s also going to mean my du by dz."},{"Start":"14:50.005 ","End":"14:52.510","Text":"Again, because I don\u0027t have any z terms,"},{"Start":"14:52.510 ","End":"14:55.960","Text":"so when I take the differential of that or I"},{"Start":"14:55.960 ","End":"15:01.820","Text":"differentiate that in terms of Theta and in terms of z, I\u0027ll get 0."},{"Start":"15:02.640 ","End":"15:06.845","Text":"In that case, my potential energy,"},{"Start":"15:06.845 ","End":"15:10.580","Text":"which can be a function of r, Theta and z."},{"Start":"15:10.580 ","End":"15:16.865","Text":"So if my derivatives with respect to Theta and z are equal to 0,"},{"Start":"15:16.865 ","End":"15:25.235","Text":"then that means that my u function is not dependent on my Theta or my z,"},{"Start":"15:25.235 ","End":"15:30.950","Text":"which means that it\u0027s just dependent on my r. Then I can"},{"Start":"15:30.950 ","End":"15:37.120","Text":"see that this term will cancel out and this term will cancel out,"},{"Start":"15:37.120 ","End":"15:41.960","Text":"and all I\u0027m left with is this term over here."},{"Start":"15:41.960 ","End":"15:51.920","Text":"Then I will get that my negative F is equal to my du by dr in the r-direction."},{"Start":"15:52.050 ","End":"15:55.130","Text":"This is exactly what I wanted to get."},{"Start":"15:55.130 ","End":"16:00.215","Text":"Then, in order to find my potential energy u,"},{"Start":"16:00.215 ","End":"16:03.150","Text":"which will, of course, be as a function of r,"},{"Start":"16:04.950 ","End":"16:08.080","Text":"in here it\u0027s going to be my f(r),"},{"Start":"16:08.080 ","End":"16:12.995","Text":"so it\u0027s going to be the negative integral of my F as a function of"},{"Start":"16:12.995 ","End":"16:20.375","Text":"r dr. An example is our gravitational force field."},{"Start":"16:20.375 ","End":"16:23.390","Text":"Our gravitational force field is something like this."},{"Start":"16:23.390 ","End":"16:25.565","Text":"We have our f(r), our function,"},{"Start":"16:25.565 ","End":"16:31.390","Text":"which is equal to negative some constant divided by r^2."},{"Start":"16:31.390 ","End":"16:35.704","Text":"In order to find our potential energy of our gravity,"},{"Start":"16:35.704 ","End":"16:39.890","Text":"it\u0027s going to be the negative integral of our f(r),"},{"Start":"16:39.890 ","End":"16:46.905","Text":"which is negative A divided by r^2 without this negative over here,"},{"Start":"16:46.905 ","End":"16:51.520","Text":"and then dr. Then this is going to"},{"Start":"16:51.520 ","End":"16:56.830","Text":"equal to our negative A divided by negative 1 divided by r,"},{"Start":"16:56.830 ","End":"17:02.465","Text":"the integral plus some integrating constant."},{"Start":"17:02.465 ","End":"17:05.945","Text":"Now, usually this constant is set to equal to 0,"},{"Start":"17:05.945 ","End":"17:07.580","Text":"and as we\u0027ve seen with our energy,"},{"Start":"17:07.580 ","End":"17:13.114","Text":"we can always set the constant equal to 0 and not take it into account,"},{"Start":"17:13.114 ","End":"17:16.230","Text":"as we\u0027ve seen with other types of energy."},{"Start":"17:16.470 ","End":"17:19.315","Text":"That\u0027s the end of our lesson."},{"Start":"17:19.315 ","End":"17:23.390","Text":"Just as a brief conclusion from this lesson,"},{"Start":"17:23.390 ","End":"17:24.890","Text":"you should take away this;"},{"Start":"17:24.890 ","End":"17:28.360","Text":"the characteristics of a central force and also,"},{"Start":"17:28.360 ","End":"17:30.325","Text":"what a central force looks like."},{"Start":"17:30.325 ","End":"17:35.780","Text":"A central force is only dependent on our r and it\u0027s in our r-direction,"},{"Start":"17:35.780 ","End":"17:37.565","Text":"so it\u0027s of this form."},{"Start":"17:37.565 ","End":"17:39.755","Text":"The characteristics are super,"},{"Start":"17:39.755 ","End":"17:44.940","Text":"super important because they\u0027re going to help you so much when answering questions."},{"Start":"17:44.940 ","End":"17:47.510","Text":"That\u0027s the end of our lesson."}],"ID":9453},{"Watched":false,"Name":"Gravitiational Force And Free Fall","Duration":"7m 37s","ChapterTopicVideoID":9184,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.710","Text":"Hello. In this lesson,"},{"Start":"00:01.710 ","End":"00:04.250","Text":"we\u0027re going to be speaking about gravitational force,"},{"Start":"00:04.250 ","End":"00:07.635","Text":"and we\u0027re going to show that it\u0027s a type of central force."},{"Start":"00:07.635 ","End":"00:11.325","Text":"Gravitational force, not just on Earth but in general,"},{"Start":"00:11.325 ","End":"00:13.545","Text":"is a force which looks like this."},{"Start":"00:13.545 ","End":"00:16.050","Text":"It\u0027s equal to negative G,"},{"Start":"00:16.050 ","End":"00:18.840","Text":"which is some kind of constant which is written over here,"},{"Start":"00:18.840 ","End":"00:21.555","Text":"multiplied by M times m,"},{"Start":"00:21.555 ","End":"00:25.960","Text":"so the mass of 2 bodies divided by r^2 and it\u0027s,"},{"Start":"00:25.960 ","End":"00:29.810","Text":"of course, going in the radial direction."},{"Start":"00:29.810 ","End":"00:35.455","Text":"It\u0027s the attractive force between any 2 bodies which have a mass."},{"Start":"00:35.455 ","End":"00:39.140","Text":"Now, a lot of the time we are going to be working with 1 mass,"},{"Start":"00:39.140 ","End":"00:41.840","Text":"which has a huge mass,"},{"Start":"00:41.840 ","End":"00:44.105","Text":"and another one which is very small."},{"Start":"00:44.105 ","End":"00:49.514","Text":"That\u0027s why they\u0027re represented by M and the small m. For example,"},{"Start":"00:49.514 ","End":"00:51.630","Text":"when working with Earth."},{"Start":"00:51.630 ","End":"00:54.380","Text":"We can see Earth has a huge mass and"},{"Start":"00:54.380 ","End":"00:57.380","Text":"then the person standing on Earth has a very tiny mass."},{"Start":"00:57.380 ","End":"01:01.615","Text":"Then this is the equation to work out the force between the 2."},{"Start":"01:01.615 ","End":"01:04.185","Text":"Between every 2 bodies,"},{"Start":"01:04.185 ","End":"01:05.640","Text":"there\u0027s this type of force."},{"Start":"01:05.640 ","End":"01:09.675","Text":"If another person was standing over here, for instance,"},{"Start":"01:09.675 ","End":"01:12.260","Text":"so there will still be this gravitational force"},{"Start":"01:12.260 ","End":"01:15.920","Text":"between these 2 people or these 2 small masses."},{"Start":"01:15.920 ","End":"01:19.475","Text":"But because the force is going to be so tiny that it\u0027s something"},{"Start":"01:19.475 ","End":"01:23.870","Text":"that is negligible and we don\u0027t really take that into consideration."},{"Start":"01:23.870 ","End":"01:26.580","Text":"Now here we have our r^2."},{"Start":"01:26.580 ","End":"01:30.650","Text":"Our r represents the distance between the 2 masses."},{"Start":"01:30.650 ","End":"01:33.740","Text":"The distance is measured from the center of mass."},{"Start":"01:33.740 ","End":"01:35.375","Text":"For instance, from Earth,"},{"Start":"01:35.375 ","End":"01:38.900","Text":"our center of mass is somewhere in the center of the sphere."},{"Start":"01:38.900 ","End":"01:45.995","Text":"Our r is going to go from the center of the sphere until the center of mass of the human."},{"Start":"01:45.995 ","End":"01:49.710","Text":"This is our r,"},{"Start":"01:49.970 ","End":"01:53.470","Text":"and of course, we have our r-hat over here to show"},{"Start":"01:53.470 ","End":"01:57.220","Text":"that our force is acting in the radial direction."},{"Start":"01:57.220 ","End":"02:00.595","Text":"We can see that there\u0027s a negative sign over here,"},{"Start":"02:00.595 ","End":"02:06.535","Text":"meaning that our origin is located over here at the center"},{"Start":"02:06.535 ","End":"02:12.909","Text":"of the Earth and that human is being pulled towards the center of the Earth."},{"Start":"02:12.909 ","End":"02:18.080","Text":"I can rub that out and draw it like this."},{"Start":"02:18.840 ","End":"02:22.720","Text":"Now we can see that our gravitational force is essential force"},{"Start":"02:22.720 ","End":"02:25.885","Text":"because we can see that all of this"},{"Start":"02:25.885 ","End":"02:32.580","Text":"over here is some function with respect to r. That\u0027s our f r,"},{"Start":"02:32.580 ","End":"02:35.309","Text":"and then it\u0027s acting in the radial direction."},{"Start":"02:35.309 ","End":"02:36.405","Text":"From the previous lesson,"},{"Start":"02:36.405 ","End":"02:40.735","Text":"we can see that that\u0027s the format for central force."},{"Start":"02:40.735 ","End":"02:45.140","Text":"We now know that this is a central force because it\u0027s in this format."},{"Start":"02:45.140 ","End":"02:47.405","Text":"Therefore, if this is a central force,"},{"Start":"02:47.405 ","End":"02:54.150","Text":"then we know that we\u0027re going to have conservation of energy and of angular momentum."},{"Start":"02:54.430 ","End":"02:58.325","Text":"If we have conservation of energy and of angular momentum,"},{"Start":"02:58.325 ","End":"03:01.730","Text":"that means that our energy is equal to some constant and"},{"Start":"03:01.730 ","End":"03:06.090","Text":"our angular momentum is equal to another constant."},{"Start":"03:06.230 ","End":"03:12.500","Text":"We know that our gravitational force is denoted by mg,"},{"Start":"03:12.500 ","End":"03:14.645","Text":"and it\u0027s something that\u0027s constant."},{"Start":"03:14.645 ","End":"03:18.110","Text":"Suddenly, we see over here that are gravitational force is"},{"Start":"03:18.110 ","End":"03:21.810","Text":"dependent on the distance, our r^2."},{"Start":"03:21.810 ","End":"03:25.100","Text":"Let\u0027s take a look at how we connect this equation to"},{"Start":"03:25.100 ","End":"03:28.343","Text":"our mg. How we got this constant from it."},{"Start":"03:28.343 ","End":"03:32.285","Text":"Usually, when we\u0027re working with our gravitational force,"},{"Start":"03:32.285 ","End":"03:34.862","Text":"we\u0027re working with it close to Earth."},{"Start":"03:34.862 ","End":"03:40.925","Text":"That means that our Delta h is significantly smaller than the radius of Earth."},{"Start":"03:40.925 ","End":"03:43.415","Text":"The radius of Earth,"},{"Start":"03:43.415 ","End":"03:54.145","Text":"let\u0027s write our R_E is equal to something like 6,400 kilometers."},{"Start":"03:54.145 ","End":"03:57.495","Text":"That\u0027s a roundabout the radius of Earth."},{"Start":"03:57.495 ","End":"04:01.459","Text":"When we\u0027re working out the gravitational force of some building,"},{"Start":"04:01.459 ","End":"04:05.840","Text":"where its Delta h of the building is equal to,"},{"Start":"04:05.840 ","End":"04:07.625","Text":"let\u0027s say, 20 meters."},{"Start":"04:07.625 ","End":"04:13.700","Text":"We can see 20 meters is significantly smaller than 6,400 kilometers."},{"Start":"04:13.700 ","End":"04:17.405","Text":"We\u0027re doing this approximation."},{"Start":"04:17.405 ","End":"04:25.325","Text":"Then what I have to do is I can write out that my central force, my gravitational force,"},{"Start":"04:25.325 ","End":"04:27.185","Text":"using this general equation,"},{"Start":"04:27.185 ","End":"04:29.615","Text":"is equal to my negative G,"},{"Start":"04:29.615 ","End":"04:33.065","Text":"this constant multiplied by my mass of the Earth,"},{"Start":"04:33.065 ","End":"04:34.895","Text":"which is also unknown constant."},{"Start":"04:34.895 ","End":"04:38.090","Text":"You can look it up on the Internet multiplied by m,"},{"Start":"04:38.090 ","End":"04:41.300","Text":"the mass of the person standing on the building,"},{"Start":"04:41.300 ","End":"04:44.970","Text":"let\u0027s say, and then divide it by our r^2."},{"Start":"04:44.970 ","End":"04:51.250","Text":"What is our r^2? It\u0027s the radius of the Earth plus this extra 20 meter height."},{"Start":"04:51.250 ","End":"04:54.515","Text":"That\u0027s R_E, radius of the Earth plus"},{"Start":"04:54.515 ","End":"04:57.700","Text":"or minus if we\u0027re going 20 meters then it doesn\u0027t matter."},{"Start":"04:57.700 ","End":"04:59.245","Text":"Our Delta h,"},{"Start":"04:59.245 ","End":"05:01.360","Text":"so that\u0027s the height of our building."},{"Start":"05:01.360 ","End":"05:04.370","Text":"All of this squared, just like over here."},{"Start":"05:04.370 ","End":"05:07.969","Text":"Now that of course in the radial direction."},{"Start":"05:07.969 ","End":"05:10.760","Text":"Now that is approximately equal to."},{"Start":"05:10.760 ","End":"05:13.190","Text":"Because we said we\u0027re doing this approximation where"},{"Start":"05:13.190 ","End":"05:16.550","Text":"our Delta h is significantly smaller than our radius of the Earth,"},{"Start":"05:16.550 ","End":"05:20.605","Text":"we can see that 20 meters is very"},{"Start":"05:20.605 ","End":"05:26.435","Text":"negligible considering that the radius of the Earth is 6,400 kilometers."},{"Start":"05:26.435 ","End":"05:30.620","Text":"That means that we can cross this out and not take it into account,"},{"Start":"05:30.620 ","End":"05:33.550","Text":"and then we\u0027ll be left with something like this."},{"Start":"05:33.550 ","End":"05:35.105","Text":"Here it\u0027s been written out."},{"Start":"05:35.105 ","End":"05:36.965","Text":"We have our negative GM,"},{"Start":"05:36.965 ","End":"05:39.605","Text":"the mass of the Earth multiplied by the mass of the person,"},{"Start":"05:39.605 ","End":"05:42.245","Text":"divided by the radius of Earth squared,"},{"Start":"05:42.245 ","End":"05:43.985","Text":"which is this exact equation,"},{"Start":"05:43.985 ","End":"05:50.975","Text":"and then that\u0027s equal to mg. What have we defined as g, a gravitational force?"},{"Start":"05:50.975 ","End":"05:56.345","Text":"It\u0027s simply this our G constant multiplied by the mass of the Earth,"},{"Start":"05:56.345 ","End":"06:00.095","Text":"which is a constant, divided by the radius of the Earth squared,"},{"Start":"06:00.095 ","End":"06:01.625","Text":"which is also a constant."},{"Start":"06:01.625 ","End":"06:04.040","Text":"That\u0027s over here. Here you can see"},{"Start":"06:04.040 ","End":"06:08.240","Text":"the substitution of all the values for our constant G,"},{"Start":"06:08.240 ","End":"06:10.100","Text":"the mass of our Earth,"},{"Start":"06:10.100 ","End":"06:13.000","Text":"and the radius of Earth over here."},{"Start":"06:13.000 ","End":"06:16.320","Text":"Now we can see that we got that our small g value,"},{"Start":"06:16.320 ","End":"06:21.515","Text":"which we know and have is equal to 9.8 meters per second squared."},{"Start":"06:21.515 ","End":"06:24.383","Text":"This is where it comes from."},{"Start":"06:24.383 ","End":"06:27.210","Text":"We can see that this is our G constant,"},{"Start":"06:27.210 ","End":"06:29.240","Text":"this is our M_E, mass of the Earth,"},{"Start":"06:29.240 ","End":"06:31.910","Text":"and this is our radius of the Earth squared."},{"Start":"06:31.910 ","End":"06:36.410","Text":"Now we can see where we got our equation mg and what our g is,"},{"Start":"06:36.410 ","End":"06:39.515","Text":"and how we got to 9.8 meters per second squared."},{"Start":"06:39.515 ","End":"06:43.430","Text":"Whenever we\u0027re speaking about some kind of body that\u0027s"},{"Start":"06:43.430 ","End":"06:47.990","Text":"acting at this type of approximation where it\u0027s Delta h,"},{"Start":"06:47.990 ","End":"06:52.820","Text":"its height above the Earth\u0027s surface is smaller than the radius of the Earth."},{"Start":"06:52.820 ","End":"06:58.150","Text":"Then we can do this approximation and that\u0027s how we got our mg. Of course,"},{"Start":"06:58.150 ","End":"07:00.575","Text":"later on when we\u0027re going to be dealing with,"},{"Start":"07:00.575 ","End":"07:03.350","Text":"let\u0027s say our Earth and its gravitational force,"},{"Start":"07:03.350 ","End":"07:09.020","Text":"where the stars or the gravitational force between different stars, so then, of course,"},{"Start":"07:09.020 ","End":"07:12.634","Text":"we\u0027re not going to be able to use this approximation, this mg,"},{"Start":"07:12.634 ","End":"07:15.380","Text":"because this Delta h is not going to"},{"Start":"07:15.380 ","End":"07:18.005","Text":"be significantly smaller than the radius of the Earth."},{"Start":"07:18.005 ","End":"07:21.470","Text":"In some cases may be significantly bigger than the radius of the Earth."},{"Start":"07:21.470 ","End":"07:27.500","Text":"In those cases when we\u0027re dealing with the gravitational force between 2 stars,"},{"Start":"07:27.500 ","End":"07:31.370","Text":"then we\u0027re going to be using this general equation again."},{"Start":"07:31.370 ","End":"07:34.710","Text":"Using this will be incorrect."},{"Start":"07:34.760 ","End":"07:38.050","Text":"That\u0027s the end of this lesson."}],"ID":9454},{"Watched":false,"Name":"Gravitational Force And Trajectory","Duration":"6m 2s","ChapterTopicVideoID":9185,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"Hello. In our previous lesson,"},{"Start":"00:01.920 ","End":"00:06.300","Text":"we learned about our gravitational force and we showed that it was a central force."},{"Start":"00:06.300 ","End":"00:15.900","Text":"We showed that this over here is our F function of r and it\u0027s acting in the r direction."},{"Start":"00:15.900 ","End":"00:18.780","Text":"Now, because we showed that this was a central force,"},{"Start":"00:18.780 ","End":"00:21.610","Text":"that our gravitational force was central,"},{"Start":"00:21.610 ","End":"00:25.710","Text":"that meant that our angular momentum was going to be equal to a"},{"Start":"00:25.710 ","End":"00:30.480","Text":"constant and that our energy was going to be equal to another constant because,"},{"Start":"00:30.480 ","End":"00:35.175","Text":"with central forces, both our angular momentum and our energy are conserved."},{"Start":"00:35.175 ","End":"00:39.725","Text":"Now in this video, we\u0027re going to see what we can do with this."},{"Start":"00:39.725 ","End":"00:46.130","Text":"Now we\u0027re going to speak about how we find the potential energy of our central force."},{"Start":"00:46.130 ","End":"00:51.054","Text":"Here our gravitational force between any 2 bodies with a mass."},{"Start":"00:51.054 ","End":"00:55.760","Text":"It\u0027s going to be equal to negative the integral of r force,"},{"Start":"00:55.760 ","End":"00:57.530","Text":"which is a function only of r,"},{"Start":"00:57.530 ","End":"01:04.330","Text":"dot dr. Now we can substitute in our F as a function of r,"},{"Start":"01:04.330 ","End":"01:06.090","Text":"which is this over here."},{"Start":"01:06.090 ","End":"01:11.780","Text":"We\u0027ll have negative the integral of negative GMm divided by r^2 dr."},{"Start":"01:11.780 ","End":"01:18.230","Text":"What we will get when we do this integration divided by r^2,"},{"Start":"01:18.230 ","End":"01:20.115","Text":"so we\u0027ll get another minus."},{"Start":"01:20.115 ","End":"01:24.770","Text":"That minus and the minus from here will cancel out and then we have this minus as well."},{"Start":"01:24.770 ","End":"01:27.905","Text":"There\u0027s a negative sign over here and then we have GMm"},{"Start":"01:27.905 ","End":"01:32.605","Text":"divided by r. Now notice in our potential energy,"},{"Start":"01:32.605 ","End":"01:36.230","Text":"our numerator will be the same as our force."},{"Start":"01:36.230 ","End":"01:40.400","Text":"However, our denominator is just r, it\u0027s not r^2."},{"Start":"01:40.400 ","End":"01:42.485","Text":"It\u0027s very important to remember that."},{"Start":"01:42.485 ","End":"01:49.170","Text":"Then we can say that our GMm is equal to some constant called Alpha."},{"Start":"01:49.870 ","End":"01:54.710","Text":"Now, in the meantime, let\u0027s just assume that I\u0027m writing as Alpha in order"},{"Start":"01:54.710 ","End":"01:58.940","Text":"to just not always have to write out my GMm because obviously,"},{"Start":"01:58.940 ","End":"02:02.135","Text":"my values over here don\u0027t change throughout the question."},{"Start":"02:02.135 ","End":"02:07.615","Text":"But later, we\u0027ll see that there\u0027s another reason for writing Alpha over here."},{"Start":"02:07.615 ","End":"02:10.910","Text":"Given this potential energy, if I have,"},{"Start":"02:10.910 ","End":"02:14.570","Text":"let\u0027s say an asteroid hurtling towards earth."},{"Start":"02:14.570 ","End":"02:18.700","Text":"The way that I can find the total energy of the system, my E,"},{"Start":"02:18.700 ","End":"02:22.520","Text":"it\u0027s going to be equal to my kinetic energy of the asteroid,"},{"Start":"02:22.520 ","End":"02:24.365","Text":"which is half mv^2,"},{"Start":"02:24.365 ","End":"02:26.690","Text":"plus its potential energy."},{"Start":"02:26.690 ","End":"02:31.070","Text":"My potential energy is negative GMm divided by"},{"Start":"02:31.070 ","End":"02:36.880","Text":"r. This equation is super important."},{"Start":"02:36.880 ","End":"02:42.280","Text":"If I\u0027m answering a question where I\u0027m using the idea of my conservation of energy,"},{"Start":"02:42.280 ","End":"02:46.510","Text":"I can write that my energy before is equal or my energy after,"},{"Start":"02:46.510 ","End":"02:49.000","Text":"whichever, is equal to this."},{"Start":"02:49.000 ","End":"02:53.015","Text":"My kinetic energy plus potential energy,"},{"Start":"02:53.015 ","End":"02:55.095","Text":"which has a minus sign."},{"Start":"02:55.095 ","End":"02:58.245","Text":"Now we\u0027ve dealt with our energy,"},{"Start":"02:58.245 ","End":"03:02.890","Text":"the next thing that we can do is that we can know how our body is going to move."},{"Start":"03:02.890 ","End":"03:04.615","Text":"We can know its track,"},{"Start":"03:04.615 ","End":"03:06.820","Text":"its course, its trajectory."},{"Start":"03:06.820 ","End":"03:10.960","Text":"It\u0027s track equation is given by this over here."},{"Start":"03:10.960 ","End":"03:13.225","Text":"Now, how I got to this equation,"},{"Start":"03:13.225 ","End":"03:15.355","Text":"I\u0027ll show you in one of the later lessons."},{"Start":"03:15.355 ","End":"03:18.620","Text":"But in the meantime, what\u0027s important to know is that there is"},{"Start":"03:18.620 ","End":"03:23.040","Text":"this equation that we know it and to try and understand it."},{"Start":"03:23.300 ","End":"03:29.750","Text":"Our equation for r as a function of Theta is equal to r_0 divided"},{"Start":"03:29.750 ","End":"03:35.745","Text":"by 1 plus Epsilon multiplied by cosine of Theta,"},{"Start":"03:35.745 ","End":"03:41.625","Text":"where r_0 and our Epsilon are these over here, these constants."},{"Start":"03:41.625 ","End":"03:48.320","Text":"We can see that all the terms in our r_0s are constants and similarly with our Epsilon."},{"Start":"03:48.320 ","End":"03:51.515","Text":"What these constants are for the meantime, doesn\u0027t matter."},{"Start":"03:51.515 ","End":"03:56.050","Text":"All we have to know right now is that these 2 are constants."},{"Start":"03:56.050 ","End":"04:00.495","Text":"We can see that our equation is our r as a function of Theta."},{"Start":"04:00.495 ","End":"04:04.505","Text":"We\u0027re working in polar coordinates where r represents"},{"Start":"04:04.505 ","End":"04:11.320","Text":"the distance that the body is from origin, from here."},{"Start":"04:12.620 ","End":"04:18.510","Text":"The distance is going to be our r and our Theta is"},{"Start":"04:18.510 ","End":"04:24.590","Text":"the angle that our r vector is in relation to the x-axis."},{"Start":"04:24.590 ","End":"04:30.475","Text":"If over here I labeled that this axis is my x-axis and this is my y-axis."},{"Start":"04:30.475 ","End":"04:35.035","Text":"My vector going from my body until the origin,"},{"Start":"04:35.035 ","End":"04:36.775","Text":"imagine that that\u0027s a straight line,"},{"Start":"04:36.775 ","End":"04:40.405","Text":"is going to be my r and that\u0027s the size."},{"Start":"04:40.405 ","End":"04:49.500","Text":"Then my angle is going to be this angle over here in gray and that is my Theta."},{"Start":"04:49.840 ","End":"04:52.205","Text":"If I know my angle,"},{"Start":"04:52.205 ","End":"04:58.195","Text":"then I can find my r. All I have to do is find my Theta,"},{"Start":"04:58.195 ","End":"05:01.940","Text":"I substitute it into this equation and I can know the distance"},{"Start":"05:01.940 ","End":"05:06.870","Text":"that the first body is from the second body."},{"Start":"05:07.060 ","End":"05:09.589","Text":"Now, with this equation,"},{"Start":"05:09.589 ","End":"05:15.830","Text":"we have a few options for the motion and they\u0027re dependent on our Epsilon over here."},{"Start":"05:15.830 ","End":"05:21.235","Text":"We can see that our Epsilon is dependent on the energy of the body."},{"Start":"05:21.235 ","End":"05:25.145","Text":"Now, remember that our energy, this over here,"},{"Start":"05:25.145 ","End":"05:30.485","Text":"is constant throughout the entire period of motion."},{"Start":"05:30.485 ","End":"05:32.300","Text":"Because this is a central force,"},{"Start":"05:32.300 ","End":"05:34.555","Text":"our energy is constant,"},{"Start":"05:34.555 ","End":"05:41.030","Text":"which means that if I know the energy of the body even at one point in the motion,"},{"Start":"05:41.030 ","End":"05:43.190","Text":"because it\u0027s constant throughout all the motion,"},{"Start":"05:43.190 ","End":"05:49.540","Text":"then I can know the trajectory or the course that my body will be moving in."},{"Start":"05:49.540 ","End":"05:53.690","Text":"In the next videos, we\u0027re going to see our different tracks or"},{"Start":"05:53.690 ","End":"05:55.280","Text":"the different trajectories of"},{"Start":"05:55.280 ","End":"05:59.645","Text":"different bodies and we\u0027ll see what happens with this equation."},{"Start":"05:59.645 ","End":"06:03.000","Text":"That\u0027s the end of this lesson."}],"ID":9455},{"Watched":false,"Name":"Circle And Ellipse 4","Duration":"17m 11s","ChapterTopicVideoID":9186,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.215","Text":"The first option that we have for this body to move is in a circular course."},{"Start":"00:07.215 ","End":"00:10.500","Text":"This happens when our Epsilon over here is equal to 0."},{"Start":"00:10.500 ","End":"00:13.890","Text":"When we set this equal to 0,"},{"Start":"00:13.890 ","End":"00:18.900","Text":"our Epsilon will see that our r as a function of Theta is going to be our r_0."},{"Start":"00:18.900 ","End":"00:20.820","Text":"If this is our track equation,"},{"Start":"00:20.820 ","End":"00:25.875","Text":"this is just a simple circle of radius r_0."},{"Start":"00:25.875 ","End":"00:30.720","Text":"Every time that we have a track equation where our distance from the origin is the same."},{"Start":"00:30.720 ","End":"00:34.230","Text":"That means that we\u0027re moving in a perfect circle,"},{"Start":"00:34.230 ","End":"00:37.290","Text":"and that happens when our Epsilon is equal to 0."},{"Start":"00:37.290 ","End":"00:39.255","Text":"This can be given by,"},{"Start":"00:39.255 ","End":"00:42.190","Text":"for example, a satellite orbiting Earth."},{"Start":"00:42.190 ","End":"00:44.150","Text":"A lot of the time when we\u0027re speaking about"},{"Start":"00:44.150 ","End":"00:47.030","Text":"some body which is moving in a perfect circle,"},{"Start":"00:47.030 ","End":"00:53.660","Text":"we\u0027re going to use this equation for our force is equal to our gravitational force,"},{"Start":"00:53.660 ","End":"00:56.180","Text":"our GMm divided by r^2."},{"Start":"00:56.180 ","End":"01:00.035","Text":"Now the R is capital just to show that this is a constant."},{"Start":"01:00.035 ","End":"01:05.015","Text":"The sum of our forces in the radial direction is going to be this."},{"Start":"01:05.015 ","End":"01:09.065","Text":"As you can see, we only have this force pointing to the center of the circle."},{"Start":"01:09.065 ","End":"01:12.560","Text":"Then because we\u0027re moving in circular motion,"},{"Start":"01:12.560 ","End":"01:14.960","Text":"we can say that our sum of all of the forces is also"},{"Start":"01:14.960 ","End":"01:17.705","Text":"going to be equal to mv^2 divided by our"},{"Start":"01:17.705 ","End":"01:23.750","Text":"R. We can isolate outer v to find what it is and it\u0027s this."},{"Start":"01:23.750 ","End":"01:28.100","Text":"Now notice this m is the mass of our Earth."},{"Start":"01:28.100 ","End":"01:33.560","Text":"The small m represents the mass of our satellite over here."},{"Start":"01:33.560 ","End":"01:39.215","Text":"But as we can see, because I was calculating forces acting on the satellite,"},{"Start":"01:39.215 ","End":"01:42.440","Text":"so my 2 small ms will cancel out."},{"Start":"01:42.440 ","End":"01:44.695","Text":"This m is our Earth."},{"Start":"01:44.695 ","End":"01:48.260","Text":"A lot of the times we\u0027re going to use this equation, or just straightaway,"},{"Start":"01:48.260 ","End":"01:53.600","Text":"write that our v is equal to the square root of GM divided by"},{"Start":"01:53.600 ","End":"01:59.210","Text":"R. Now something interesting that we can see from this equation is that if I let say,"},{"Start":"01:59.210 ","End":"02:04.090","Text":"send a satellite up to space and I wanted to travel a certain distance away from Earth,"},{"Start":"02:04.090 ","End":"02:08.240","Text":"we can see that all I have to do is increase or decrease"},{"Start":"02:08.240 ","End":"02:13.650","Text":"its velocity that it will be traveling at accordingly in order to get this equation."},{"Start":"02:13.910 ","End":"02:16.790","Text":"You\u0027ll see that in this equation,"},{"Start":"02:16.790 ","End":"02:19.520","Text":"there\u0027s nothing to do with the satellite itself,"},{"Start":"02:19.520 ","End":"02:22.520","Text":"so its size or its mass or anything."},{"Start":"02:22.520 ","End":"02:25.880","Text":"It\u0027s only the velocity with which it is traveling at,"},{"Start":"02:25.880 ","End":"02:30.835","Text":"and that will decide the distance that it can be away from the Earth."},{"Start":"02:30.835 ","End":"02:35.540","Text":"This was the easiest type of movement than we can have when we"},{"Start":"02:35.540 ","End":"02:40.325","Text":"have an object moving in a perfect circle and that is when our Epsilon is equal to 0."},{"Start":"02:40.325 ","End":"02:44.240","Text":"Let\u0027s see the other type of motion that this equation can describe."},{"Start":"02:44.240 ","End":"02:47.190","Text":"Our next shape is an ellipse."},{"Start":"02:47.190 ","End":"02:52.225","Text":"It looks like this and that happens when our Epsilon is between 0 and 1."},{"Start":"02:52.225 ","End":"02:54.930","Text":"This Epsilon over here is between 0 and 1."},{"Start":"02:54.930 ","End":"02:56.865","Text":"Now when does that happen?"},{"Start":"02:56.865 ","End":"02:58.805","Text":"It\u0027s when our energy,"},{"Start":"02:58.805 ","End":"03:00.815","Text":"our value for E over here,"},{"Start":"03:00.815 ","End":"03:04.045","Text":"is some negative value."},{"Start":"03:04.045 ","End":"03:07.340","Text":"All the other values over here are positive so"},{"Start":"03:07.340 ","End":"03:11.150","Text":"the only thing that can be is that our energy over here is negative,"},{"Start":"03:11.150 ","End":"03:13.700","Text":"which means that we\u0027re going to have the square root of"},{"Start":"03:13.700 ","End":"03:15.800","Text":"something which is smaller than 1,"},{"Start":"03:15.800 ","End":"03:20.590","Text":"which is also going to be smaller than 1 and bigger than 0."},{"Start":"03:20.590 ","End":"03:24.340","Text":"When our Epsilon value is between 0 and 1,"},{"Start":"03:24.340 ","End":"03:28.835","Text":"it means that our motion is going to be in the shape of an ellipse."},{"Start":"03:28.835 ","End":"03:32.810","Text":"Now an example for something in our universe which moves in"},{"Start":"03:32.810 ","End":"03:37.265","Text":"this motion is our Earth around the sun."},{"Start":"03:37.265 ","End":"03:39.095","Text":"Of course not just Earth,"},{"Start":"03:39.095 ","End":"03:40.865","Text":"but all the other planets in"},{"Start":"03:40.865 ","End":"03:44.165","Text":"our solar system also move in then the ellipse around the sun."},{"Start":"03:44.165 ","End":"03:46.595","Text":"Now with this elliptical trajectory,"},{"Start":"03:46.595 ","End":"03:51.800","Text":"we have some very important and interesting characteristics."},{"Start":"03:51.800 ","End":"03:58.280","Text":"We can see this type of trajectory is the most common in our work."},{"Start":"03:58.280 ","End":"04:01.505","Text":"Now before we speak about these special characteristics,"},{"Start":"04:01.505 ","End":"04:05.765","Text":"let\u0027s first speak about how we characterize an ellipse."},{"Start":"04:05.765 ","End":"04:11.300","Text":"What defines an ellipse is that every single point on the ellipse,"},{"Start":"04:11.300 ","End":"04:15.065","Text":"so every single point on this curve,"},{"Start":"04:15.065 ","End":"04:18.865","Text":"which is surrounding 2 focal points in the center of the ellipse,"},{"Start":"04:18.865 ","End":"04:21.980","Text":"that the sum of the distances to"},{"Start":"04:21.980 ","End":"04:27.725","Text":"the 2 focal points is constant for every single point on the curve."},{"Start":"04:27.725 ","End":"04:30.140","Text":"What does it mean, what I just said?"},{"Start":"04:30.140 ","End":"04:32.135","Text":"In an ellipse, there are"},{"Start":"04:32.135 ","End":"04:37.510","Text":"2 very important points inside the ellipse which are called focal points."},{"Start":"04:37.510 ","End":"04:42.395","Text":"What does that mean especially when we\u0027re speaking about planets orbiting?"},{"Start":"04:42.395 ","End":"04:47.270","Text":"The large body will be at one of the focal points so that will be over here."},{"Start":"04:47.270 ","End":"04:51.770","Text":"The next focal point will be on the other side symmetrically,"},{"Start":"04:51.770 ","End":"04:54.860","Text":"and it will be around about over here."},{"Start":"04:54.860 ","End":"04:58.865","Text":"Then if I choose any point on the curve of the ellipse,"},{"Start":"04:58.865 ","End":"05:01.220","Text":"so let\u0027s say this point over here,"},{"Start":"05:01.220 ","End":"05:02.840","Text":"but it can be any other point,"},{"Start":"05:02.840 ","End":"05:07.380","Text":"then the total distance to each focal points,"},{"Start":"05:07.380 ","End":"05:11.015","Text":"so that means this distance plus"},{"Start":"05:11.015 ","End":"05:13.490","Text":"this distance is going to be some"},{"Start":"05:13.490 ","End":"05:17.495","Text":"constant and it\u0027s going to be the same for every single point."},{"Start":"05:17.495 ","End":"05:20.615","Text":"That means if I chose a point over here,"},{"Start":"05:20.615 ","End":"05:24.265","Text":"this distance to that focal point plus"},{"Start":"05:24.265 ","End":"05:31.775","Text":"this distance to this focal point will be the same as the sum of these 2 distances."},{"Start":"05:31.775 ","End":"05:37.500","Text":"Our central body is always going to be at one of the focal points."},{"Start":"05:37.500 ","End":"05:41.765","Text":"Then we can define all measurements and"},{"Start":"05:41.765 ","End":"05:47.785","Text":"distances which are interesting to us and can help us to work out different equations."},{"Start":"05:47.785 ","End":"05:51.770","Text":"The first interesting measurement is this red arrow over here,"},{"Start":"05:51.770 ","End":"05:53.525","Text":"which is r min."},{"Start":"05:53.525 ","End":"05:58.250","Text":"This is our minimal distance and it goes from one of the focal points"},{"Start":"05:58.250 ","End":"06:03.765","Text":"until the outermost point of the ellipse."},{"Start":"06:03.765 ","End":"06:06.260","Text":"What does this distance represent?"},{"Start":"06:06.260 ","End":"06:07.970","Text":"That if, for instance, over here,"},{"Start":"06:07.970 ","End":"06:13.010","Text":"Earth carried on its orbit and stopped over here at this point so"},{"Start":"06:13.010 ","End":"06:16.715","Text":"this distance I\u0027m in is the minimal distance"},{"Start":"06:16.715 ","End":"06:20.960","Text":"that our Earth will ever be away from this focal point."},{"Start":"06:20.960 ","End":"06:23.720","Text":"The smallest distance away from the focal point,"},{"Start":"06:23.720 ","End":"06:26.075","Text":"the smallest distance that will be away from the sun."},{"Start":"06:26.075 ","End":"06:27.980","Text":"Then we also have r max,"},{"Start":"06:27.980 ","End":"06:30.320","Text":"which is our maximum distance."},{"Start":"06:30.320 ","End":"06:33.875","Text":"If our Earth continues in its orbit to this point,"},{"Start":"06:33.875 ","End":"06:40.395","Text":"so it will be the furthest away from focal point, from our sun."},{"Start":"06:40.395 ","End":"06:42.375","Text":"That\u0027s our r max."},{"Start":"06:42.375 ","End":"06:47.660","Text":"We have r, which is our distance that our Earth is from our sun,"},{"Start":"06:47.660 ","End":"06:50.570","Text":"which is at one of the focal points of the ellipse."},{"Start":"06:50.570 ","End":"06:55.094","Text":"R_min is the minimum distance that our Earth can be from the sun"},{"Start":"06:55.094 ","End":"06:59.375","Text":"and our r_max is the maximum distance it will be from the sun."},{"Start":"06:59.375 ","End":"07:03.470","Text":"The next thing over here is r_0. Now what is r_0?"},{"Start":"07:03.470 ","End":"07:09.080","Text":"It\u0027s a line which goes perpendicular from r_min and r_max."},{"Start":"07:09.080 ","End":"07:13.940","Text":"The angle over here is going to be 90 degrees and it\u0027s"},{"Start":"07:13.940 ","End":"07:19.055","Text":"this distance until we reach the curve of our ellipse."},{"Start":"07:19.055 ","End":"07:21.110","Text":"Now, when our Earth is here,"},{"Start":"07:21.110 ","End":"07:26.890","Text":"we can see that our angle Theta is going to be 90 degrees."},{"Start":"07:26.890 ","End":"07:30.124","Text":"Going back over here to our track equation,"},{"Start":"07:30.124 ","End":"07:33.880","Text":"cosine of 90 is going to be equal to 0."},{"Start":"07:33.880 ","End":"07:39.645","Text":"Then we\u0027ll get that our r at 90 degrees is equal to our r_0."},{"Start":"07:39.645 ","End":"07:43.820","Text":"That means that our Earth is going to be at this point here when we\u0027re"},{"Start":"07:43.820 ","End":"07:49.260","Text":"perpendicular to this line of r_min and r_max."},{"Start":"07:49.260 ","End":"07:53.325","Text":"That is what this constant r_0 is."},{"Start":"07:53.325 ","End":"07:56.390","Text":"It\u0027s this line over here, this distance over here,"},{"Start":"07:56.390 ","End":"08:00.325","Text":"which is at 90 degrees to r_min or r_max."},{"Start":"08:00.325 ","End":"08:03.995","Text":"Now what I want to do is I want to talk about these 2 points."},{"Start":"08:03.995 ","End":"08:07.070","Text":"When our Earth is located at r_minimum,"},{"Start":"08:07.070 ","End":"08:11.215","Text":"and when our Earth is located at r_maximum."},{"Start":"08:11.215 ","End":"08:14.375","Text":"Let\u0027s first write out our equation for energy."},{"Start":"08:14.375 ","End":"08:18.920","Text":"The equation for the energy of my Earth when it\u0027s at either of these points"},{"Start":"08:18.920 ","End":"08:23.430","Text":"or any other point on our ellipse is going to be equal to."},{"Start":"08:23.430 ","End":"08:27.740","Text":"We\u0027re going to have our E is equal to, the kinetic energy,"},{"Start":"08:27.740 ","End":"08:36.240","Text":"which is 1/2 mv^2 and then minus our force going in."},{"Start":"08:36.240 ","End":"08:39.080","Text":"That\u0027s going to be our gravitational force,"},{"Start":"08:39.080 ","End":"08:46.385","Text":"which is GMm divided by r. Now if I rearrange that,"},{"Start":"08:46.385 ","End":"08:55.155","Text":"so I will get that my 1/2mv^2 is going to be equal to E plus"},{"Start":"08:55.155 ","End":"09:03.395","Text":"my GMm divided by r. Now the reason that I\u0027ve rearranged it like so is because"},{"Start":"09:03.395 ","End":"09:06.380","Text":"I want to show something very interesting and important"},{"Start":"09:06.380 ","End":"09:12.035","Text":"specifically when Earth is located at r_min and at r_max."},{"Start":"09:12.035 ","End":"09:17.010","Text":"Our energy over here is of course a constant."},{"Start":"09:18.580 ","End":"09:22.640","Text":"It\u0027s always constant with the movement never changing."},{"Start":"09:22.640 ","End":"09:24.770","Text":"Why does this interest us?"},{"Start":"09:24.770 ","End":"09:28.640","Text":"Let\u0027s take a look at our denominator over here our r. This is"},{"Start":"09:28.640 ","End":"09:33.875","Text":"our radius and we\u0027re specifically talking about r_min and r_max."},{"Start":"09:33.875 ","End":"09:37.300","Text":"If my r is bigger,"},{"Start":"09:37.300 ","End":"09:40.750","Text":"so I\u0027ll write r is big,"},{"Start":"09:40.750 ","End":"09:47.210","Text":"so then this entire expression over here."},{"Start":"09:47.280 ","End":"09:50.515","Text":"I\u0027ll draw it in blue,"},{"Start":"09:50.515 ","End":"09:54.445","Text":"so that is going to be small."},{"Start":"09:54.445 ","End":"09:56.290","Text":"The larger the denominator,"},{"Start":"09:56.290 ","End":"09:58.615","Text":"the smaller this expression."},{"Start":"09:58.615 ","End":"10:01.510","Text":"Now because these 2 sides of the equation have to be"},{"Start":"10:01.510 ","End":"10:04.705","Text":"equal and my E over here as a constant,"},{"Start":"10:04.705 ","End":"10:06.370","Text":"so it\u0027s never changing."},{"Start":"10:06.370 ","End":"10:08.635","Text":"If this becomes smaller,"},{"Start":"10:08.635 ","End":"10:14.140","Text":"then that means that my velocity also has to become smaller."},{"Start":"10:14.140 ","End":"10:18.190","Text":"That means that the velocity with which my earth or my body,"},{"Start":"10:18.190 ","End":"10:20.500","Text":"whichever it is, is orbiting,"},{"Start":"10:20.500 ","End":"10:30.280","Text":"this focal point is also going to become smaller in order to keep this equation equal."},{"Start":"10:30.280 ","End":"10:37.240","Text":"What does that mean? My velocity at r_max,"},{"Start":"10:37.240 ","End":"10:39.250","Text":"so at my biggest radius,"},{"Start":"10:39.250 ","End":"10:41.470","Text":"when my radius is as big as possible,"},{"Start":"10:41.470 ","End":"10:45.940","Text":"is going to be a smaller velocity."},{"Start":"10:45.940 ","End":"10:49.345","Text":"That\u0027s going to be my minimum velocity."},{"Start":"10:49.345 ","End":"10:53.450","Text":"Similarly, when my r is small,"},{"Start":"10:54.780 ","End":"10:57.430","Text":"then my denominator is small,"},{"Start":"10:57.430 ","End":"11:03.490","Text":"which means that the whole expression in blue is going to be big."},{"Start":"11:03.490 ","End":"11:05.800","Text":"Then in order to keep,"},{"Start":"11:05.800 ","End":"11:08.035","Text":"because my E over here is a constant,"},{"Start":"11:08.035 ","End":"11:11.889","Text":"in order to keep both sides of the equation equal,"},{"Start":"11:11.889 ","End":"11:18.445","Text":"then that means that my velocity also has to increase."},{"Start":"11:18.445 ","End":"11:20.800","Text":"That means that therefore,"},{"Start":"11:20.800 ","End":"11:23.425","Text":"my velocity at r_min,"},{"Start":"11:23.425 ","End":"11:26.170","Text":"my minimum over here,"},{"Start":"11:26.170 ","End":"11:32.030","Text":"so that\u0027s going to be my maximum velocity."},{"Start":"11:32.100 ","End":"11:34.240","Text":"That\u0027s how it changes,"},{"Start":"11:34.240 ","End":"11:40.720","Text":"the closer that my earth is or my orbiting body is to the large body at the focal point,"},{"Start":"11:40.720 ","End":"11:43.555","Text":"the faster and the greater its velocity,"},{"Start":"11:43.555 ","End":"11:48.489","Text":"and the further that my orbiting body is from the focal point,"},{"Start":"11:48.489 ","End":"11:49.990","Text":"from the large body,"},{"Start":"11:49.990 ","End":"11:54.235","Text":"then the slower its velocity will be."},{"Start":"11:54.235 ","End":"11:56.995","Text":"Over here, at this point,"},{"Start":"11:56.995 ","End":"11:58.330","Text":"let\u0027s draw it like this,"},{"Start":"11:58.330 ","End":"12:02.740","Text":"so here my velocity is going to be a minimum and over here,"},{"Start":"12:02.740 ","End":"12:08.365","Text":"at this point, my velocity is going to be at a maximum."},{"Start":"12:08.365 ","End":"12:10.600","Text":"Now another important thing to look at,"},{"Start":"12:10.600 ","End":"12:15.145","Text":"at these specific points at my r_min and my r_max is that the angle"},{"Start":"12:15.145 ","End":"12:21.745","Text":"between my radius vector and my velocity vector is going to be 90 degrees."},{"Start":"12:21.745 ","End":"12:26.830","Text":"Specifically, only at these 2 points and another points,"},{"Start":"12:26.830 ","End":"12:31.075","Text":"so my r vector will be like this."},{"Start":"12:31.075 ","End":"12:36.145","Text":"This is some r and my velocity will be in this direction."},{"Start":"12:36.145 ","End":"12:39.460","Text":"As we can see, the angle is not 90 degrees."},{"Start":"12:39.460 ","End":"12:41.665","Text":"Only at r_min and r_max,"},{"Start":"12:41.665 ","End":"12:46.990","Text":"the angle between the radius and the velocity will be 90 degrees."},{"Start":"12:46.990 ","End":"12:50.485","Text":"Now, why is this interesting and important to know?"},{"Start":"12:50.485 ","End":"12:55.345","Text":"It\u0027s because it makes it easier for us to work out its angular momentum."},{"Start":"12:55.345 ","End":"12:59.005","Text":"Let\u0027s remember our equation for angular momentum."},{"Start":"12:59.005 ","End":"13:02.500","Text":"If we\u0027re going to take just the size of our angular momentum,"},{"Start":"13:02.500 ","End":"13:07.000","Text":"it\u0027s going to be equal to the size of our r vector,"},{"Start":"13:07.000 ","End":"13:10.690","Text":"multiplied by the size of our mass,"},{"Start":"13:10.690 ","End":"13:13.270","Text":"multiplied by our velocity vector,"},{"Start":"13:13.270 ","End":"13:21.250","Text":"and then multiplied by sine of the angle between our radius and our velocity."},{"Start":"13:21.250 ","End":"13:25.480","Text":"Now why is my angular momentum really interesting to me over here?"},{"Start":"13:25.480 ","End":"13:27.355","Text":"That\u0027s because over here,"},{"Start":"13:27.355 ","End":"13:30.475","Text":"it\u0027s size is something which is constant."},{"Start":"13:30.475 ","End":"13:32.815","Text":"My angular momentum, the size of it,"},{"Start":"13:32.815 ","End":"13:37.190","Text":"not the direction, the size is constant throughout."},{"Start":"13:37.560 ","End":"13:39.805","Text":"In order to work it out,"},{"Start":"13:39.805 ","End":"13:44.065","Text":"I have my r vector multiplied by my mv, just to remind you,"},{"Start":"13:44.065 ","End":"13:47.695","Text":"this my mv is my angular momentum,"},{"Start":"13:47.695 ","End":"13:51.715","Text":"and then multiplied by sine of the angle between the 2."},{"Start":"13:51.715 ","End":"13:54.130","Text":"In that case, at these specific points,"},{"Start":"13:54.130 ","End":"14:00.730","Text":"we can see that the angle between my r vector and my v vector is 90 degrees."},{"Start":"14:00.730 ","End":"14:03.760","Text":"That means that my Phi,"},{"Start":"14:03.760 ","End":"14:11.050","Text":"my angle at r_min is equal to my angle at r_max,"},{"Start":"14:11.050 ","End":"14:15.265","Text":"and that is equal to 90 degrees."},{"Start":"14:15.265 ","End":"14:16.795","Text":"Then what happens?"},{"Start":"14:16.795 ","End":"14:20.335","Text":"I can use this in order to get rid of this variable."},{"Start":"14:20.335 ","End":"14:22.930","Text":"Because this variable, I won\u0027t always know."},{"Start":"14:22.930 ","End":"14:25.750","Text":"How will I know what this angle fears over here?"},{"Start":"14:25.750 ","End":"14:27.475","Text":"I can\u0027t tell what that is."},{"Start":"14:27.475 ","End":"14:31.885","Text":"If I take at these 2 points, at my extremes,"},{"Start":"14:31.885 ","End":"14:33.805","Text":"my r_min and my r_max,"},{"Start":"14:33.805 ","End":"14:39.970","Text":"then I know that the angle is 90 and I know what my sine of 90 degrees is equal to."},{"Start":"14:39.970 ","End":"14:43.900","Text":"Then we can just rewrite this because we know that"},{"Start":"14:43.900 ","End":"14:47.649","Text":"our angular momentum is always constant and is never changing."},{"Start":"14:47.649 ","End":"14:50.905","Text":"Then we can write out this equation."},{"Start":"14:50.905 ","End":"14:55.030","Text":"Then I can write, let\u0027s write it here,"},{"Start":"14:55.030 ","End":"14:59.305","Text":"that my angular momentum at my minimum radius,"},{"Start":"14:59.305 ","End":"15:01.930","Text":"at my r_min, over here,"},{"Start":"15:01.930 ","End":"15:08.485","Text":"is going to be equal to my mass multiplied by my v_max."},{"Start":"15:08.485 ","End":"15:10.675","Text":"Because at the minimum radius,"},{"Start":"15:10.675 ","End":"15:14.515","Text":"my velocity is at a maximum multiplied by my r_min,"},{"Start":"15:14.515 ","End":"15:18.280","Text":"because it\u0027s going to be r_min multiplied by m multiplied"},{"Start":"15:18.280 ","End":"15:22.750","Text":"by v_max and sine of 90 is of course 1."},{"Start":"15:22.750 ","End":"15:29.155","Text":"Then that is going to be equal to my angular momentum and my r_max."},{"Start":"15:29.155 ","End":"15:31.495","Text":"Because angular momentum is conserved,"},{"Start":"15:31.495 ","End":"15:37.900","Text":"and that\u0027s going to be my mass multiplied by my v. Over here,"},{"Start":"15:37.900 ","End":"15:38.980","Text":"because I\u0027m an r_max,"},{"Start":"15:38.980 ","End":"15:43.585","Text":"it\u0027s going to be my v_min multiplied by my r,"},{"Start":"15:43.585 ","End":"15:45.310","Text":"which is my r_max."},{"Start":"15:45.310 ","End":"15:50.440","Text":"Again, my sine of 90 is equal to 1."},{"Start":"15:50.440 ","End":"15:54.805","Text":"When I\u0027m looking at these 2 points and my r_min and my r_max,"},{"Start":"15:54.805 ","End":"15:58.075","Text":"then that means in my equation for my angular momentum,"},{"Start":"15:58.075 ","End":"16:01.465","Text":"I have 1 less variable to try and work out and deal with."},{"Start":"16:01.465 ","End":"16:03.895","Text":"Because angular momentum is conserved,"},{"Start":"16:03.895 ","End":"16:07.030","Text":"I can work out the angular momentum at these 2 points and"},{"Start":"16:07.030 ","End":"16:10.870","Text":"then say that that is the angular momentum everywhere."},{"Start":"16:10.870 ","End":"16:15.099","Text":"This is the basic of what we need to know with an ellipse,"},{"Start":"16:15.099 ","End":"16:18.130","Text":"and it\u0027s important to understand the characteristics of"},{"Start":"16:18.130 ","End":"16:21.850","Text":"our r_min and our r_max and what that means with regards to"},{"Start":"16:21.850 ","End":"16:30.775","Text":"velocity and then how that translates into our equation for our angular momentum."},{"Start":"16:30.775 ","End":"16:34.960","Text":"Of course, just to clear something up that"},{"Start":"16:34.960 ","End":"16:39.475","Text":"our orbiting body can rotate or orbit in both directions,"},{"Start":"16:39.475 ","End":"16:43.900","Text":"going either anticlockwise or clockwise, doesn\u0027t matter."},{"Start":"16:43.900 ","End":"16:48.595","Text":"Remember that an ellipse is when our Epsilon is between our 0 and 1,"},{"Start":"16:48.595 ","End":"16:54.130","Text":"and to remember all of these equations and how I got to them."},{"Start":"16:54.130 ","End":"16:59.305","Text":"Now in the next video I\u0027m going to show some other useful qualities and"},{"Start":"16:59.305 ","End":"17:05.319","Text":"geometric qualities as well as equations which are referring to my ellipse."},{"Start":"17:05.319 ","End":"17:08.875","Text":"But these are the most important."},{"Start":"17:08.875 ","End":"17:11.720","Text":"That\u0027s the end of this lesson."}],"ID":9456},{"Watched":false,"Name":"Ellipse And Hyperbola","Duration":"20m 17s","ChapterTopicVideoID":9187,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"Hello, over here we can see that we have a lot of equations."},{"Start":"00:04.920 ","End":"00:08.940","Text":"Don\u0027t be afraid because a lot of them are very similar to one another,"},{"Start":"00:08.940 ","End":"00:11.760","Text":"and you don\u0027t need to know them off by heart as you can write"},{"Start":"00:11.760 ","End":"00:15.645","Text":"them in your notes or equation sheets."},{"Start":"00:15.645 ","End":"00:18.090","Text":"Let\u0027s take a look at what we have."},{"Start":"00:18.090 ","End":"00:22.050","Text":"Now, we already know what r_min is and what r_max is,"},{"Start":"00:22.050 ","End":"00:24.180","Text":"we spoke about it, last lesson,"},{"Start":"00:24.180 ","End":"00:29.205","Text":"r_min is the shortest distance from a focal point,"},{"Start":"00:29.205 ","End":"00:30.555","Text":"from our large broadly,"},{"Start":"00:30.555 ","End":"00:32.550","Text":"which is this over here."},{"Start":"00:32.550 ","End":"00:37.110","Text":"If our body carries on rotating and this ellipse shape so the r_min is over"},{"Start":"00:37.110 ","End":"00:41.300","Text":"here the shortest distance from the body to our focal point,"},{"Start":"00:41.300 ","End":"00:42.860","Text":"and of course our r_max,"},{"Start":"00:42.860 ","End":"00:44.075","Text":"which is this over here."},{"Start":"00:44.075 ","End":"00:46.375","Text":"It\u0027s our longest distance,"},{"Start":"00:46.375 ","End":"00:50.270","Text":"the furthest away that we can be from the focal point."},{"Start":"00:50.270 ","End":"00:54.890","Text":"If the Earth carries on rotating over here and it\u0027s found over here,"},{"Start":"00:54.890 ","End":"01:01.190","Text":"it will be the furthest distance it can possibly be away from the sun, our focal point."},{"Start":"01:01.190 ","End":"01:07.565","Text":"It\u0027s useful to see how we get our r_max and are r_min equations from this over here."},{"Start":"01:07.565 ","End":"01:11.045","Text":"This r as a function of Theta is our track equation."},{"Start":"01:11.045 ","End":"01:14.750","Text":"Let\u0027s say we want to find for my track equation our r_max."},{"Start":"01:14.750 ","End":"01:20.120","Text":"We know that our r_0 and our Epsilon over here are constants."},{"Start":"01:20.120 ","End":"01:24.035","Text":"In order to get my r Theta being some maximum value,"},{"Start":"01:24.035 ","End":"01:29.780","Text":"I know that I have to have my denominator as a minimum value."},{"Start":"01:29.780 ","End":"01:32.810","Text":"How can I make my denominator the lowest"},{"Start":"01:32.810 ","End":"01:37.400","Text":"possible when my cosine Theta is its lowest possible value?"},{"Start":"01:37.400 ","End":"01:41.825","Text":"The lowest value that cosine Theta can be is negative 1."},{"Start":"01:41.825 ","End":"01:44.990","Text":"If I substitute in my angle Theta,"},{"Start":"01:44.990 ","End":"01:48.725","Text":"such that my cosine of Theta is equal to negative 1,"},{"Start":"01:48.725 ","End":"01:51.270","Text":"then we\u0027ll see that we get my r_max,"},{"Start":"01:51.270 ","End":"01:54.795","Text":"which will be r_0 divided by 1 minus Epsilon,"},{"Start":"01:54.795 ","End":"01:56.660","Text":"and that\u0027s this equation over here."},{"Start":"01:56.660 ","End":"01:58.800","Text":"That\u0027s how we got this."},{"Start":"01:59.060 ","End":"02:03.260","Text":"The lowest value for the denominator is 1 minus Epsilon,"},{"Start":"02:03.260 ","End":"02:08.750","Text":"and that means that our whole number itself will be large, our r_max."},{"Start":"02:08.750 ","End":"02:11.360","Text":"Similarly for our r_minimum."},{"Start":"02:11.360 ","End":"02:13.580","Text":"If we want to have the minimal r,"},{"Start":"02:13.580 ","End":"02:17.375","Text":"that means that we want our denominator to be as big as possible."},{"Start":"02:17.375 ","End":"02:21.050","Text":"Again, our r_0 and our Epsilon are constant."},{"Start":"02:21.050 ","End":"02:24.160","Text":"The only thing I can change is my angle Theta,"},{"Start":"02:24.160 ","End":"02:26.600","Text":"and I wanted to put in an angle Theta such that"},{"Start":"02:26.600 ","End":"02:30.215","Text":"my cosine of Theta is its largest value that it can possibly be,"},{"Start":"02:30.215 ","End":"02:32.975","Text":"which of course is positive 1."},{"Start":"02:32.975 ","End":"02:39.985","Text":"If I do that, then I\u0027ll have that my r_min will equal r_0 divided by 1 plus Epsilon,"},{"Start":"02:39.985 ","End":"02:43.895","Text":"and that\u0027s again, what do we get over here for r_minimum equation."},{"Start":"02:43.895 ","End":"02:50.790","Text":"1 plus Epsilon is the largest possible denominator we can get."},{"Start":"02:51.040 ","End":"02:53.975","Text":"We can also check that that makes sense."},{"Start":"02:53.975 ","End":"02:55.280","Text":"In order to get our r_max,"},{"Start":"02:55.280 ","End":"02:58.260","Text":"we need our cosine of Theta to equal negative 1."},{"Start":"02:58.260 ","End":"03:04.865","Text":"When does cosine Theta equal negative 1 is when theta is equal to a 180 degrees,"},{"Start":"03:04.865 ","End":"03:06.385","Text":"and that\u0027s over here."},{"Start":"03:06.385 ","End":"03:08.100","Text":"If this is r_0,"},{"Start":"03:08.100 ","End":"03:12.675","Text":"180 degrees is over here. That works out."},{"Start":"03:12.675 ","End":"03:15.610","Text":"For our cosine Theta to equal 1,"},{"Start":"03:15.610 ","End":"03:19.160","Text":"then that means that Theta has to be equal to 0."},{"Start":"03:19.160 ","End":"03:21.110","Text":"When we\u0027re at an angle of 0,"},{"Start":"03:21.110 ","End":"03:24.625","Text":"and then we can see that that\u0027s over here our r_min."},{"Start":"03:24.625 ","End":"03:28.210","Text":"Our r_min is where cosine Theta is equal to 1,"},{"Start":"03:28.210 ","End":"03:30.530","Text":"and that\u0027s when I Theta is equal to 0."},{"Start":"03:30.530 ","End":"03:33.210","Text":"It makes sense and it works out."},{"Start":"03:33.490 ","End":"03:37.460","Text":"Just a reminder that this axis over here"},{"Start":"03:37.460 ","End":"03:40.610","Text":"is our y-axis and this axis over here is our x-axis."},{"Start":"03:40.610 ","End":"03:44.435","Text":"Of course our angles are measured relative to our x-axis."},{"Start":"03:44.435 ","End":"03:46.190","Text":"When our body is over here,"},{"Start":"03:46.190 ","End":"03:49.505","Text":"then the angle relative to the x-axis is 0,"},{"Start":"03:49.505 ","End":"03:52.745","Text":"and when our body is all the way over here,"},{"Start":"03:52.745 ","End":"03:56.930","Text":"so it\u0027s this angle over here, which is half a circle,"},{"Start":"03:56.930 ","End":"03:59.625","Text":"which is a 180 degrees,"},{"Start":"03:59.625 ","End":"04:02.640","Text":"and that corresponds to our r_max."},{"Start":"04:02.640 ","End":"04:05.760","Text":"Now we\u0027ve seen how we can find our r_max and"},{"Start":"04:05.760 ","End":"04:09.230","Text":"our r_min by knowing our r_0 and our Epsilon."},{"Start":"04:09.230 ","End":"04:11.500","Text":"But what happens if we have our r_max and"},{"Start":"04:11.500 ","End":"04:14.765","Text":"our r_min values and we want to calculate Epsilon?"},{"Start":"04:14.765 ","End":"04:21.035","Text":"All we have to do is put these 2 equations together and try and isolate out our Epsilon."},{"Start":"04:21.035 ","End":"04:24.365","Text":"Once you do that, you\u0027ll get that your Epsilon is equal to"},{"Start":"04:24.365 ","End":"04:28.010","Text":"r_max minus r_min divided by r_max plus r_min."},{"Start":"04:28.010 ","End":"04:30.905","Text":"A little bit of algebra over here,"},{"Start":"04:30.905 ","End":"04:33.635","Text":"and it\u0027s a very useful equation."},{"Start":"04:33.635 ","End":"04:36.140","Text":"Now let\u0027s talk about our a."},{"Start":"04:36.140 ","End":"04:39.045","Text":"We can see that our a is this over here."},{"Start":"04:39.045 ","End":"04:44.365","Text":"It\u0027s this length over here from one end of the ellipse up until the center."},{"Start":"04:44.365 ","End":"04:47.060","Text":"This is called the semi-major axes."},{"Start":"04:47.060 ","End":"04:49.835","Text":"The major axis is this line,"},{"Start":"04:49.835 ","End":"04:52.500","Text":"the equator of the ellipse."},{"Start":"04:52.500 ","End":"04:58.475","Text":"It\u0027s the long line going across from one end of the ellipse to the other, the long line."},{"Start":"04:58.475 ","End":"05:02.540","Text":"That\u0027s the major axes and semi obviously means in the center,"},{"Start":"05:02.540 ","End":"05:04.280","Text":"so here would be the center."},{"Start":"05:04.280 ","End":"05:07.970","Text":"Now, notice that we\u0027re not finding"},{"Start":"05:07.970 ","End":"05:13.190","Text":"the distance to our points over here where everybody is,"},{"Start":"05:13.190 ","End":"05:17.365","Text":"or to the focal points which are located here and here."},{"Start":"05:17.365 ","End":"05:21.180","Text":"It\u0027s right in the center of the ellipse."},{"Start":"05:21.180 ","End":"05:23.535","Text":"That\u0027s our a. How we find this,"},{"Start":"05:23.535 ","End":"05:26.210","Text":"it\u0027s our length of r_min plus r_max."},{"Start":"05:26.210 ","End":"05:30.040","Text":"Because notice that is equal to the whole major axis."},{"Start":"05:30.040 ","End":"05:33.400","Text":"Our r_min, is up until this focal point."},{"Start":"05:33.400 ","End":"05:36.790","Text":"Then from here until the end is our r_max,"},{"Start":"05:36.790 ","End":"05:39.385","Text":"and then divided by 2 will give us the center."},{"Start":"05:39.385 ","End":"05:44.290","Text":"Then when we substitute in our r_min and our r_max into this equation divided by 2,"},{"Start":"05:44.290 ","End":"05:45.980","Text":"we\u0027ll get this value,"},{"Start":"05:45.980 ","End":"05:49.820","Text":"r_0 divided by 1 minus Epsilon^2."},{"Start":"05:49.820 ","End":"05:54.715","Text":"Then when I substitute in my values for r_0 and Epsilon,"},{"Start":"05:54.715 ","End":"05:57.115","Text":"so that\u0027s these values over here,"},{"Start":"05:57.115 ","End":"06:04.860","Text":"then I\u0027ll get that my a this over here is equal to negative Alpha divided by 2E."},{"Start":"06:05.230 ","End":"06:08.540","Text":"Again, a reminder for what our Alpha is when we\u0027re"},{"Start":"06:08.540 ","End":"06:10.985","Text":"dealing with this case over here with the Earth and the Sun,"},{"Start":"06:10.985 ","End":"06:14.670","Text":"our Alpha is this value over here, GMm."},{"Start":"06:14.840 ","End":"06:17.450","Text":"All of these equations are useful."},{"Start":"06:17.450 ","End":"06:20.090","Text":"This is just a simplified version of this."},{"Start":"06:20.090 ","End":"06:23.000","Text":"With this section with the Alpha,"},{"Start":"06:23.000 ","End":"06:28.580","Text":"if our Alpha is going to be a constant when we\u0027re dealing with Earth, for instance."},{"Start":"06:28.580 ","End":"06:31.085","Text":"If we know our energy of the system,"},{"Start":"06:31.085 ","End":"06:37.534","Text":"then we can know what this length over here, 1/2 the ellipses."},{"Start":"06:37.534 ","End":"06:39.755","Text":"Then we have our b,"},{"Start":"06:39.755 ","End":"06:43.520","Text":"that\u0027s this over here, and that is our semi-minor axis."},{"Start":"06:43.520 ","End":"06:49.910","Text":"Our minor axis is the axis that goes perpendicular to our major axis,"},{"Start":"06:49.910 ","End":"06:54.190","Text":"our major axis being alongside and minor axis being the short side."},{"Start":"06:54.190 ","End":"06:58.845","Text":"A semi-minor is 1/2 of this, of the short."},{"Start":"06:58.845 ","End":"07:05.920","Text":"It\u0027s going to be of course, at 90 degrees to our a or at 90 degrees to our major axis,"},{"Start":"07:05.920 ","End":"07:09.859","Text":"and the equation for this is just going to be r_0"},{"Start":"07:09.859 ","End":"07:14.705","Text":"divided by the square root of 1 minus Epsilon^2."},{"Start":"07:14.705 ","End":"07:18.800","Text":"Then the last useful equation that we"},{"Start":"07:18.800 ","End":"07:22.295","Text":"should know is the equation to find the area of an ellipse."},{"Start":"07:22.295 ","End":"07:23.660","Text":"The area of an ellipse,"},{"Start":"07:23.660 ","End":"07:27.725","Text":"this is this S is equal to Pi times a times b,"},{"Start":"07:27.725 ","End":"07:30.035","Text":"where this is our a and this is b."},{"Start":"07:30.035 ","End":"07:32.650","Text":"Our a over here, and our b."},{"Start":"07:32.650 ","End":"07:35.885","Text":"This is the area, soon we\u0027re going to see why this equation is very"},{"Start":"07:35.885 ","End":"07:39.440","Text":"useful when we get into capillary equations."},{"Start":"07:39.440 ","End":"07:42.000","Text":"In the meantime, let\u0027s take a look."},{"Start":"07:42.000 ","End":"07:48.410","Text":"If we have our area of an ellipse is equal to Pi times a times b,"},{"Start":"07:48.410 ","End":"07:50.960","Text":"if we\u0027re going to look at a circle,"},{"Start":"07:50.960 ","End":"07:54.650","Text":"which is a specific type of ellipse."},{"Start":"07:54.650 ","End":"07:58.070","Text":"In a cycle, how can we define it in terms of an ellipse?"},{"Start":"07:58.070 ","End":"08:01.480","Text":"It\u0027s when our a and our b are equal."},{"Start":"08:01.480 ","End":"08:05.040","Text":"Because our radius is constant in a circle."},{"Start":"08:05.040 ","End":"08:06.720","Text":"Our a and our b are equal,"},{"Start":"08:06.720 ","End":"08:09.945","Text":"and then that\u0027s called, as we know in a circle, our radius."},{"Start":"08:09.945 ","End":"08:12.545","Text":"Then we\u0027ll have Pi times r times r,"},{"Start":"08:12.545 ","End":"08:14.585","Text":"which is simply Pir^2."},{"Start":"08:14.585 ","End":"08:18.960","Text":"As we can see, that is in fact the area of a circle."},{"Start":"08:19.130 ","End":"08:25.520","Text":"This is also an important equation to learn and that is it for this section,"},{"Start":"08:25.520 ","End":"08:28.910","Text":"the next thing that we\u0027re going to speak about is the hyperbola,"},{"Start":"08:28.910 ","End":"08:35.365","Text":"which is our third option for motion when we\u0027re dealing with our orbitals."},{"Start":"08:35.365 ","End":"08:37.475","Text":"When we\u0027re speaking about our motion,"},{"Start":"08:37.475 ","End":"08:39.950","Text":"when dealing with the shape of a hyperbola,"},{"Start":"08:39.950 ","End":"08:45.620","Text":"that means that our Epsilon must be bigger than 1."},{"Start":"08:45.620 ","End":"08:50.135","Text":"Our hyperbolic motion, this type of motion,"},{"Start":"08:50.135 ","End":"08:52.445","Text":"is significantly different to"},{"Start":"08:52.445 ","End":"08:56.225","Text":"our circular motion or a motion when dealing with an ellipse."},{"Start":"08:56.225 ","End":"08:58.910","Text":"Because in circular motion with the ellipse,"},{"Start":"08:58.910 ","End":"09:00.890","Text":"our motion was periodical."},{"Start":"09:00.890 ","End":"09:02.720","Text":"Now here with the hyperbola,"},{"Start":"09:02.720 ","End":"09:08.015","Text":"we can see that if we have some asteroid which is coming close to Earth,"},{"Start":"09:08.015 ","End":"09:11.360","Text":"the asteroid will go around and then fly off,"},{"Start":"09:11.360 ","End":"09:16.510","Text":"it\u0027s not going to continue at circle and carry on orbiting the Earth."},{"Start":"09:16.510 ","End":"09:20.720","Text":"When we were dealing with our circle orbitals or ellipse,"},{"Start":"09:20.720 ","End":"09:26.065","Text":"we could see that our body would be orbiting around some shape,"},{"Start":"09:26.065 ","End":"09:31.535","Text":"some center, and we could work out the time per orbital,"},{"Start":"09:31.535 ","End":"09:35.320","Text":"the velocity that was rotating around and it would carry on indefinitely."},{"Start":"09:35.320 ","End":"09:36.830","Text":"Here with a hyperbola,"},{"Start":"09:36.830 ","End":"09:38.990","Text":"we can see that some body,"},{"Start":"09:38.990 ","End":"09:42.350","Text":"let\u0027s say this asteroid comes from some area in space,"},{"Start":"09:42.350 ","End":"09:45.215","Text":"from infinity, comes towards Earth,"},{"Start":"09:45.215 ","End":"09:47.165","Text":"gets really close to Earth."},{"Start":"09:47.165 ","End":"09:49.250","Text":"We can see this r_min over here,"},{"Start":"09:49.250 ","End":"09:56.125","Text":"and then flies back out and returns to infinity somewhere in space."},{"Start":"09:56.125 ","End":"10:03.740","Text":"Again, this type of motion happens when our Epsilon is bigger or equal to 1."},{"Start":"10:03.740 ","End":"10:06.590","Text":"That means that this over here is bigger or equal to 1."},{"Start":"10:06.590 ","End":"10:07.835","Text":"What does that mean?"},{"Start":"10:07.835 ","End":"10:10.555","Text":"That means something to do with energy."},{"Start":"10:10.555 ","End":"10:18.185","Text":"This is the same as saying that our energy of the system is bigger or equal to 0."},{"Start":"10:18.185 ","End":"10:21.785","Text":"When that happens, when our energy over here is bigger or equal to 0,"},{"Start":"10:21.785 ","End":"10:25.920","Text":"then our Epsilon will be bigger or equal to 1."},{"Start":"10:26.410 ","End":"10:32.450","Text":"We can really see that when we substitute that our energy is bigger than 0 over"},{"Start":"10:32.450 ","End":"10:37.640","Text":"here over here this expression will be some positive expression,"},{"Start":"10:37.640 ","End":"10:41.509","Text":"and then our Epsilon will be the square root of 1 plus this expression,"},{"Start":"10:41.509 ","End":"10:44.640","Text":"which will be bigger than 1."},{"Start":"10:44.640 ","End":"10:46.690","Text":"That\u0027s our Epsilon which is bigger than 1,"},{"Start":"10:46.690 ","End":"10:51.410","Text":"and when we substitute in that our energy is exactly equal to 0,"},{"Start":"10:51.410 ","End":"10:54.530","Text":"so we can see here that this then will equal 0 and then"},{"Start":"10:54.530 ","End":"10:58.190","Text":"our Epsilon is just going to be equal to 1."},{"Start":"10:58.190 ","End":"11:00.650","Text":"Then what we get is a parabola."},{"Start":"11:00.650 ","End":"11:06.676","Text":"A parabola is just a type of hyperbola."},{"Start":"11:06.676 ","End":"11:13.333","Text":"But they\u0027re very similar."},{"Start":"11:13.333 ","End":"11:23.189","Text":"I don\u0027t really right now want to go into"},{"Start":"11:23.189 ","End":"11:32.464","Text":"explaining the difference between"},{"Start":"11:32.464 ","End":"11:39.597","Text":"a hyperbola and a parabola,"},{"Start":"11:39.597 ","End":"11:49.865","Text":"just know that they\u0027re very similar and"},{"Start":"11:49.865 ","End":"11:56.739","Text":"that if you have an Epsilon"},{"Start":"11:56.739 ","End":"12:04.265","Text":"which is bigger or equal to 1,"},{"Start":"12:04.265 ","End":"12:11.508","Text":"that means that your energy"},{"Start":"12:11.508 ","End":"12:17.512","Text":"is bigger or equal to 0,"},{"Start":"12:17.512 ","End":"12:21.633","Text":"and then you get"},{"Start":"12:21.633 ","End":"12:28.631","Text":"a hyperbola or a parabola,"},{"Start":"12:28.631 ","End":"12:36.617","Text":"which is a type of a hyperbola."},{"Start":"12:36.617 ","End":"12:39.822","Text":"This is what"},{"Start":"12:39.822 ","End":"12:48.498","Text":"that track equation looks like."},{"Start":"12:48.498 ","End":"12:58.182","Text":"The main things to know when dealing"},{"Start":"12:58.182 ","End":"13:07.908","Text":"with this hyperbolic motion is that"},{"Start":"13:07.908 ","End":"13:13.766","Text":"because a asteroid or"},{"Start":"13:13.766 ","End":"13:23.243","Text":"some body is coming from infinity,"},{"Start":"13:23.243 ","End":"13:29.803","Text":"doing some semi-orbit,"},{"Start":"13:29.803 ","End":"13:38.692","Text":"and then going back to infinity,"},{"Start":"13:38.692 ","End":"13:45.636","Text":"this is some open motion."},{"Start":"13:45.636 ","End":"13:54.611","Text":"Over here we can see it\u0027s closed,"},{"Start":"13:54.611 ","End":"14:03.977","Text":"but we don\u0027t have an end over here."},{"Start":"14:03.977 ","End":"14:10.031","Text":"That means that we\u0027re"},{"Start":"14:10.031 ","End":"14:17.803","Text":"not going to have our r_max,"},{"Start":"14:17.803 ","End":"14:24.251","Text":"because our r_max will"},{"Start":"14:24.251 ","End":"14:30.046","Text":"just go to infinity,"},{"Start":"14:30.046 ","End":"14:37.582","Text":"but we will have our r_min."},{"Start":"14:37.582 ","End":"14:46.190","Text":"That\u0027s when our asteroid is as"},{"Start":"14:46.190 ","End":"14:55.562","Text":"close as it will ever be to earth."},{"Start":"14:55.562 ","End":"15:03.948","Text":"That\u0027s over here, and again,"},{"Start":"15:03.948 ","End":"15:13.088","Text":"it\u0027s located at this point over"},{"Start":"15:13.088 ","End":"15:18.747","Text":"here when our angle"},{"Start":"15:18.747 ","End":"15:24.434","Text":"Theta is equal to 0."},{"Start":"15:24.434 ","End":"15:28.007","Text":"Now, we have"},{"Start":"15:28.007 ","End":"15:37.351","Text":"the same type of conservation,"},{"Start":"15:37.351 ","End":"15:47.475","Text":"we have conservation of momentum"},{"Start":"15:47.475 ","End":"15:55.837","Text":"and conservation of energy"},{"Start":"15:55.837 ","End":"15:59.857","Text":"in this case,"},{"Start":"15:59.857 ","End":"16:05.731","Text":"and that\u0027s all that"},{"Start":"16:05.731 ","End":"16:12.383","Text":"you have to remember."},{"Start":"16:12.383 ","End":"16:16.095","Text":"Now let\u0027s do"},{"Start":"16:16.095 ","End":"16:22.814","Text":"a little conclusion."},{"Start":"16:22.814 ","End":"16:31.829","Text":"The first option that we have"},{"Start":"16:31.829 ","End":"16:42.092","Text":"is when our Epsilon is equal to 0."},{"Start":"16:42.092 ","End":"16:48.624","Text":"What does that mean?"},{"Start":"16:48.624 ","End":"16:56.759","Text":"That means that let\u0027s say"},{"Start":"16:56.759 ","End":"17:05.773","Text":"we have our Earth over here,"},{"Start":"17:05.773 ","End":"17:10.916","Text":"and then we have"},{"Start":"17:10.916 ","End":"17:16.094","Text":"some satellite"},{"Start":"17:16.094 ","End":"17:21.692","Text":"orbiting Earth,"},{"Start":"17:21.692 ","End":"17:28.538","Text":"and it\u0027s orbiting in"},{"Start":"17:28.538 ","End":"17:39.955","Text":"a perfect circle where the radius"},{"Start":"17:39.955 ","End":"17:47.803","Text":"is always constant and"},{"Start":"17:47.803 ","End":"17:52.806","Text":"it\u0027s some r_0,"},{"Start":"17:52.806 ","End":"18:00.406","Text":"and we have some force"},{"Start":"18:00.406 ","End":"18:07.251","Text":"going in over here,"},{"Start":"18:07.251 ","End":"18:17.712","Text":"and we know that our equation"},{"Start":"18:17.712 ","End":"18:28.887","Text":"for the sum of all of the forces"},{"Start":"18:28.887 ","End":"18:38.048","Text":"going into the center of"},{"Start":"18:38.048 ","End":"18:45.538","Text":"a circular orbit is"},{"Start":"18:45.538 ","End":"18:52.240","Text":"going to be as so."},{"Start":"18:52.240 ","End":"19:01.649","Text":"The sum of our forces in"},{"Start":"19:01.649 ","End":"19:11.877","Text":"the radial direction is"},{"Start":"19:11.877 ","End":"19:18.512","Text":"equal to our GMm divided by our radius squared,"},{"Start":"19:18.512 ","End":"19:20.481","Text":"and that\u0027s of course, because we\u0027re dealing with"},{"Start":"19:20.481 ","End":"19:22.607","Text":"circular motion equal to mv^2 divided by our radius."},{"Start":"19:22.607 ","End":"19:24.756","Text":"We can of course workout our V and our V will be equal to"},{"Start":"19:24.756 ","End":"19:27.381","Text":"the square root of GM divided by r. We went over this last lesson."},{"Start":"19:27.381 ","End":"19:30.545","Text":"This of course happens our Epsilon is equal to 0 when our energy is equal to 0."},{"Start":"19:30.545 ","End":"19:32.687","Text":"Now, our second option that we have is our ellipse."},{"Start":"19:32.687 ","End":"19:35.582","Text":"Now our ellipse happens when our Epsilon is between 0 and between 1."},{"Start":"19:35.582 ","End":"19:36.732","Text":"Now when does this happen?"},{"Start":"19:36.732 ","End":"19:39.164","Text":"When our energy, our value for e is some negative value."},{"Start":"19:39.164 ","End":"19:41.796","Text":"With an ellipse, we have 2 focal points inside the ellipse,"},{"Start":"19:41.796 ","End":"19:43.648","Text":"one of which has large body in the center."},{"Start":"19:43.648 ","End":"19:45.692","Text":"For instance, if we\u0027re dealing with our sun,"},{"Start":"19:45.692 ","End":"19:48.847","Text":"so then we have some elliptical motion where we have that over here."},{"Start":"19:48.847 ","End":"19:51.274","Text":"We have our r_min, at an angle of Theta is equal to 0."},{"Start":"19:51.274 ","End":"19:54.677","Text":"Over here, we\u0027ll have our r_max at an angle equal to Pi or 180 degrees."},{"Start":"19:54.677 ","End":"19:56.214","Text":"We also have, if you remember,"},{"Start":"19:56.214 ","End":"19:57.459","Text":"some maximum velocity,"},{"Start":"19:57.459 ","End":"19:59.872","Text":"and we have our minimum velocity where both of"},{"Start":"19:59.872 ","End":"20:03.438","Text":"these velocities are at 90 degrees to the radius at these points."},{"Start":"20:03.438 ","End":"20:05.879","Text":"Then we had at the beginning of the lesson,"},{"Start":"20:05.879 ","End":"20:09.579","Text":"the different equations that we had relating to an ellipse."},{"Start":"20:09.579 ","End":"20:10.687","Text":"If you remember,"},{"Start":"20:10.687 ","End":"20:13.226","Text":"we had until our center lengthwise,"},{"Start":"20:13.226 ","End":"20:14.898","Text":"this length over here"},{"Start":"20:14.898 ","End":"20:16.505","Text":"was our length A,"},{"Start":"20:16.505 ","End":"20:17.050","Text":"and this was the semi-major,"},{"Start":"20:17.050 ","End":"20:17.051","Text":"and then we had until the middle from the short side,"},{"Start":"20:17.051 ","End":"20:17.052","Text":"and this was our B and this was semi-minor and they\u0027re of course,"},{"Start":"20:17.052 ","End":"20:17.053","Text":"perpendicular to one another."},{"Start":"20:17.053 ","End":"20:17.054","Text":"Then we had the equations for working them out. You can go back a bit in the video."},{"Start":"20:17.054 ","End":"20:17.055","Text":"Our third and final option was, of course, our hyperbola,"},{"Start":"20:17.055 ","End":"20:17.056","Text":"and that was when our Epsilon was bigger or equal to 1,"},{"Start":"20:17.056 ","End":"20:17.057","Text":"which corresponded to our energy being bigger or equal to 0."},{"Start":"20:17.057 ","End":"20:17.058","Text":"Here we had some shape, let\u0027s say it\u0027s again, our Earth,"},{"Start":"20:17.058 ","End":"20:17.059","Text":"and then we had some asteroid traveling towards our earth in this motion."},{"Start":"20:17.059 ","End":"20:17.060","Text":"This was some open type of motion."},{"Start":"20:17.060 ","End":"20:17.061","Text":"Our asteroid comes from infinity, comes around the Earth and returns to infinity."},{"Start":"20:17.061 ","End":"20:17.062","Text":"We didn\u0027t have our value for r_max, but we did have our value for our r_min,"},{"Start":"20:17.062 ","End":"20:17.063","Text":"again located at Theta is equal to 0."},{"Start":"20:17.063 ","End":"20:17.064","Text":"Although this motion is not periodical,"},{"Start":"20:17.064 ","End":"20:17.065","Text":"like what we had here with the ellipse or here where the circle,"},{"Start":"20:17.065 ","End":"20:17.066","Text":"we still have conservation of energy and conservation of angular momentum."},{"Start":"20:17.066 ","End":"20:17.067","Text":"We saw that all of our options can be described by this equation over here,"},{"Start":"20:17.067 ","End":"20:17.068","Text":"which is called r-track equation."},{"Start":"20:17.068 ","End":"20:17.069","Text":"It\u0027s our r as a function of Theta and it equals to our r_0"},{"Start":"20:17.069 ","End":"20:17.070","Text":"divided by plus Epsilon multiplied by"},{"Start":"20:17.070 ","End":"20:17.071","Text":"cosine of Theta where we saw that our r_0 and our Epsilon were constants."},{"Start":"20:17.071 ","End":"20:17.072","Text":"Over here in our equation for our Epsilon, we see that we have this Alpha."},{"Start":"20:17.072 ","End":"20:17.073","Text":"Our Alpha is always going to be some constant value,"},{"Start":"20:17.073 ","End":"20:17.074","Text":"and it\u0027s going to be our value for big G,"},{"Start":"20:17.074 ","End":"20:17.075","Text":"which is this value over here,"},{"Start":"20:17.075 ","End":"20:17.076","Text":"multiplied by the mass of the large body and multiplied by the mass of the smaller body."},{"Start":"20:17.076 ","End":"20:17.077","Text":"Our Alpha is something that we can know immediately but in order to"},{"Start":"20:17.077 ","End":"20:17.078","Text":"find out my values for my r_0 and my Epsilon,"},{"Start":"20:17.078 ","End":"20:17.079","Text":"I\u0027m going to need to know the values for my energy and for my angular momentum,"},{"Start":"20:17.079 ","End":"20:17.080","Text":"my L over here."},{"Start":"20:17.080 ","End":"20:17.081","Text":"In our first lesson where we were speaking about this,"},{"Start":"20:17.081 ","End":"20:17.082","Text":"we saw that we can solve these types of questions by using the idea of conservation of"},{"Start":"20:17.082 ","End":"20:17.083","Text":"energy and conservation of angular momentum,"},{"Start":"20:17.083 ","End":"20:17.084","Text":"as we can see in our equations."},{"Start":"20:17.084 ","End":"20:17.085","Text":"We saw that if I can find my energy in the system at some time,"},{"Start":"20:17.085 ","End":"20:17.086","Text":"and if I can find my angular momentum of the system at any time,"},{"Start":"20:17.086 ","End":"20:17.087","Text":"that means that that\u0027s the energy and the angular momentum at"},{"Start":"20:17.087 ","End":"20:17.088","Text":"every single moment because we"},{"Start":"20:17.088 ","End":"20:17.089","Text":"have conservation of energy"},{"Start":"20:17.089 ","End":"20:17.090","Text":"and conservation of angular momentum."},{"Start":"20:17.090 ","End":"20:17.091","Text":"Then I could substitute in my values for my e and my L into my equations for my Epsilon,"},{"Start":"20:17.091 ","End":"20:17.092","Text":"and my r_0,"},{"Start":"20:17.092 ","End":"20:17.093","Text":"find out what their values are,"},{"Start":"20:17.093 ","End":"20:17.094","Text":"and then substitute them into my track equation over here and then find out"},{"Start":"20:17.094 ","End":"20:17.095","Text":"which shape I have."},{"Start":"20:17.095 ","End":"20:17.096","Text":"Then I\u0027ll know what my trajectory looks like exactly."},{"Start":"20:17.096 ","End":"20:17.097","Text":"How would I find out what my value for energy is?"},{"Start":"20:17.097 ","End":"20:17.098","Text":"If we look at our energy equation, I can see that if I can find some point in my orbit,"},{"Start":"20:17.098 ","End":"20:17.099","Text":"whichever type of orbit it is,"},{"Start":"20:17.099 ","End":"20:17.100","Text":"where I can know my value for my V and my value for my r,"},{"Start":"20:17.100 ","End":"20:17.101","Text":"then I can substitute them in and find out my energy."},{"Start":"20:17.101 ","End":"20:17.102","Text":"Then of course, for my angular momentum,"},{"Start":"20:17.102 ","End":"20:17.103","Text":"I\u0027d have to know my r cross my mass times my velocity,"},{"Start":"20:17.103 ","End":"20:17.104","Text":"which means that I would also have to know the angle between my radius"},{"Start":"20:17.104 ","End":"20:17.105","Text":"and my velocity vectors."},{"Start":"20:17.105 ","End":"20:17.106","Text":"That\u0027s why a lot of the time I\u0027m going to want to try and find my angular momentum from"},{"Start":"20:17.106 ","End":"20:17.107","Text":"my point where I have my r minimum,"},{"Start":"20:17.107 ","End":"20:17.108","Text":"because there I know that my r minimum is perpendicular to my velocity."},{"Start":"20:17.108 ","End":"20:17.109","Text":"The angle between my r_min and my velocity is always going to be 90 degrees."},{"Start":"20:17.109 ","End":"20:17.110","Text":"That\u0027s another way to solve these types of questions,"},{"Start":"20:17.110 ","End":"20:17.111","Text":"is to try and find my energy by trying to find a specific point on my orbit that I"},{"Start":"20:17.111 ","End":"20:17.112","Text":"can know my velocity and my radius,"},{"Start":"20:17.112 ","End":"20:17.113","Text":"and to find my angular momentum."},{"Start":"20:17.113 ","End":"20:17.114","Text":"Usually, I\u0027ll use my r_min because then I will also know my r,"},{"Start":"20:17.114 ","End":"20:17.115","Text":"I\u0027ll also know my velocity and I know the angle between the."},{"Start":"20:17.115 ","End":"20:17.116","Text":"Then I can just substitute in those values into these 2 equations and then"},{"Start":"20:17.116 ","End":"20:17.117","Text":"substitute it in to my track equation."},{"Start":"20:17.117 ","End":"20:18.050","Text":"That\u0027s the end of the lesson."}],"ID":9457},{"Watched":false,"Name":"Rocket Fired From Earth Returns","Duration":"14m 18s","ChapterTopicVideoID":9188,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.875","Text":"Hello. In this question,"},{"Start":"00:01.875 ","End":"00:06.765","Text":"we\u0027re going to be answering a question about gravitational force."},{"Start":"00:06.765 ","End":"00:12.375","Text":"Here we have our Earth and from Earth they shoot out some rocket."},{"Start":"00:12.375 ","End":"00:20.115","Text":"The rocket does some trajectory like so and returns to Earth."},{"Start":"00:20.115 ","End":"00:25.365","Text":"We\u0027re being told that there\u0027s some point on the trajectory."},{"Start":"00:25.365 ","End":"00:30.690","Text":"Let\u0027s say it\u0027s over here and its distance from the center of the Earth,"},{"Start":"00:30.690 ","End":"00:35.925","Text":"this distance over here is equal to r_1."},{"Start":"00:35.925 ","End":"00:41.310","Text":"We\u0027re also being told that the angle between this r_1 vector,"},{"Start":"00:41.310 ","End":"00:46.190","Text":"and the velocity at that point is equal to 30 degrees."},{"Start":"00:46.190 ","End":"00:51.870","Text":"What does that mean? That means that we have some velocity over here."},{"Start":"00:51.870 ","End":"00:54.495","Text":"Let\u0027s call this V_1."},{"Start":"00:54.495 ","End":"01:00.790","Text":"Of course, V_1 is tangential to the trajectory."},{"Start":"01:00.950 ","End":"01:06.105","Text":"We\u0027re being told that the angle between our r_1 and our V_1,"},{"Start":"01:06.105 ","End":"01:07.780","Text":"which we don\u0027t know."},{"Start":"01:07.780 ","End":"01:12.302","Text":"This angle over here is 30 degrees."},{"Start":"01:12.302 ","End":"01:16.760","Text":"We\u0027re also given the radius of our Earth."},{"Start":"01:16.760 ","End":"01:19.415","Text":"That\u0027s called our Re,"},{"Start":"01:19.415 ","End":"01:22.640","Text":"the radius of the Earth and we\u0027re being told that"},{"Start":"01:22.640 ","End":"01:27.765","Text":"the rocket is hitting Earth with some velocity,"},{"Start":"01:27.765 ","End":"01:29.620","Text":"let\u0027s call it V_2."},{"Start":"01:29.620 ","End":"01:35.345","Text":"That if we take this to be some axes,"},{"Start":"01:35.345 ","End":"01:40.360","Text":"then we\u0027re being told that the angle between our V_2 and"},{"Start":"01:40.360 ","End":"01:46.790","Text":"this axes is equal to Theta and our Theta is also given."},{"Start":"01:47.000 ","End":"01:50.785","Text":"Again, our V_2, we don\u0027t know."},{"Start":"01:50.785 ","End":"01:53.935","Text":"What we do know is our radius of the Earth,"},{"Start":"01:53.935 ","End":"01:56.545","Text":"our r_1 over here,"},{"Start":"01:56.545 ","End":"01:58.570","Text":"our angle over here,"},{"Start":"01:58.570 ","End":"02:01.314","Text":"which is 30 degrees,"},{"Start":"02:01.314 ","End":"02:04.990","Text":"and our angle between our V_2 in this axis,"},{"Start":"02:04.990 ","End":"02:08.150","Text":"which is equal to Theta."},{"Start":"02:08.180 ","End":"02:10.585","Text":"Question number 1,"},{"Start":"02:10.585 ","End":"02:18.635","Text":"we\u0027re being asked to find what is our Theta_0 and what is our V_0, so what is that?"},{"Start":"02:18.635 ","End":"02:23.299","Text":"If we still have our axes over here,"},{"Start":"02:23.299 ","End":"02:28.535","Text":"going like so, so our angle with which our rocket leaves."},{"Start":"02:28.535 ","End":"02:31.625","Text":"This is our Theta_0 which we\u0027re trying to find."},{"Start":"02:31.625 ","End":"02:36.395","Text":"Of course our rocket is going to have some kind of velocity V_0,"},{"Start":"02:36.395 ","End":"02:40.445","Text":"which is of course tangential to the trajectory."},{"Start":"02:40.445 ","End":"02:43.730","Text":"We\u0027re being asked to find what is our Theta_0 and our V_0,"},{"Start":"02:43.730 ","End":"02:50.185","Text":"our velocity at exiting the Earth and the angle with which the rocket exits the Earth."},{"Start":"02:50.185 ","End":"02:54.650","Text":"The next things that we\u0027re being asked to find are our V_1."},{"Start":"02:54.650 ","End":"03:01.980","Text":"That\u0027s that over here and our V_2 over here when the rocket comes back to Earth."},{"Start":"03:02.150 ","End":"03:06.165","Text":"What we have are 4 unknowns that we need to find."},{"Start":"03:06.165 ","End":"03:07.951","Text":"Let\u0027s see how we do this."},{"Start":"03:07.951 ","End":"03:12.230","Text":"Now because we\u0027re dealing with some question with gravity."},{"Start":"03:12.230 ","End":"03:15.020","Text":"We have a very strong force of gravity over here."},{"Start":"03:15.020 ","End":"03:18.455","Text":"The first thing that we should do is use the idea of"},{"Start":"03:18.455 ","End":"03:22.960","Text":"our central force because gravity is a type of central force."},{"Start":"03:22.960 ","End":"03:25.840","Text":"We know our central force due to"},{"Start":"03:25.840 ","End":"03:28.975","Text":"gravity and we know that when dealing with central forces,"},{"Start":"03:28.975 ","End":"03:34.360","Text":"we have conservation of energy and conservation of angular momentum."},{"Start":"03:34.360 ","End":"03:38.995","Text":"I wrote over here that we have conservation of energy and angular momentum."},{"Start":"03:38.995 ","End":"03:42.580","Text":"Now let\u0027s write our conservation equations."},{"Start":"03:42.580 ","End":"03:45.685","Text":"In this question, we have 3 points of interest."},{"Start":"03:45.685 ","End":"03:49.495","Text":"We have our 0 where rocket exits the Earth."},{"Start":"03:49.495 ","End":"03:51.805","Text":"We have our Point number 1,"},{"Start":"03:51.805 ","End":"03:54.925","Text":"which is the point where we know our value for r_1,"},{"Start":"03:54.925 ","End":"03:58.183","Text":"its distance from the center of the Earth,"},{"Start":"03:58.183 ","End":"04:00.228","Text":"and this angle over here,"},{"Start":"04:00.228 ","End":"04:02.240","Text":"and our Point number 2,"},{"Start":"04:02.240 ","End":"04:05.545","Text":"which is over here when our rocket returns to Earth."},{"Start":"04:05.545 ","End":"04:08.770","Text":"Let\u0027s write equations for each of these points."},{"Start":"04:08.770 ","End":"04:14.411","Text":"Let\u0027s first start writing the equation for conservation of energy at our Point 0,"},{"Start":"04:14.411 ","End":"04:16.640","Text":"where rocket exits the Earth."},{"Start":"04:16.640 ","End":"04:22.435","Text":"I forgot to say that mass of the Earth is also given."},{"Start":"04:22.435 ","End":"04:26.570","Text":"Our energy when our rocket is leaving the Earth is equal to,"},{"Start":"04:26.570 ","End":"04:30.320","Text":"of course, 1/2 multiplied by the mass of the rocket,"},{"Start":"04:30.320 ","End":"04:37.775","Text":"multiplied by its exiting velocity minus our potential energy."},{"Start":"04:37.775 ","End":"04:43.470","Text":"Why minus? Because our potential energy is pulling us back down closer to Earth."},{"Start":"04:43.470 ","End":"04:52.370","Text":"Our potential energy is going to be GM_m divided by the radius of the Earth."},{"Start":"04:52.370 ","End":"04:55.325","Text":"Because when I start, I\u0027m a distance"},{"Start":"04:55.325 ","End":"04:58.955","Text":"from the center of the Earth equal to the radius of the Earth."},{"Start":"04:58.955 ","End":"05:01.650","Text":"I\u0027m starting right from here."},{"Start":"05:02.240 ","End":"05:06.865","Text":"This capital M over here is our mass of the Earth."},{"Start":"05:06.865 ","End":"05:12.390","Text":"Then our equation for our energy at Point number 1 over here."},{"Start":"05:12.390 ","End":"05:15.843","Text":"That\u0027s going to be 1/2 multiplied by the mass of the rocket"},{"Start":"05:15.843 ","End":"05:19.545","Text":"multiply by its velocity at that point squared."},{"Start":"05:19.545 ","End":"05:22.225","Text":"Then again minus the potential energy,"},{"Start":"05:22.225 ","End":"05:25.435","Text":"again minus because we\u0027re being pulled back to Earth."},{"Start":"05:25.435 ","End":"05:29.305","Text":"Our potential energy is going to be G multiplied by the mass of the Earth,"},{"Start":"05:29.305 ","End":"05:36.100","Text":"multiplied by the mass of the rocket and this time our radius is equal to our r_1."},{"Start":"05:36.230 ","End":"05:38.950","Text":"For our third point of interest,"},{"Start":"05:38.950 ","End":"05:44.005","Text":"which is at Point number 2 over here when the rocket is returning to Earth."},{"Start":"05:44.005 ","End":"05:47.330","Text":"Our energy is going to be 1/2 the mass of the rocket multiplied by"},{"Start":"05:47.330 ","End":"05:51.560","Text":"its velocity at that point squared minus its potential energy,"},{"Start":"05:51.560 ","End":"05:56.540","Text":"which is again going to be GM_E multiplied by m divided by,"},{"Start":"05:56.540 ","End":"06:01.175","Text":"again because it\u0027s returned to Earth, so RE."},{"Start":"06:01.175 ","End":"06:04.030","Text":"Now what I can do is,"},{"Start":"06:04.030 ","End":"06:05.840","Text":"because I have conservation of energy,"},{"Start":"06:05.840 ","End":"06:09.470","Text":"I can equate any 2 of these equations to each other."},{"Start":"06:09.470 ","End":"06:14.150","Text":"Easier on E_1 or with E_2 or any which combination that you want."},{"Start":"06:14.150 ","End":"06:18.170","Text":"Then we\u0027ll be left with 2 equations. Why not 3?"},{"Start":"06:18.170 ","End":"06:20.600","Text":"Because if I compare, let\u0027s say E_0 to E_1,"},{"Start":"06:20.600 ","End":"06:22.745","Text":"and then I compare that equation to my E_2,"},{"Start":"06:22.745 ","End":"06:26.900","Text":"I\u0027m going to get some linear multiplication or"},{"Start":"06:26.900 ","End":"06:33.554","Text":"some linear dependence on 1 of the other equations."},{"Start":"06:33.554 ","End":"06:37.010","Text":"I\u0027ll equate 2 of these equations 1 to another,"},{"Start":"06:37.010 ","End":"06:43.230","Text":"and I\u0027ll be left with 2 equations. Let\u0027s see what I can do."},{"Start":"06:43.230 ","End":"06:50.210","Text":"If I compare my E_0 to my E_2, why am I doing this?"},{"Start":"06:50.210 ","End":"06:51.440","Text":"Because I can see that both have"},{"Start":"06:51.440 ","End":"06:56.295","Text":"the exact same potential energy so it\u0027s going to be easy to calculate."},{"Start":"06:56.295 ","End":"06:57.900","Text":"Once I compare the 2,"},{"Start":"06:57.900 ","End":"07:06.333","Text":"I\u0027ll see that my velocity exiting is going to be equal to my velocity upon my return."},{"Start":"07:06.333 ","End":"07:09.675","Text":"Now I\u0027m left with 2 equations."},{"Start":"07:09.675 ","End":"07:14.250","Text":"My first equation being my E_1 over here,"},{"Start":"07:14.250 ","End":"07:19.185","Text":"and my second equation being that my V_0 is equal to V_2."},{"Start":"07:19.185 ","End":"07:23.115","Text":"Now let\u0027s try and work out our angular momentum."},{"Start":"07:23.115 ","End":"07:28.740","Text":"My angular momentum at Point 0, so my L_0."},{"Start":"07:28.740 ","End":"07:34.724","Text":"That\u0027s of course going to be equal to the mass of the rocket multiplied by its velocity."},{"Start":"07:34.724 ","End":"07:37.410","Text":"At this point is going to be V_0 multiplied"},{"Start":"07:37.410 ","End":"07:41.415","Text":"by its distance from the center of the Earth, which is R_E."},{"Start":"07:41.415 ","End":"07:50.535","Text":"Then multiplied by sine of the angle between my radius and my velocity."},{"Start":"07:50.535 ","End":"07:54.100","Text":"As we know, that\u0027s going to be my Theta_0."},{"Start":"07:55.250 ","End":"08:01.650","Text":"I\u0027m multiplying all of this by sine of the angle between my velocity and my radius."},{"Start":"08:01.650 ","End":"08:03.540","Text":"That\u0027s from my Point 0,"},{"Start":"08:03.540 ","End":"08:07.810","Text":"let\u0027s look at my angular momentum for Point number 1."},{"Start":"08:08.300 ","End":"08:11.700","Text":"My L_1 is going to be the mass of"},{"Start":"08:11.700 ","End":"08:15.225","Text":"the rocket multiplied by its velocity at the point which is V_1,"},{"Start":"08:15.225 ","End":"08:18.195","Text":"multiplied by its distance from the center of the Earth,"},{"Start":"08:18.195 ","End":"08:20.040","Text":"which is my r_1,"},{"Start":"08:20.040 ","End":"08:24.360","Text":"and then multiplied by Sine of the angle between my radius and my velocity,"},{"Start":"08:24.360 ","End":"08:29.520","Text":"which over here is 30 degrees, was given."},{"Start":"08:29.520 ","End":"08:34.530","Text":"Then my angular momentum and my last point when the rocket is returning."},{"Start":"08:34.530 ","End":"08:38.730","Text":"I have the mass of the rocket multiplied by its velocity,"},{"Start":"08:38.730 ","End":"08:41.925","Text":"multiplied by its distance from the center of the Earth,"},{"Start":"08:41.925 ","End":"08:44.550","Text":"which is R_E, and then multiplied by"},{"Start":"08:44.550 ","End":"08:47.595","Text":"Sine of the angle between my radius and my velocity,"},{"Start":"08:47.595 ","End":"08:50.895","Text":"which we said was given, it\u0027s Theta_2."},{"Start":"08:50.895 ","End":"08:55.540","Text":"We know this, so that\u0027s Theta_2."},{"Start":"08:56.570 ","End":"09:00.045","Text":"Again, we have conservation of angular momentum."},{"Start":"09:00.045 ","End":"09:05.040","Text":"That means that I can equate any one of these 3 equations"},{"Start":"09:05.040 ","End":"09:10.020","Text":"to one of the other equations and I will be able to cancel out some terms,"},{"Start":"09:10.020 ","End":"09:14.100","Text":"and find out the relation between 2 of the unknowns."},{"Start":"09:14.100 ","End":"09:17.700","Text":"Just like with the energy I\u0027m going to be left from all of"},{"Start":"09:17.700 ","End":"09:22.275","Text":"these 3 equations with just 2 equations which are useful to me."},{"Start":"09:22.275 ","End":"09:24.810","Text":"Because again, my third equation,"},{"Start":"09:24.810 ","End":"09:29.174","Text":"we\u0027ll just have some linear dependence on 1 of the other equations."},{"Start":"09:29.174 ","End":"09:33.690","Text":"Let\u0027s say that I\u0027m going to compare my L_0 with my L_2."},{"Start":"09:33.690 ","End":"09:35.325","Text":"Because I already know that my V_0,"},{"Start":"09:35.325 ","End":"09:37.315","Text":"and V_2 are equal."},{"Start":"09:37.315 ","End":"09:41.115","Text":"This when I\u0027m comparing to my L_2,"},{"Start":"09:41.115 ","End":"09:50.480","Text":"so I\u0027ll have my MV_2 multiplied it by my R_E multiplied by my sine of Theta_2."},{"Start":"09:50.480 ","End":"09:52.730","Text":"Then my m on both sides cancel out."},{"Start":"09:52.730 ","End":"09:55.265","Text":"My V_0 is equal to my V_2,"},{"Start":"09:55.265 ","End":"09:57.290","Text":"as we found out. They can cancel out."},{"Start":"09:57.290 ","End":"09:59.420","Text":"My R_E is equal to my R_E,"},{"Start":"09:59.420 ","End":"10:05.045","Text":"and then I\u0027m left with my sine Theta_0 is equal to my sine Theta_2."},{"Start":"10:05.045 ","End":"10:13.500","Text":"I\u0027m then left with that my Theta_0 is equal to Theta_2,"},{"Start":"10:13.500 ","End":"10:17.145","Text":"and my Theta_2 is of course, not an unknown."},{"Start":"10:17.145 ","End":"10:18.510","Text":"I\u0027m given this value,"},{"Start":"10:18.510 ","End":"10:22.240","Text":"so now I also know what my Theta_0 is equal to."},{"Start":"10:22.820 ","End":"10:26.025","Text":"Now I have 2 more equations,"},{"Start":"10:26.025 ","End":"10:29.025","Text":"so I have this equation,"},{"Start":"10:29.025 ","End":"10:32.325","Text":"this is my equation 3 and this is my equation 4,"},{"Start":"10:32.325 ","End":"10:35.880","Text":"so I had 4 unknowns and I have 4 equations,"},{"Start":"10:35.880 ","End":"10:37.845","Text":"which means that I can solve each of them,"},{"Start":"10:37.845 ","End":"10:39.945","Text":"1 has already been solved."},{"Start":"10:39.945 ","End":"10:43.245","Text":"I\u0027m not going to do all of the algebra over here."},{"Start":"10:43.245 ","End":"10:46.080","Text":"All you have to do is substitute each one of these equations"},{"Start":"10:46.080 ","End":"10:49.185","Text":"into the other in order to find out your unknowns."},{"Start":"10:49.185 ","End":"10:54.840","Text":"We\u0027ve already found that our Theta_0 is equal to our Theta_2,"},{"Start":"10:54.840 ","End":"10:58.920","Text":"which we know, and now we just have to do this for the rest of them."},{"Start":"10:58.920 ","End":"11:00.540","Text":"You can do this on your own,"},{"Start":"11:00.540 ","End":"11:01.815","Text":"I\u0027m not going to waste some time."},{"Start":"11:01.815 ","End":"11:04.410","Text":"Let\u0027s go on to Question number 2."},{"Start":"11:04.410 ","End":"11:09.060","Text":"In Question number 2, I\u0027m being asked to find what my r_max is"},{"Start":"11:09.060 ","End":"11:13.095","Text":"equal to and what my V_min is equal to. What does that mean?"},{"Start":"11:13.095 ","End":"11:15.555","Text":"I\u0027m trying to find my greatest radius,"},{"Start":"11:15.555 ","End":"11:18.266","Text":"when the rocket is furthest away from the Earth,"},{"Start":"11:18.266 ","End":"11:21.885","Text":"and the corresponding velocity at that point."},{"Start":"11:21.885 ","End":"11:24.870","Text":"If you remember back to one of the previous lessons,"},{"Start":"11:24.870 ","End":"11:32.867","Text":"we saw that at a maximum distance from my Earth or from my center of rotation,"},{"Start":"11:32.867 ","End":"11:38.070","Text":"so the velocity at that point is going to be a minimum velocity."},{"Start":"11:38.070 ","End":"11:43.400","Text":"Let\u0027s say that over here is our point,"},{"Start":"11:43.400 ","End":"11:50.630","Text":"and this distance over here is going to be our r_max."},{"Start":"11:50.630 ","End":"11:53.120","Text":"Then we know that our velocity over here,"},{"Start":"11:53.120 ","End":"11:56.490","Text":"as we said, is going to be V_min."},{"Start":"11:56.490 ","End":"12:00.140","Text":"We also know that the angle between our r_max and"},{"Start":"12:00.140 ","End":"12:04.770","Text":"our V_min is always going to be 90 degrees."},{"Start":"12:04.820 ","End":"12:08.550","Text":"Now what we\u0027re going to do is we\u0027re going to write out"},{"Start":"12:08.550 ","End":"12:12.420","Text":"our equation for our angular momentum at this point,"},{"Start":"12:12.420 ","End":"12:19.030","Text":"at a point where our rocket is at a distance r_max from the center of the Earth."},{"Start":"12:19.310 ","End":"12:25.335","Text":"We\u0027ll have that our angular momentum at our position r_max"},{"Start":"12:25.335 ","End":"12:30.960","Text":"is going to be the mass of the rocket multiplied by our velocity at that point,"},{"Start":"12:30.960 ","End":"12:32.910","Text":"which we know is V_min,"},{"Start":"12:32.910 ","End":"12:35.730","Text":"multiplied by our radius,"},{"Start":"12:35.730 ","End":"12:37.590","Text":"which we know is r_max,"},{"Start":"12:37.590 ","End":"12:43.560","Text":"and then multiplied by sine of the angle between our r_max and our V_min,"},{"Start":"12:43.560 ","End":"12:45.315","Text":"which is 90 degrees."},{"Start":"12:45.315 ","End":"12:50.490","Text":"Sine of 90 is just 1."},{"Start":"12:50.490 ","End":"12:53.250","Text":"Because we have conservation of angular momentum,"},{"Start":"12:53.250 ","End":"12:58.845","Text":"if I equate my L of my r_max to one of my angular momentums over here,"},{"Start":"12:58.845 ","End":"13:02.400","Text":"I\u0027ll be left with 1 equation,"},{"Start":"13:02.400 ","End":"13:05.770","Text":"and I\u0027m still going to have 2 unknowns."},{"Start":"13:05.870 ","End":"13:10.540","Text":"My V_min is unknown and my r_max is unknown."},{"Start":"13:10.540 ","End":"13:12.510","Text":"That will be my 1 equation,"},{"Start":"13:12.510 ","End":"13:14.520","Text":"and then I have to find my second equation,"},{"Start":"13:14.520 ","End":"13:18.765","Text":"which is of course going to come from the idea of conservation of energy."},{"Start":"13:18.765 ","End":"13:28.125","Text":"My energy at my point r_max is simply going to be equal to 1/2 of its kinetic energy."},{"Start":"13:28.125 ","End":"13:33.300","Text":"It\u0027s 1/2 the mass of the rocket multiplied by V_min^2."},{"Start":"13:33.300 ","End":"13:35.880","Text":"Then plus the potential energy,"},{"Start":"13:35.880 ","End":"13:37.350","Text":"which is again a negative."},{"Start":"13:37.350 ","End":"13:39.780","Text":"G multiplied by the mass of the Earth,"},{"Start":"13:39.780 ","End":"13:42.555","Text":"multiplied by the mass of the rocket,"},{"Start":"13:42.555 ","End":"13:45.165","Text":"divided by its distance from the center,"},{"Start":"13:45.165 ","End":"13:48.105","Text":"which is a distance of r_max."},{"Start":"13:48.105 ","End":"13:54.359","Text":"Then all I have to do is I can say that this is 1 equation, 2 equations."},{"Start":"13:54.359 ","End":"13:56.864","Text":"I have 2 equations and 2 unknowns."},{"Start":"13:56.864 ","End":"14:00.615","Text":"I\u0027ll equate my energy to 1 of these energy equations,"},{"Start":"14:00.615 ","End":"14:04.710","Text":"and my angular momentum to 1 of these angular momentum equations."},{"Start":"14:04.710 ","End":"14:08.145","Text":"Then I can isolate out and I can find out what my V_min,"},{"Start":"14:08.145 ","End":"14:09.765","Text":"and my r_max is."},{"Start":"14:09.765 ","End":"14:12.260","Text":"Again, I\u0027m not going to waste time in doing the algebra,"},{"Start":"14:12.260 ","End":"14:14.480","Text":"you can do this alone on a piece of paper."},{"Start":"14:14.480 ","End":"14:18.330","Text":"That is the end of this lesson."}],"ID":9458},{"Watched":false,"Name":"Escape Velocity","Duration":"8m 27s","ChapterTopicVideoID":9189,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.560","Text":"Hello. In this video,"},{"Start":"00:01.560 ","End":"00:05.270","Text":"we\u0027re going to be speaking about what is escape velocity."},{"Start":"00:05.270 ","End":"00:07.425","Text":"Let\u0027s take a look over here."},{"Start":"00:07.425 ","End":"00:09.480","Text":"Over here we have Earth and we have"},{"Start":"00:09.480 ","End":"00:13.980","Text":"some rocket which wants to escape from Earth\u0027s atmosphere."},{"Start":"00:13.980 ","End":"00:15.675","Text":"Now, this doesn\u0027t have to be Earth,"},{"Start":"00:15.675 ","End":"00:17.250","Text":"this can be any large body,"},{"Start":"00:17.250 ","End":"00:19.290","Text":"the Sun, certain miles, whatever,"},{"Start":"00:19.290 ","End":"00:25.570","Text":"and, of course also doesn\u0027t necessarily have to be a rocket for this example."},{"Start":"00:25.730 ","End":"00:33.990","Text":"Our question is, how much the minimum amount,"},{"Start":"00:33.990 ","End":"00:37.610","Text":"a velocity does this rocket have to have such that it will"},{"Start":"00:37.610 ","End":"00:43.175","Text":"fly in out away from the Earth and it will never return?"},{"Start":"00:43.175 ","End":"00:47.880","Text":"We\u0027re trying to find what V_min is."},{"Start":"00:48.170 ","End":"00:51.275","Text":"This V_min is our escape velocity,"},{"Start":"00:51.275 ","End":"00:52.670","Text":"and we want to find it."},{"Start":"00:52.670 ","End":"00:57.815","Text":"How can we do this? We\u0027re going to use the idea of conservation of energy."},{"Start":"00:57.815 ","End":"01:00.095","Text":"Let\u0027s take a look at how we do this."},{"Start":"01:00.095 ","End":"01:06.260","Text":"We can say that our energy right at the beginning as the rocket is leaving Earth,"},{"Start":"01:06.260 ","End":"01:11.705","Text":"our initial energy is going to heat 1/2MV^2."},{"Start":"01:11.705 ","End":"01:13.550","Text":"This is going to be our V_min,"},{"Start":"01:13.550 ","End":"01:16.810","Text":"but in the meantime, let\u0027s call it I V_initial."},{"Start":"01:16.810 ","End":"01:21.510","Text":"Our kinetic energy minus our potential energy."},{"Start":"01:21.510 ","End":"01:24.225","Text":"It\u0027s going to be GM,"},{"Start":"01:24.225 ","End":"01:26.465","Text":"and then to make it more general,"},{"Start":"01:26.465 ","End":"01:30.005","Text":"not just for Earth but for any body or star,"},{"Start":"01:30.005 ","End":"01:32.720","Text":"let\u0027s call it b. M_b,"},{"Start":"01:32.720 ","End":"01:37.205","Text":"the mass of the star multiplied by the mass of the rocket or whatever it is,"},{"Start":"01:37.205 ","End":"01:43.440","Text":"divided by the radius of the body."},{"Start":"01:44.690 ","End":"01:48.890","Text":"Here specifically it will be the radius of the Earth."},{"Start":"01:48.890 ","End":"01:50.330","Text":"But, of course, if it\u0027s the Sun,"},{"Start":"01:50.330 ","End":"01:53.335","Text":"the radius will be different. That\u0027s the body."},{"Start":"01:53.335 ","End":"01:55.080","Text":"Of course, just to remind,"},{"Start":"01:55.080 ","End":"02:00.065","Text":"our equation for energy is going to be the kinetic energy plus our potential energy."},{"Start":"02:00.065 ","End":"02:04.895","Text":"But here, our potential energy is pointing in the opposite direction to"},{"Start":"02:04.895 ","End":"02:10.335","Text":"our kinetic energy because our V is going out in this direction,"},{"Start":"02:10.335 ","End":"02:13.640","Text":"and the potential energy is what we\u0027re"},{"Start":"02:13.640 ","End":"02:17.360","Text":"experiencing when we\u0027re trying to return back to Earth."},{"Start":"02:17.360 ","End":"02:20.315","Text":"That\u0027s where this minus comes from."},{"Start":"02:20.315 ","End":"02:22.190","Text":"That\u0027s our initial energy."},{"Start":"02:22.190 ","End":"02:27.195","Text":"Then our final energy is going to be equal to."},{"Start":"02:27.195 ","End":"02:28.835","Text":"Again, we have our kinetic energy,"},{"Start":"02:28.835 ","End":"02:36.590","Text":"it\u0027s going to be 1\\2m V_final^2, again,"},{"Start":"02:36.590 ","End":"02:37.955","Text":"plus our potential energy,"},{"Start":"02:37.955 ","End":"02:41.840","Text":"minus G the mass of the body,"},{"Start":"02:41.840 ","End":"02:47.130","Text":"the mass of the rocket divided by our radius."},{"Start":"02:47.240 ","End":"02:51.680","Text":"If our rocket is now located here somewhere,"},{"Start":"02:51.680 ","End":"02:53.915","Text":"this is a very long distance away,"},{"Start":"02:53.915 ","End":"02:57.725","Text":"our radius is going to be something like infinity."},{"Start":"02:57.725 ","End":"02:59.120","Text":"It\u0027s going to be very far,"},{"Start":"02:59.120 ","End":"03:06.005","Text":"and then that means that this whole term is going to be approximately equal to 0."},{"Start":"03:06.005 ","End":"03:09.455","Text":"Now going with the idea of conservation of energy,"},{"Start":"03:09.455 ","End":"03:13.820","Text":"I can say that my E_i is equals to my E_final."},{"Start":"03:13.820 ","End":"03:19.350","Text":"We can see that I can cross out everywhere my mass of the rocket,"},{"Start":"03:19.350 ","End":"03:20.730","Text":"and now let\u0027s do this."},{"Start":"03:20.730 ","End":"03:26.360","Text":"I\u0027ll get that my 1/2 V_i^2 minus G,"},{"Start":"03:26.360 ","End":"03:28.385","Text":"the mass of the large body,"},{"Start":"03:28.385 ","End":"03:37.015","Text":"divided by the radius of the large body is equal to 1/2 of my final velocity squared."},{"Start":"03:37.015 ","End":"03:40.880","Text":"Now all we have to do is we have to isolate out our V_i."},{"Start":"03:40.880 ","End":"03:42.740","Text":"I\u0027ll get that our V_i,"},{"Start":"03:42.740 ","End":"03:44.285","Text":"which is what we\u0027re trying to find,"},{"Start":"03:44.285 ","End":"03:50.750","Text":"is equal to the square root of 2G multiplied by the mass of the large body,"},{"Start":"03:50.750 ","End":"03:59.130","Text":"divided by the radius of the large body plus our V_final^2."},{"Start":"04:00.740 ","End":"04:06.900","Text":"That means that I need to have at least this velocity over here."},{"Start":"04:06.900 ","End":"04:12.370","Text":"Now I want to find the minimum value that this V_i can be."},{"Start":"04:12.370 ","End":"04:16.255","Text":"When is this V_i value at its minimum,"},{"Start":"04:16.255 ","End":"04:21.900","Text":"and that\u0027s how we\u0027re going to find our V_min. How do I find it?"},{"Start":"04:21.900 ","End":"04:26.850","Text":"What I have over here, let\u0027s highlight this."},{"Start":"04:26.850 ","End":"04:30.750","Text":"This expression over here,"},{"Start":"04:30.750 ","End":"04:32.870","Text":"this is all constant, 2,"},{"Start":"04:32.870 ","End":"04:34.075","Text":"my G is a constant,"},{"Start":"04:34.075 ","End":"04:35.470","Text":"my mass of the body is constant,"},{"Start":"04:35.470 ","End":"04:37.195","Text":"and so is its radius."},{"Start":"04:37.195 ","End":"04:42.655","Text":"That means that the only thing that I can play around with is my V_final."},{"Start":"04:42.655 ","End":"04:48.455","Text":"In order to have my V_i at its most minimal value,"},{"Start":"04:48.455 ","End":"04:53.875","Text":"that means that I have to have my V_f at the smallest value that it can possibly be."},{"Start":"04:53.875 ","End":"04:58.045","Text":"That\u0027s the velocity of the rocket when it\u0027s very far away."},{"Start":"04:58.045 ","End":"05:01.595","Text":"What value can my V_f have?"},{"Start":"05:01.595 ","End":"05:05.075","Text":"The lowest value that it can have is 0."},{"Start":"05:05.075 ","End":"05:08.435","Text":"If I say that my V_final=0,"},{"Start":"05:08.435 ","End":"05:12.740","Text":"then my V_i will be the lowest possible value it can be,"},{"Start":"05:12.740 ","End":"05:17.285","Text":"which is the square root of this over here in yellow."},{"Start":"05:17.285 ","End":"05:20.560","Text":"What does that mean if my V_final=0?"},{"Start":"05:20.560 ","End":"05:26.190","Text":"That means that as my rocket shoots out of Earth into space,"},{"Start":"05:26.190 ","End":"05:30.830","Text":"to some very large distance away from my Earth here,"},{"Start":"05:30.830 ","End":"05:33.739","Text":"for example, at a distance of infinity,"},{"Start":"05:33.739 ","End":"05:37.640","Text":"so it reaches infinity and then stops."},{"Start":"05:37.640 ","End":"05:43.730","Text":"At infinity when it reaches a very far distance away from my body or my sky,"},{"Start":"05:43.730 ","End":"05:47.360","Text":"it comes to a standstill."},{"Start":"05:47.360 ","End":"05:56.750","Text":"In that case, I can write that my V_min is going to be equal to the square root of 2G,"},{"Start":"05:56.750 ","End":"06:02.675","Text":"the mass of the large body divided by the radius of the large body,"},{"Start":"06:02.675 ","End":"06:05.430","Text":"and square root of all of that."},{"Start":"06:05.510 ","End":"06:11.900","Text":"Any velocity equal to this V_min or above is going to be my escape velocity,"},{"Start":"06:11.900 ","End":"06:15.530","Text":"and that\u0027s the minimum velocity or the velocity that I need"},{"Start":"06:15.530 ","End":"06:19.820","Text":"in order to escape Earth\u0027s atmosphere."},{"Start":"06:19.820 ","End":"06:23.345","Text":"Now, here in this example, specifically,"},{"Start":"06:23.345 ","End":"06:30.170","Text":"we were speaking about my initial starting position leaving Earth,"},{"Start":"06:30.170 ","End":"06:34.355","Text":"but we can also take a rocket at this position"},{"Start":"06:34.355 ","End":"06:39.260","Text":"somewhere in the Earth\u0027s atmosphere or wherever it is,"},{"Start":"06:39.260 ","End":"06:41.870","Text":"anywhere, and then we\u0027ll just work out"},{"Start":"06:41.870 ","End":"06:46.375","Text":"our energy equation according to this starting spot and not on Earth."},{"Start":"06:46.375 ","End":"06:50.360","Text":"We can find the same equation for something over here, and again,"},{"Start":"06:50.360 ","End":"06:53.360","Text":"reminding you that it\u0027s not just when dealing with Earth and a rocket,"},{"Start":"06:53.360 ","End":"06:57.965","Text":"but any planet or star and anything leaving it."},{"Start":"06:57.965 ","End":"07:01.700","Text":"Another interesting thing to take a look at,"},{"Start":"07:01.700 ","End":"07:03.650","Text":"so we can see that my V_min,"},{"Start":"07:03.650 ","End":"07:10.970","Text":"my escape velocity is not dependent on the direction that my rocket is in."},{"Start":"07:10.970 ","End":"07:14.075","Text":"Here, specifically, we had that my rocket is shooting"},{"Start":"07:14.075 ","End":"07:16.700","Text":"up in the radial direction out of Earth."},{"Start":"07:16.700 ","End":"07:19.430","Text":"But I could also have that my rocket is on Earth and it\u0027s"},{"Start":"07:19.430 ","End":"07:23.670","Text":"shooting straight like this or any which way,"},{"Start":"07:23.670 ","End":"07:25.275","Text":"any other direction,"},{"Start":"07:25.275 ","End":"07:29.235","Text":"and we\u0027ll see that my V_min still has to equal the same thing."},{"Start":"07:29.235 ","End":"07:31.755","Text":"It\u0027s not dependent on the direction,"},{"Start":"07:31.755 ","End":"07:35.825","Text":"it\u0027s just dependent on the size of the velocity."},{"Start":"07:35.825 ","End":"07:37.670","Text":"If I would shoot over here,"},{"Start":"07:37.670 ","End":"07:42.410","Text":"the only difference would be that when we\u0027re using our conservation of energy equations,"},{"Start":"07:42.410 ","End":"07:46.080","Text":"my V_final instead of being over here,"},{"Start":"07:46.080 ","End":"07:49.365","Text":"my V_final might be over here or over here,"},{"Start":"07:49.365 ","End":"07:54.950","Text":"if my rocket does some hyperbolic trajectory and ends up over here."},{"Start":"07:54.950 ","End":"07:58.940","Text":"The trick to remember is just that it reaches infinity,"},{"Start":"07:58.940 ","End":"08:01.925","Text":"and there it\u0027s velocity is approximately equal to 0,"},{"Start":"08:01.925 ","End":"08:05.280","Text":"and then we can calculate these equations."},{"Start":"08:05.420 ","End":"08:08.225","Text":"This is our escape velocity."},{"Start":"08:08.225 ","End":"08:10.750","Text":"It\u0027s only some size,"},{"Start":"08:10.750 ","End":"08:13.245","Text":"it\u0027s independent of direction,"},{"Start":"08:13.245 ","End":"08:16.850","Text":"and the only thing that will change if you change the direction with which it\u0027s in,"},{"Start":"08:16.850 ","End":"08:18.560","Text":"it\u0027s that in infinity,"},{"Start":"08:18.560 ","End":"08:20.615","Text":"your rocket or whatever it might be,"},{"Start":"08:20.615 ","End":"08:26.055","Text":"will end up at a different part of space and not necessarily over here."},{"Start":"08:26.055 ","End":"08:28.810","Text":"That\u0027s the end of this lesson."}],"ID":9459},{"Watched":false,"Name":"Part Of Rocket At Escape Velocity","Duration":"19m 18s","ChapterTopicVideoID":9190,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.235","Text":"Hello. In this question over here,"},{"Start":"00:02.235 ","End":"00:03.630","Text":"we have our Earth,"},{"Start":"00:03.630 ","End":"00:08.310","Text":"and we have a rocket which is orbiting Earth in a perfect circle."},{"Start":"00:08.310 ","End":"00:12.075","Text":"The radius of this orbit is given to us."},{"Start":"00:12.075 ","End":"00:15.420","Text":"It\u0027s this over here, and it\u0027s a radius R."},{"Start":"00:15.420 ","End":"00:19.410","Text":"We`re being told that at some point in this rocket\u0027s motion."},{"Start":"00:19.410 ","End":"00:21.360","Text":"At some point in the orbit,"},{"Start":"00:21.360 ","End":"00:25.040","Text":"the rocket splits into 2 parts."},{"Start":"00:25.040 ","End":"00:30.225","Text":"So 1 part goes in this direction."},{"Start":"00:30.225 ","End":"00:37.820","Text":"It\u0027s going in the radial direction out from Earth with a velocity of V escape."},{"Start":"00:37.820 ","End":"00:46.625","Text":"The other part is flying in some direction here with a velocity of V_2."},{"Start":"00:46.625 ","End":"00:50.420","Text":"We\u0027re also being told that our piece of the rocket,"},{"Start":"00:50.420 ","End":"00:52.295","Text":"which is in the radial direction."},{"Start":"00:52.295 ","End":"00:56.320","Text":"I\u0027ll write it over here, has a mass of 1/3m."},{"Start":"00:56.320 ","End":"01:01.250","Text":"When our whole rocket is of mass m. That means that the mass of the rocket with"},{"Start":"01:01.250 ","End":"01:06.690","Text":"a velocity V_2 has a mass of 2/3."},{"Start":"01:07.480 ","End":"01:11.645","Text":"Our first question, question number 1,"},{"Start":"01:11.645 ","End":"01:14.650","Text":"is to find our minimum radius,"},{"Start":"01:14.650 ","End":"01:18.490","Text":"and our maximum radius of the orbit of the piece which has left."},{"Start":"01:18.490 ","End":"01:21.100","Text":"If this piece over here,"},{"Start":"01:21.100 ","End":"01:24.155","Text":"which is traveling at a velocity of V_2."},{"Start":"01:24.155 ","End":"01:26.760","Text":"Let\u0027s see how we find our r_min,"},{"Start":"01:26.760 ","End":"01:31.420","Text":"and our r_max of this piece over here, which has left."},{"Start":"01:31.420 ","End":"01:33.486","Text":"When a rocket splits into 2,"},{"Start":"01:33.486 ","End":"01:36.700","Text":"which happens at approximately this point over here."},{"Start":"01:36.700 ","End":"01:40.565","Text":"We can use the idea of conservation of momentum."},{"Start":"01:40.565 ","End":"01:44.560","Text":"What I have to do is I have to draw in my axis"},{"Start":"01:44.560 ","End":"01:48.685","Text":"just like I would in a normal conservation of momentum question."},{"Start":"01:48.685 ","End":"01:52.225","Text":"Over here I can have my radial axis."},{"Start":"01:52.225 ","End":"01:55.950","Text":"Over here I have my tangential axis,"},{"Start":"01:55.950 ","End":"01:59.850","Text":"and it\u0027s meant to be touching this orbit."},{"Start":"02:00.050 ","End":"02:06.435","Text":"I can call my axis that this is my r-axis, my radial direction."},{"Start":"02:06.435 ","End":"02:09.555","Text":"This is the tangential axis."},{"Start":"02:09.555 ","End":"02:11.270","Text":"The tangential to the circle,"},{"Start":"02:11.270 ","End":"02:13.835","Text":"or I can just name this my y-axis,"},{"Start":"02:13.835 ","End":"02:16.795","Text":"and this my x-axis. It makes no difference."},{"Start":"02:16.795 ","End":"02:22.310","Text":"We have a rocket before it reaches this point where our rocket splits."},{"Start":"02:22.310 ","End":"02:25.700","Text":"Let\u0027s draw it over here."},{"Start":"02:25.700 ","End":"02:30.660","Text":"It has some velocity in this direction."},{"Start":"02:30.660 ","End":"02:35.025","Text":"Our V_0, and of course our V_0, we don\u0027t know."},{"Start":"02:35.025 ","End":"02:39.530","Text":"Now let\u0027s write out our equation for conservation of momentum."},{"Start":"02:39.530 ","End":"02:44.090","Text":"Let\u0027s write our momentum in the X-direction."},{"Start":"02:44.090 ","End":"02:47.750","Text":"That\u0027s going to be equal to initially the mass of"},{"Start":"02:47.750 ","End":"02:53.645","Text":"the rocket m multiplied by its initial velocity V_0, which we don\u0027t know."},{"Start":"02:53.645 ","End":"02:57.984","Text":"That\u0027s going to be equal to the momentum after the explosion."},{"Start":"02:57.984 ","End":"03:00.492","Text":"Now, in the momentum after the explosion,"},{"Start":"03:00.492 ","End":"03:04.400","Text":"we can see that only this section of the rocket with"},{"Start":"03:04.400 ","End":"03:09.230","Text":"a velocity of V_2 remains with some x-component."},{"Start":"03:09.230 ","End":"03:12.080","Text":"As we can see, a piece of rocket with"},{"Start":"03:12.080 ","End":"03:17.630","Text":"the escape velocity is moving in the radial direction or along the y-axis,"},{"Start":"03:17.630 ","End":"03:21.135","Text":"which means that its x-component is equal to 0."},{"Start":"03:21.135 ","End":"03:25.355","Text":"Then we have to write the mass of this section of the rocket only."},{"Start":"03:25.355 ","End":"03:31.510","Text":"As we know, 2/3 M multiplied by its velocity, which is V_2."},{"Start":"03:31.510 ","End":"03:33.020","Text":"Of course this V_2,"},{"Start":"03:33.020 ","End":"03:36.665","Text":"we\u0027re referring to only the x-component of our V_2."},{"Start":"03:36.665 ","End":"03:39.265","Text":"We\u0027re only dealing with the x-axis right now."},{"Start":"03:39.265 ","End":"03:43.590","Text":"Of course I\u0027m just reminding that before we have the explosion."},{"Start":"03:43.590 ","End":"03:47.195","Text":"We can see that our rocket is very close to this point over here,"},{"Start":"03:47.195 ","End":"03:48.960","Text":"where the explosion happens."},{"Start":"03:48.960 ","End":"03:51.480","Text":"Where the rocket splits into 2."},{"Start":"03:51.480 ","End":"03:56.070","Text":"We can know that my V_0 over here is only in the x-direction."},{"Start":"03:56.070 ","End":"03:58.125","Text":"It only has an x-component."},{"Start":"03:58.125 ","End":"04:02.635","Text":"Now the momentum on the y-axis."},{"Start":"04:02.635 ","End":"04:06.080","Text":"Because we just said that my velocity over here just before"},{"Start":"04:06.080 ","End":"04:09.320","Text":"the explosion is only on this axis,"},{"Start":"04:09.320 ","End":"04:12.245","Text":"the tangential axis, the x-axis."},{"Start":"04:12.245 ","End":"04:18.740","Text":"My mass times velocity in the y-direction here will be equal to 0."},{"Start":"04:18.740 ","End":"04:22.300","Text":"That\u0027s going to be equal to the momentum after."},{"Start":"04:22.300 ","End":"04:24.645","Text":"That\u0027s going to be equal to,"},{"Start":"04:24.645 ","End":"04:27.960","Text":"let\u0027s do the y-component of the momentum of this."},{"Start":"04:27.960 ","End":"04:30.070","Text":"We know it\u0027s 1/3m,"},{"Start":"04:30.070 ","End":"04:33.320","Text":"and its velocity is equal to V escape."},{"Start":"04:33.320 ","End":"04:35.425","Text":"Soon we\u0027ll see what our V escape is."},{"Start":"04:35.425 ","End":"04:42.050","Text":"Then we can add in y-component of the momentum on this piece of the rocket."},{"Start":"04:42.050 ","End":"04:44.060","Text":"I can do plus or minus."},{"Start":"04:44.060 ","End":"04:45.620","Text":"It doesn\u0027t really matter."},{"Start":"04:45.620 ","End":"04:48.740","Text":"I\u0027ll do plus, and then I\u0027ll just get my answer in the end."},{"Start":"04:48.740 ","End":"04:50.800","Text":"For my V_2 as a negative number."},{"Start":"04:50.800 ","End":"04:53.390","Text":"It doesn\u0027t matter, it will balance out in the end."},{"Start":"04:53.390 ","End":"04:55.715","Text":"Plus 2/3 M,"},{"Start":"04:55.715 ","End":"04:58.025","Text":"which is the mass of this rocket."},{"Start":"04:58.025 ","End":"05:01.220","Text":"Then multiplied by the velocity of the rocket,"},{"Start":"05:01.220 ","End":"05:04.405","Text":"but its y-component this time."},{"Start":"05:04.405 ","End":"05:06.575","Text":"As far as V escape,"},{"Start":"05:06.575 ","End":"05:09.980","Text":"we discussed it in the last lesson."},{"Start":"05:09.980 ","End":"05:16.155","Text":"As we know, it\u0027s going to be the square root of 2G, which is a constant."},{"Start":"05:16.155 ","End":"05:20.990","Text":"Multiplied by the mass of the Earth because that\u0027s the planet that we\u0027re working with."},{"Start":"05:20.990 ","End":"05:26.570","Text":"All of this divided by the distance the rocket was from the center of the Earth."},{"Start":"05:26.570 ","End":"05:30.960","Text":"It\u0027s distance was this radius r over here."},{"Start":"05:31.310 ","End":"05:36.060","Text":"Now we\u0027re going to try, and find what our V_0 is equal to."},{"Start":"05:36.060 ","End":"05:39.800","Text":"We know that because the rocket is traveling in a circle,"},{"Start":"05:39.800 ","End":"05:41.750","Text":"it\u0027s orbiting Earth in a circle."},{"Start":"05:41.750 ","End":"05:45.890","Text":"We know that our central force F is going to be equal"},{"Start":"05:45.890 ","End":"05:50.090","Text":"to our G constant multiplied by the mass of the Earth."},{"Start":"05:50.090 ","End":"05:51.785","Text":"The planet that we\u0027re orbiting."},{"Start":"05:51.785 ","End":"05:54.095","Text":"Multiplied by the mass of the rocket,"},{"Start":"05:54.095 ","End":"05:59.169","Text":"divided by the distance that the rocket was from the center of the Earth squared."},{"Start":"05:59.169 ","End":"06:04.325","Text":"It\u0027s R^2 and then that is equal to the mass of the rocket"},{"Start":"06:04.325 ","End":"06:09.725","Text":"multiplied by its velocity squared divided by its radius."},{"Start":"06:09.725 ","End":"06:12.420","Text":"Its distance from the center of the Earth."},{"Start":"06:13.010 ","End":"06:19.385","Text":"What I\u0027m doing right now in this equation over here is I\u0027m comparing the force that"},{"Start":"06:19.385 ","End":"06:25.895","Text":"Earth is exerting on the rocket to my angular acceleration."},{"Start":"06:25.895 ","End":"06:30.275","Text":"Now this is a very useful equation which you can use a lot."},{"Start":"06:30.275 ","End":"06:33.050","Text":"I\u0027ll underline it in red."},{"Start":"06:33.050 ","End":"06:34.490","Text":"You can use this a lot,"},{"Start":"06:34.490 ","End":"06:39.370","Text":"and it\u0027s very easy to find these types of unknowns,"},{"Start":"06:39.370 ","End":"06:42.319","Text":"and then therefore to answer the whole question."},{"Start":"06:42.319 ","End":"06:48.860","Text":"Remember this. Now what I want to do is I want to find out what my V_0 is."},{"Start":"06:48.860 ","End":"06:51.905","Text":"All I have to do is I have to isolate this out."},{"Start":"06:51.905 ","End":"06:57.875","Text":"It\u0027s going to be the square root of G multiplied by the mass of the Earth"},{"Start":"06:57.875 ","End":"07:04.385","Text":"divided by R. Now we know what our V_0 is."},{"Start":"07:04.385 ","End":"07:06.545","Text":"We know what V escape is."},{"Start":"07:06.545 ","End":"07:09.620","Text":"It\u0027s over here, which means that it\u0027s going to"},{"Start":"07:09.620 ","End":"07:12.650","Text":"be very easy for us to find out what our V_2x is,"},{"Start":"07:12.650 ","End":"07:14.360","Text":"and what our V_2y is."},{"Start":"07:14.360 ","End":"07:17.510","Text":"Let\u0027s put text over here to show that we\u0027ve found them."},{"Start":"07:17.510 ","End":"07:20.255","Text":"We just have to isolate them out now."},{"Start":"07:20.255 ","End":"07:25.050","Text":"Now when we know what V_2 x-component is,"},{"Start":"07:25.050 ","End":"07:27.560","Text":"and what our V_2 y-component is."},{"Start":"07:27.560 ","End":"07:32.225","Text":"We can use these values in order to find our r_min,"},{"Start":"07:32.225 ","End":"07:38.370","Text":"and our r_max."},{"Start":"07:38.370 ","End":"07:39.915","Text":"How do we do this?"},{"Start":"07:39.915 ","End":"07:41.855","Text":"Let\u0027s take a look at this motion again."},{"Start":"07:41.855 ","End":"07:44.630","Text":"We\u0027re being told that our section of V_2."},{"Start":"07:44.630 ","End":"07:49.915","Text":"Let\u0027s call this piece number 2."},{"Start":"07:49.915 ","End":"07:56.050","Text":"We know that our piece number 2 is orbiting the Earth in some trajectory."},{"Start":"07:56.050 ","End":"07:58.820","Text":"Now because we\u0027re being told that we have an r_min, and r_max."},{"Start":"07:58.820 ","End":"08:03.440","Text":"We can assume that it\u0027s going to be an elliptical trajectory."},{"Start":"08:03.440 ","End":"08:05.090","Text":"It could also be circular."},{"Start":"08:05.090 ","End":"08:08.600","Text":"That\u0027s a private case where r_min is equal to r_max."},{"Start":"08:08.600 ","End":"08:11.300","Text":"But that\u0027s a private case of an ellipse."},{"Start":"08:11.300 ","End":"08:14.950","Text":"Either way we\u0027re working with some ellipse over here."},{"Start":"08:14.950 ","End":"08:18.200","Text":"When we\u0027re dealing with a trajectory which is"},{"Start":"08:18.200 ","End":"08:22.160","Text":"either a perfect circle or ellipse also hyperbola."},{"Start":"08:22.160 ","End":"08:26.480","Text":"We know that we\u0027re going to be dealing with some central force,"},{"Start":"08:26.480 ","End":"08:33.100","Text":"which means that we have conservation of angular momentum, and of energy."},{"Start":"08:33.100 ","End":"08:35.500","Text":"That\u0027s what I wrote over here."},{"Start":"08:35.500 ","End":"08:37.435","Text":"When we have an elliptical trajectory,"},{"Start":"08:37.435 ","End":"08:40.330","Text":"then we know that we have central force and therefore we have"},{"Start":"08:40.330 ","End":"08:43.945","Text":"conservation of energy and angular momentum."},{"Start":"08:43.945 ","End":"08:48.410","Text":"Let\u0027s start off with our equations for conservation of energy."},{"Start":"08:49.080 ","End":"08:58.000","Text":"Let\u0027s start off with our energy when we\u0027re located at our maximum radius, at r_max."},{"Start":"08:58.000 ","End":"09:01.510","Text":"That\u0027s going to be equal to kinetic energy at that point."},{"Start":"09:01.510 ","End":"09:03.985","Text":"So 1/2 of r_m,"},{"Start":"09:03.985 ","End":"09:11.740","Text":"so we know that our M is 2/3 M. The mass piece number 2 of the rocket."},{"Start":"09:11.740 ","End":"09:16.690","Text":"So this is our mass multiplied by its velocity."},{"Start":"09:16.690 ","End":"09:18.729","Text":"Now we know at our maximum radius,"},{"Start":"09:18.729 ","End":"09:25.120","Text":"we\u0027re going to have a minimum velocity so it\u0027s going to be equal to V_min^2,"},{"Start":"09:25.120 ","End":"09:28.195","Text":"and then we have to add our potential energy."},{"Start":"09:28.195 ","End":"09:31.300","Text":"Now because our potential energy is trying to go"},{"Start":"09:31.300 ","End":"09:34.570","Text":"in the opposite direction to our kinetic energy so it\u0027s negative."},{"Start":"09:34.570 ","End":"09:39.760","Text":"So negative in the potential energy is G multiplied by mass of the Earth,"},{"Start":"09:39.760 ","End":"09:44.374","Text":"multiplied by mass piece number 2 of the rockets,"},{"Start":"09:44.374 ","End":"09:47.515","Text":"so that\u0027s 2/3 of our original rocket mass,"},{"Start":"09:47.515 ","End":"09:52.110","Text":"multiplied by our r_max."},{"Start":"09:52.110 ","End":"09:57.915","Text":"Then we can work out our energy when we\u0027re located at our r_min."},{"Start":"09:57.915 ","End":"10:07.000","Text":"That\u0027s going to be 1/2 of our mass multiplied by the velocity at our minimum radius."},{"Start":"10:07.000 ","End":"10:10.060","Text":"As we know, the velocity at our minimum radius is"},{"Start":"10:10.060 ","End":"10:15.160","Text":"our maximum velocity squared and then again plus our potential energy,"},{"Start":"10:15.160 ","End":"10:18.430","Text":"which here are potential energy is in the negative direction."},{"Start":"10:18.430 ","End":"10:21.445","Text":"So G multiplied by mass of the Earth,"},{"Start":"10:21.445 ","End":"10:28.180","Text":"multiplied by the mass of this piece of rocket divided by its radius,"},{"Start":"10:28.180 ","End":"10:30.070","Text":"it\u0027s the distance from the center of the Earth,"},{"Start":"10:30.070 ","End":"10:33.865","Text":"which is going to be our r_min."},{"Start":"10:33.865 ","End":"10:37.615","Text":"Now what I want to do is I want to find the size of the energy."},{"Start":"10:37.615 ","End":"10:40.615","Text":"I need to equate them to something."},{"Start":"10:40.615 ","End":"10:46.720","Text":"What I can do is find the size of the energy a moment after the explosion,"},{"Start":"10:46.720 ","End":"10:49.705","Text":"a moment after the rocket splits."},{"Start":"10:49.705 ","End":"10:54.040","Text":"Of course, at the explosion, when the rocket splits,"},{"Start":"10:54.040 ","End":"10:56.530","Text":"I\u0027m going to have some kind of energy law,"},{"Start":"10:56.530 ","End":"10:58.854","Text":"but a moment after the explosion,"},{"Start":"10:58.854 ","End":"11:06.025","Text":"when I have this velocity over here,"},{"Start":"11:06.025 ","End":"11:08.350","Text":"I can work out this energy."},{"Start":"11:08.350 ","End":"11:13.075","Text":"Now what we want to do is we want to see what energy we have at that moment,"},{"Start":"11:13.075 ","End":"11:18.025","Text":"a moment after the rocket splits into 2. Let\u0027s call that E_0."},{"Start":"11:18.025 ","End":"11:22.825","Text":"So it\u0027s going to be equal to 1/2,"},{"Start":"11:22.825 ","End":"11:24.445","Text":"and then the mass."},{"Start":"11:24.445 ","End":"11:30.880","Text":"We know that that\u0027s 2/3 M multiplied by the velocity squared."},{"Start":"11:30.880 ","End":"11:34.045","Text":"What\u0027s our velocity squared at this moment?"},{"Start":"11:34.045 ","End":"11:41.980","Text":"We have our V_2 x-component squared plus our V_2 y-component squared."},{"Start":"11:41.980 ","End":"11:48.715","Text":"Remember through Pythagoras, that all of this is equal to V_2^2."},{"Start":"11:48.715 ","End":"11:51.880","Text":"Then of course we have to add in our potential energy,"},{"Start":"11:51.880 ","End":"11:55.795","Text":"so again, our potential energy is in the opposite direction, so it\u0027s negative."},{"Start":"11:55.795 ","End":"12:01.420","Text":"We\u0027re assuming that at the moment of explosion over here,"},{"Start":"12:01.420 ","End":"12:04.404","Text":"our velocity is going to change."},{"Start":"12:04.404 ","End":"12:08.860","Text":"However, its position isn\u0027t going to change that drastically,"},{"Start":"12:08.860 ","End":"12:10.975","Text":"at the moment of explosion."},{"Start":"12:10.975 ","End":"12:14.860","Text":"It hasn\u0027t had time to change position drastically."},{"Start":"12:14.860 ","End":"12:20.485","Text":"So its position is going to be G multiplied by the mass of the Earth,"},{"Start":"12:20.485 ","End":"12:23.140","Text":"multiplied by its mass."},{"Start":"12:23.140 ","End":"12:28.510","Text":"Then its position is going to be a distance r from the center of the Earth."},{"Start":"12:28.510 ","End":"12:33.985","Text":"So it\u0027s still going to be around about this point over here."},{"Start":"12:33.985 ","End":"12:38.350","Text":"As we can see, it\u0027s not much less than our R over here."},{"Start":"12:38.350 ","End":"12:42.055","Text":"Now we have these equations."},{"Start":"12:42.055 ","End":"12:44.095","Text":"Now when I equate one to the other,"},{"Start":"12:44.095 ","End":"12:47.080","Text":"we\u0027re going to be left with only 2 equations that we can"},{"Start":"12:47.080 ","End":"12:51.175","Text":"actually work with and we still have 4 unknowns."},{"Start":"12:51.175 ","End":"12:55.960","Text":"Our V_min and our r_max and our V_max, and our r_min."},{"Start":"12:55.960 ","End":"13:00.130","Text":"We have to come up with another 2 equations and that is"},{"Start":"13:00.130 ","End":"13:04.420","Text":"going to come from the idea of conservation of angular momentum."},{"Start":"13:04.420 ","End":"13:06.775","Text":"Let\u0027s write it over here."},{"Start":"13:06.775 ","End":"13:13.795","Text":"I\u0027ll have my angular momentum at my point of r_ max."},{"Start":"13:13.795 ","End":"13:14.935","Text":"So as we know,"},{"Start":"13:14.935 ","End":"13:18.745","Text":"that\u0027s going to be equal to the mass of my piece of rocket,"},{"Start":"13:18.745 ","End":"13:20.440","Text":"which is 2/3 M,"},{"Start":"13:20.440 ","End":"13:25.360","Text":"multiplied by the velocity at my maximum radius,"},{"Start":"13:25.360 ","End":"13:30.400","Text":"I have a minimum velocity multiplied by my radius there,"},{"Start":"13:30.400 ","End":"13:32.050","Text":"which is r_max,"},{"Start":"13:32.050 ","End":"13:35.470","Text":"and then multiply by sine of the angle between the 2."},{"Start":"13:35.470 ","End":"13:38.650","Text":"Now as we remember at my point r_max,"},{"Start":"13:38.650 ","End":"13:43.440","Text":"the angle between the radius and the velocity is equal to 90 degrees,"},{"Start":"13:43.440 ","End":"13:47.040","Text":"and sine of 90 is equal to 1 so I can leave it like that."},{"Start":"13:47.040 ","End":"13:52.950","Text":"Then the angular momentum of my r_min is going to be again,"},{"Start":"13:52.950 ","End":"13:57.699","Text":"equal to 2/3 M multiplied by the velocity."},{"Start":"13:57.699 ","End":"14:01.090","Text":"Now at r_min, we know that we have a maximum velocity"},{"Start":"14:01.090 ","End":"14:05.200","Text":"so V_max multiplied by the radius, which is r_min."},{"Start":"14:05.200 ","End":"14:08.515","Text":"Then again, at this point, at r_min,"},{"Start":"14:08.515 ","End":"14:13.045","Text":"we know that my velocity and my radius are perpendicular to 1 another."},{"Start":"14:13.045 ","End":"14:19.135","Text":"When we multiply this expression over here by sine of 90 degrees, we get 1."},{"Start":"14:19.135 ","End":"14:22.690","Text":"Then we can equate these 2 to each other."},{"Start":"14:22.690 ","End":"14:27.700","Text":"Now I need to find my angular momentum of my point L_0,"},{"Start":"14:27.700 ","End":"14:32.410","Text":"so that is, I\u0027m reminding you a moment after the explosion."},{"Start":"14:32.410 ","End":"14:36.640","Text":"That is going to be equal to first our mass,"},{"Start":"14:36.640 ","End":"14:38.545","Text":"so 2/3 M,"},{"Start":"14:38.545 ","End":"14:45.175","Text":"and then we have to find the angular momentum at this exact point over here."},{"Start":"14:45.175 ","End":"14:49.060","Text":"Now, when I do my calculation of r cross p,"},{"Start":"14:49.060 ","End":"14:53.845","Text":"I\u0027ll see that my r and my v aren\u0027t perpendicular to one another."},{"Start":"14:53.845 ","End":"14:56.470","Text":"I have some kind of angle here, Alpha,"},{"Start":"14:56.470 ","End":"15:02.005","Text":"and I have to multiply this by sine of this Alpha and I don\u0027t know what this Alpha is."},{"Start":"15:02.005 ","End":"15:10.660","Text":"What I do have is I can see that my V_2 over here has both an x and a y-component."},{"Start":"15:10.660 ","End":"15:17.185","Text":"What I need when solving this equation is I need to take my component of my V_2,"},{"Start":"15:17.185 ","End":"15:21.640","Text":"which is perpendicular to my R. As we said before,"},{"Start":"15:21.640 ","End":"15:25.675","Text":"if my radial direction is going like this, that\u0027s my y-axis."},{"Start":"15:25.675 ","End":"15:30.115","Text":"So that means what\u0027s my perpendicular component to my y-component?"},{"Start":"15:30.115 ","End":"15:31.720","Text":"That\u0027s my x-component."},{"Start":"15:31.720 ","End":"15:36.710","Text":"So what I need to do is I have to find my x-component of my V_2."},{"Start":"15:36.720 ","End":"15:39.670","Text":"My rocket over here,"},{"Start":"15:39.670 ","End":"15:42.475","Text":"my piece number 2 has some velocity V_2,"},{"Start":"15:42.475 ","End":"15:44.950","Text":"which has both an x and a y-component."},{"Start":"15:44.950 ","End":"15:47.935","Text":"Now, if I take the y-component, when I do,"},{"Start":"15:47.935 ","End":"15:56.470","Text":"R cross V because my R-component is parallel to my radius in the radial direction."},{"Start":"15:56.470 ","End":"16:00.325","Text":"So its angular momentum is going to be equal to 0,"},{"Start":"16:00.325 ","End":"16:09.520","Text":"because sine of the angle between my V_2y and my R is 0 degrees and sine of 0 is 0."},{"Start":"16:09.520 ","End":"16:13.270","Text":"What I have to do is I have to take my perpendicular component,"},{"Start":"16:13.270 ","End":"16:14.770","Text":"which is my x-component."},{"Start":"16:14.770 ","End":"16:22.010","Text":"So I have to take my V_2x and then sine of the angle between is going to be sine of 90."},{"Start":"16:22.170 ","End":"16:24.580","Text":"Now we\u0027ve solved that,"},{"Start":"16:24.580 ","End":"16:26.920","Text":"we don\u0027t have to find this angle."},{"Start":"16:26.920 ","End":"16:29.320","Text":"So if I carry on this slide,"},{"Start":"16:29.320 ","End":"16:31.330","Text":"we don\u0027t have to find this angle Alpha."},{"Start":"16:31.330 ","End":"16:35.485","Text":"We can simply just take our x-component."},{"Start":"16:35.485 ","End":"16:37.240","Text":"Let\u0027s go over here."},{"Start":"16:37.240 ","End":"16:41.755","Text":"Then we can see that our angular momentum at our point of explosion is going to be"},{"Start":"16:41.755 ","End":"16:47.125","Text":"our mass multiplied by our velocity x-component,"},{"Start":"16:47.125 ","End":"16:50.905","Text":"and then multiplied by our radius at that point,"},{"Start":"16:50.905 ","End":"16:54.910","Text":"which is equal to R multiplied by sine of the angle between the 2,"},{"Start":"16:54.910 ","End":"16:56.560","Text":"which is sine of 90,"},{"Start":"16:56.560 ","End":"16:58.720","Text":"which is equal to 1."},{"Start":"16:58.720 ","End":"17:01.735","Text":"Now what I have is I have 4 unknowns,"},{"Start":"17:01.735 ","End":"17:04.000","Text":"that\u0027s my V_min over here,"},{"Start":"17:04.000 ","End":"17:06.490","Text":"my V_max, my r_max, and my r_min."},{"Start":"17:06.490 ","End":"17:10.465","Text":"Then I\u0027m going to have 4 equations."},{"Start":"17:10.465 ","End":"17:15.545","Text":"They\u0027re going to be, when I equate this to this,"},{"Start":"17:15.545 ","End":"17:16.940","Text":"this to this,"},{"Start":"17:16.940 ","End":"17:18.575","Text":"so that\u0027s 2 equations."},{"Start":"17:18.575 ","End":"17:23.975","Text":"Then when I equate this equation to this and this equation to this,"},{"Start":"17:23.975 ","End":"17:29.250","Text":"so then we have 4 equations and 4 unknowns and then we can solve."},{"Start":"17:29.430 ","End":"17:32.870","Text":"Again, I\u0027m not going to go through all of the algebra,"},{"Start":"17:32.870 ","End":"17:34.385","Text":"you can do this on your own."},{"Start":"17:34.385 ","End":"17:36.950","Text":"Remember first you have to solve the algebra to find out what"},{"Start":"17:36.950 ","End":"17:40.085","Text":"your V_2x and your V_2y components are,"},{"Start":"17:40.085 ","End":"17:44.805","Text":"and then you can substitute them in over here and do the algebra this section."},{"Start":"17:44.805 ","End":"17:47.869","Text":"Just a little conclusion of how we solve this."},{"Start":"17:47.869 ","End":"17:53.030","Text":"We looked at our question and we wrote down first of all conservation of"},{"Start":"17:53.030 ","End":"17:55.235","Text":"momentum because we\u0027re working in"},{"Start":"17:55.235 ","End":"17:59.500","Text":"a circular trajectory so we have conservation of momentum."},{"Start":"17:59.500 ","End":"18:02.555","Text":"Then we work this out."},{"Start":"18:02.555 ","End":"18:04.970","Text":"Then we saw that we had this V_0 over here,"},{"Start":"18:04.970 ","End":"18:07.475","Text":"so that was our velocity a moment before"},{"Start":"18:07.475 ","End":"18:12.520","Text":"the explosion and then we wanted to work out the momentum a moment after the explosion."},{"Start":"18:12.520 ","End":"18:17.600","Text":"An easy way of finding the velocity a moment before the acceleration is"},{"Start":"18:17.600 ","End":"18:22.565","Text":"to know that because we\u0027re working in circular motion about some large body,"},{"Start":"18:22.565 ","End":"18:26.860","Text":"we\u0027re going to have some central force applied for that body"},{"Start":"18:26.860 ","End":"18:31.040","Text":"and we compare that central force to our angular acceleration,"},{"Start":"18:31.040 ","End":"18:32.715","Text":"which is this, where"},{"Start":"18:32.715 ","End":"18:39.155","Text":"the velocity component of our angular acceleration is this V_0 over here,"},{"Start":"18:39.155 ","End":"18:40.805","Text":"our velocity before it."},{"Start":"18:40.805 ","End":"18:43.730","Text":"Then we could find our V_0 and of course,"},{"Start":"18:43.730 ","End":"18:45.140","Text":"we have to know our escape velocity,"},{"Start":"18:45.140 ","End":"18:47.674","Text":"which is some constant."},{"Start":"18:47.674 ","End":"18:53.340","Text":"Then, after we found out that we could solve to find our V_2x and our V_2y."},{"Start":"18:53.340 ","End":"18:59.150","Text":"Then what we did is we noticed that we\u0027re going to have some elliptical trajectory,"},{"Start":"18:59.150 ","End":"19:01.910","Text":"which means that we\u0027re going to have some central force and if we have"},{"Start":"19:01.910 ","End":"19:05.885","Text":"a central force then we have conservation of energy and momentum."},{"Start":"19:05.885 ","End":"19:10.490","Text":"Then we found all of our energy equations and momentum equations and then we"},{"Start":"19:10.490 ","End":"19:15.215","Text":"simply have to do our algebra in order to find our unknowns."},{"Start":"19:15.215 ","End":"19:18.510","Text":"Okay, that\u0027s the end of our lesson."}],"ID":9460},{"Watched":false,"Name":"Effective Potential","Duration":"37m 54s","ChapterTopicVideoID":9191,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.725","Text":"Hello. In this lesson,"},{"Start":"00:01.725 ","End":"00:06.610","Text":"we\u0027re going to speak about the effective potential energy."},{"Start":"00:06.950 ","End":"00:09.494","Text":"This thing over here."},{"Start":"00:09.494 ","End":"00:16.995","Text":"Now, the point of this lesson is to see that when we have some central force,"},{"Start":"00:16.995 ","End":"00:20.894","Text":"not necessarily gravity, but any central force,"},{"Start":"00:20.894 ","End":"00:28.170","Text":"then we can change this equation over here to an equation with just one unknown."},{"Start":"00:28.170 ","End":"00:31.845","Text":"What we have right now are two unknowns."},{"Start":"00:31.845 ","End":"00:34.410","Text":"Our unknowns are r and r Theta."},{"Start":"00:34.410 ","End":"00:36.600","Text":"With any central force,"},{"Start":"00:36.600 ","End":"00:39.370","Text":"let\u0027s say here when we\u0027re dealing with gravity."},{"Start":"00:39.370 ","End":"00:44.165","Text":"In this lesson, we\u0027re going to show that we can rewrite this equation"},{"Start":"00:44.165 ","End":"00:49.505","Text":"that has two unknowns of r and Theta into this equation over here."},{"Start":"00:49.505 ","End":"00:51.230","Text":"Now as we can see over here,"},{"Start":"00:51.230 ","End":"00:52.489","Text":"we only have one unknown."},{"Start":"00:52.489 ","End":"00:54.440","Text":"We have our unknown r and then,"},{"Start":"00:54.440 ","End":"00:56.860","Text":"of course, the first derivative of it."},{"Start":"00:56.860 ","End":"01:00.035","Text":"Then it\u0027s a lot easier to solve these types of questions,"},{"Start":"01:00.035 ","End":"01:04.010","Text":"because it\u0027s like solving questions in just one dimension or when we have,"},{"Start":"01:04.010 ","End":"01:07.465","Text":"let\u0027s say, just the variable x and x dot."},{"Start":"01:07.465 ","End":"01:11.959","Text":"Our goal is to transform our energy and equation"},{"Start":"01:11.959 ","End":"01:17.629","Text":"into an equation of just one variable and let\u0027s see how we do that."},{"Start":"01:17.629 ","End":"01:20.029","Text":"We\u0027re going to start this equation,"},{"Start":"01:20.029 ","End":"01:21.410","Text":"deriving the equation,"},{"Start":"01:21.410 ","End":"01:26.254","Text":"by using any type of potential energy and later on in this lesson,"},{"Start":"01:26.254 ","End":"01:30.170","Text":"we\u0027ll discuss specifically the case dealing with gravity."},{"Start":"01:30.170 ","End":"01:33.185","Text":"Right now it\u0027s the most general form."},{"Start":"01:33.185 ","End":"01:35.179","Text":"We have our energy equation,"},{"Start":"01:35.179 ","End":"01:39.565","Text":"which is going to be our kinetic energy plus our potential energy."},{"Start":"01:39.565 ","End":"01:45.379","Text":"Let\u0027s take a look, so our v over here in polar coordinates is going to be equal"},{"Start":"01:45.379 ","End":"01:51.244","Text":"to r dot in the r direction plus r Theta dot in the Theta direction."},{"Start":"01:51.244 ","End":"01:54.574","Text":"Now, in the chapter speaking about polar coordinates,"},{"Start":"01:54.574 ","End":"01:57.514","Text":"we showed that this is the equation."},{"Start":"01:57.514 ","End":"01:59.194","Text":"If you don\u0027t remember this,"},{"Start":"01:59.194 ","End":"02:01.115","Text":"go and take a look at that lesson again."},{"Start":"02:01.115 ","End":"02:05.644","Text":"This is always true when we\u0027re dealing with polar coordinates."},{"Start":"02:05.644 ","End":"02:07.819","Text":"If we\u0027re dealing with Cartesian coordinates,"},{"Start":"02:07.819 ","End":"02:11.249","Text":"so our velocity would just be"},{"Start":"02:11.249 ","End":"02:19.800","Text":"our vx in the x direction plus vy in the y direction."},{"Start":"02:19.800 ","End":"02:23.870","Text":"This is simply the polar coordinate version of that."},{"Start":"02:23.870 ","End":"02:27.370","Text":"Now in our equation we have v^2."},{"Start":"02:27.370 ","End":"02:30.724","Text":"When we square this expression over here,"},{"Start":"02:30.724 ","End":"02:36.025","Text":"we\u0027ll get r dot squared plus r Theta dot squared."},{"Start":"02:36.025 ","End":"02:40.400","Text":"As we can see, if we just square this,"},{"Start":"02:40.400 ","End":"02:44.660","Text":"we\u0027ll get this, and then we\u0027re meant to get some other components."},{"Start":"02:44.660 ","End":"02:47.950","Text":"Now, that is canceled out and why is it canceled out?"},{"Start":"02:47.950 ","End":"02:54.480","Text":"It\u0027s because it\u0027s going to be 2 times our r dot multiplied by"},{"Start":"02:54.480 ","End":"03:03.420","Text":"r Theta dot multiplied by our r hat dot r Theta hat dot."},{"Start":"03:03.420 ","End":"03:12.504","Text":"Let\u0027s just write that. We\u0027re going to have 2 times r dot r Theta dot,"},{"Start":"03:12.504 ","End":"03:17.429","Text":"and then multiplied by r hat dot Theta hat."},{"Start":"03:17.429 ","End":"03:20.509","Text":"Now, if we remember our r hat vector,"},{"Start":"03:20.509 ","End":"03:22.880","Text":"the unit vector in our radial direction,"},{"Start":"03:22.880 ","End":"03:31.000","Text":"is going to be perpendicular to our unit vector in our Theta direction."},{"Start":"03:31.000 ","End":"03:35.854","Text":"When we have two perpendicular vectors, dot each other,"},{"Start":"03:35.854 ","End":"03:38.690","Text":"scalar multiplied by one another,"},{"Start":"03:38.690 ","End":"03:42.585","Text":"so that is going to be equal to 0."},{"Start":"03:42.585 ","End":"03:47.165","Text":"That\u0027s why we\u0027re left simply with this and without this component"},{"Start":"03:47.165 ","End":"03:51.860","Text":"when we square our V. Now we have our expression for our V^2,"},{"Start":"03:51.860 ","End":"03:55.770","Text":"so we\u0027re done with our kinetic energy section."},{"Start":"03:55.770 ","End":"03:58.190","Text":"Now let\u0027s move on to our potential energy."},{"Start":"03:58.190 ","End":"04:01.429","Text":"Let\u0027s start first of all with our angular momentum."},{"Start":"04:01.429 ","End":"04:06.110","Text":"We have that our equation for angular momentum is r cross p,"},{"Start":"04:06.110 ","End":"04:08.869","Text":"where p is our momentum,"},{"Start":"04:08.869 ","End":"04:13.090","Text":"which is mass times velocity over here and our velocity we\u0027ve already found."},{"Start":"04:13.090 ","End":"04:15.395","Text":"What is our r going to be?"},{"Start":"04:15.395 ","End":"04:17.495","Text":"Our r is simply this over here,"},{"Start":"04:17.495 ","End":"04:20.854","Text":"it\u0027s always going to be this r in the r direction,"},{"Start":"04:20.854 ","End":"04:25.735","Text":"cross multiplied by r mass, multiplied by rv."},{"Start":"04:25.735 ","End":"04:30.124","Text":"All of this is our v,"},{"Start":"04:30.124 ","End":"04:32.405","Text":"which we found over here."},{"Start":"04:32.405 ","End":"04:35.025","Text":"Now, what is this going to be equal to?"},{"Start":"04:35.025 ","End":"04:41.739","Text":"We\u0027re going to have r multiplied by r hat multiplied by this over here."},{"Start":"04:41.739 ","End":"04:48.214","Text":"As we know, when we have some vector cross product with itself,"},{"Start":"04:48.214 ","End":"04:50.089","Text":"k cross multiplied with itself,"},{"Start":"04:50.089 ","End":"04:53.495","Text":"that\u0027s always going to be equal to 0."},{"Start":"04:53.495 ","End":"05:00.424","Text":"Over here, two perpendicular vectors which are multiplied via the dot-product,"},{"Start":"05:00.424 ","End":"05:03.154","Text":"if they\u0027re perpendicular it will equal to 0."},{"Start":"05:03.154 ","End":"05:05.330","Text":"When we\u0027re dealing with a cross-product,"},{"Start":"05:05.330 ","End":"05:08.465","Text":"if we have two vectors which are parallel to one another,"},{"Start":"05:08.465 ","End":"05:11.329","Text":"so the cross-product is going to be equal to 0."},{"Start":"05:11.329 ","End":"05:13.939","Text":"Now, obviously these two vectors are the same,"},{"Start":"05:13.939 ","End":"05:16.370","Text":"which means that they\u0027re parallel to one another."},{"Start":"05:16.370 ","End":"05:19.119","Text":"That\u0027s going to be equal to 0."},{"Start":"05:19.119 ","End":"05:24.384","Text":"Then when we have our rr hat multiplied by this term over here,"},{"Start":"05:24.384 ","End":"05:27.325","Text":"our r Theta dot multiplied by Theta hat,"},{"Start":"05:27.325 ","End":"05:32.950","Text":"so we know that our r hat cross"},{"Start":"05:32.950 ","End":"05:40.285","Text":"multiplied by our Theta hat is simply going to be equal to z hat."},{"Start":"05:40.285 ","End":"05:43.060","Text":"This over here is handy to know."},{"Start":"05:43.060 ","End":"05:49.010","Text":"If we have our r Theta z."},{"Start":"05:49.880 ","End":"05:54.540","Text":"If we\u0027re going to cross-multiply our r and our Theta,"},{"Start":"05:54.540 ","End":"05:57.960","Text":"this is specifically with cross-multiplying,"},{"Start":"05:57.960 ","End":"06:00.810","Text":"r cross Theta we\u0027ll get rz."},{"Start":"06:00.810 ","End":"06:03.495","Text":"If we have Theta cross z,"},{"Start":"06:03.495 ","End":"06:08.824","Text":"we\u0027ll get our r. We\u0027re going clockwise,"},{"Start":"06:08.824 ","End":"06:10.615","Text":"and that\u0027s how we get the next term."},{"Start":"06:10.615 ","End":"06:12.718","Text":"Z cross r,"},{"Start":"06:12.718 ","End":"06:16.549","Text":"the next term in the clockwise direction is our Theta over here."},{"Start":"06:16.549 ","End":"06:19.009","Text":"If we start going anticlockwise,"},{"Start":"06:19.009 ","End":"06:21.275","Text":"it\u0027s the same rule, but with a negative."},{"Start":"06:21.275 ","End":"06:27.800","Text":"Z cross Theta will give us negative r. Theta cross r will give us negative z,"},{"Start":"06:27.800 ","End":"06:30.175","Text":"and so on and so forth."},{"Start":"06:30.175 ","End":"06:33.824","Text":"Now when we simplify out this,"},{"Start":"06:33.824 ","End":"06:38.786","Text":"given that r cross r is equal to 0 and r cross Theta is equal to z hat,"},{"Start":"06:38.786 ","End":"06:46.390","Text":"we\u0027ll see that our angular momentum is equal to mr^2 Theta dot in the z direction."},{"Start":"06:46.460 ","End":"06:53.240","Text":"Now we can see that our angular momentum is only in the z direction all the time."},{"Start":"06:53.240 ","End":"06:56.360","Text":"Whenever we\u0027re dealing with a central force,"},{"Start":"06:56.360 ","End":"07:00.055","Text":"our angular momentum only has a z component."},{"Start":"07:00.055 ","End":"07:02.615","Text":"This is always true and then we can,"},{"Start":"07:02.615 ","End":"07:07.250","Text":"instead of writing that our angular momentum is this in the z direction,"},{"Start":"07:07.250 ","End":"07:09.200","Text":"we can just write the z component,"},{"Start":"07:09.200 ","End":"07:13.114","Text":"which is L_z is equal to mr^2 Theta dot."},{"Start":"07:13.114 ","End":"07:14.719","Text":"Because we\u0027re taking the z component,"},{"Start":"07:14.719 ","End":"07:18.715","Text":"we don\u0027t have to add in over here our z hat."},{"Start":"07:18.715 ","End":"07:24.110","Text":"Now what I want to do is I want to isolate out my Theta dot over here,"},{"Start":"07:24.110 ","End":"07:28.715","Text":"because I want to remain with an equation which is just in terms of"},{"Start":"07:28.715 ","End":"07:34.479","Text":"r. When I isolate and I rearrange this equation,"},{"Start":"07:34.479 ","End":"07:37.204","Text":"in order to get my Theta dot over here,"},{"Start":"07:37.204 ","End":"07:41.700","Text":"I will have my L divided by mr^2."},{"Start":"07:42.050 ","End":"07:50.074","Text":"Then we will be left with our Theta dot over here and my L is equal to L_z."},{"Start":"07:50.074 ","End":"07:53.615","Text":"Instead of writing L_z all the time because it\u0027s long and tedious,"},{"Start":"07:53.615 ","End":"07:59.029","Text":"I\u0027ll just write my L. It\u0027s useful also to note that my L over here,"},{"Start":"07:59.029 ","End":"08:03.695","Text":"my angular momentum, is going to be some constant"},{"Start":"08:03.695 ","End":"08:10.810","Text":"and so the only variable that I have now in my equation is this r over here."},{"Start":"08:10.810 ","End":"08:13.920","Text":"Now we have all of our variables."},{"Start":"08:13.920 ","End":"08:18.154","Text":"Let\u0027s substitute it back into our original equation."},{"Start":"08:18.154 ","End":"08:22.030","Text":"Now my Ur I\u0027m still not touching, we\u0027ll get to this later."},{"Start":"08:22.030 ","End":"08:25.969","Text":"My E is going to be equal to my kinetic energy,"},{"Start":"08:25.969 ","End":"08:27.755","Text":"which is 1/2mv^2,"},{"Start":"08:27.755 ","End":"08:30.819","Text":"where my v^2 is this over here."},{"Start":"08:30.819 ","End":"08:35.600","Text":"I substituted my v^2 over here plus my Ur."},{"Start":"08:35.600 ","End":"08:39.860","Text":"Now what I can do is I can expand these brackets out."},{"Start":"08:39.860 ","End":"08:43.820","Text":"I\u0027ll have 1/2 mr dot squared, which is over here,"},{"Start":"08:43.820 ","End":"08:47.434","Text":"plus half mr^2 Theta dot squared,"},{"Start":"08:47.434 ","End":"08:50.520","Text":"which is this over here, plus my Ur."},{"Start":"08:51.360 ","End":"08:57.415","Text":"Now, my Theta dot we already know is going to be my L divided by mr ^2."},{"Start":"08:57.415 ","End":"09:02.575","Text":"Then my Theta dot is this over here,"},{"Start":"09:02.575 ","End":"09:04.420","Text":"and they substitute that in."},{"Start":"09:04.420 ","End":"09:08.035","Text":"Then I am left with this equation."},{"Start":"09:08.035 ","End":"09:13.539","Text":"Now we can see the equation for energy over here is going to be 1/2mr dot"},{"Start":"09:13.539 ","End":"09:20.020","Text":"squared plus L^2 divided by 2mr^2 plus ur."},{"Start":"09:20.020 ","End":"09:22.269","Text":"This expression over here,"},{"Start":"09:22.269 ","End":"09:32.179","Text":"my L^2 divided by 2mr^2 plus ur is going to be called my u effective potential energy."},{"Start":"09:32.250 ","End":"09:37.045","Text":"Soon we\u0027re going to speak about why we call this u effective."},{"Start":"09:37.045 ","End":"09:40.660","Text":"But in the meantime, all we need to know is that u effective,"},{"Start":"09:40.660 ","End":"09:45.655","Text":"our effective potential energy is equal to our L^2 divided by"},{"Start":"09:45.655 ","End":"09:51.655","Text":"2mr^2 plus our u as a function of r. Now,"},{"Start":"09:51.655 ","End":"09:56.950","Text":"a u as a function of r is simply the potential energy that we\u0027re working with."},{"Start":"09:56.950 ","End":"10:00.800","Text":"If it\u0027s due to gravity or whatever, it might be."},{"Start":"10:01.830 ","End":"10:04.899","Text":"The reason that we call this the effective"},{"Start":"10:04.899 ","End":"10:09.685","Text":"potential energy is because our energy equation looks something like this."},{"Start":"10:09.685 ","End":"10:11.799","Text":"We have our kinetic energy,"},{"Start":"10:11.799 ","End":"10:16.765","Text":"which is including our first derivative of our variable."},{"Start":"10:16.765 ","End":"10:21.985","Text":"Then, we have some kind of expression over here, where both terms,"},{"Start":"10:21.985 ","End":"10:28.135","Text":"both my L^2 divided by 2mr^2 plus my u as a function of r are both"},{"Start":"10:28.135 ","End":"10:34.524","Text":"only dependent on r. Here we have our derivative of our variable,"},{"Start":"10:34.524 ","End":"10:40.030","Text":"and here we have two terms which are dependent just on one variable."},{"Start":"10:40.030 ","End":"10:44.754","Text":"The reason we do this is in order to simplify our equation."},{"Start":"10:44.754 ","End":"10:47.605","Text":"Usually we have our equation for energy,"},{"Start":"10:47.605 ","End":"10:50.019","Text":"which is with our kinetic energy,"},{"Start":"10:50.019 ","End":"10:53.515","Text":"which will be 1/2mx dot squared."},{"Start":"10:53.515 ","End":"10:55.389","Text":"Then, plus our potential energy,"},{"Start":"10:55.389 ","End":"10:59.869","Text":"which will be plus our mgx or whatever it might be."},{"Start":"11:00.720 ","End":"11:03.520","Text":"We have two terms."},{"Start":"11:03.520 ","End":"11:06.975","Text":"Here our idea is to again,"},{"Start":"11:06.975 ","End":"11:10.830","Text":"split up our energy into two terms again."},{"Start":"11:10.830 ","End":"11:13.770","Text":"Our equation with our x dot here,"},{"Start":"11:13.770 ","End":"11:15.539","Text":"specifically it\u0027s r dot."},{"Start":"11:15.539 ","End":"11:20.770","Text":"Then, our term with just our x."},{"Start":"11:20.770 ","End":"11:25.120","Text":"Here, x is or r. That\u0027s the idea behind this."},{"Start":"11:25.120 ","End":"11:28.720","Text":"Then we\u0027re just working with two separate terms in this equation,"},{"Start":"11:28.720 ","End":"11:32.920","Text":"which makes it a lot easier to look at and understand."},{"Start":"11:32.920 ","End":"11:36.925","Text":"Then this is like our total potential energy."},{"Start":"11:36.925 ","End":"11:42.865","Text":"If I\u0027m looking at the problem as if it only has one parameter, one variable."},{"Start":"11:42.865 ","End":"11:48.865","Text":"That\u0027s because both these terms are dependent solely on r and not on"},{"Start":"11:48.865 ","End":"11:55.660","Text":"the first derivative of r. This term over here is our kinetic energy,"},{"Start":"11:55.660 ","End":"12:01.734","Text":"and these two terms together are potential energy."},{"Start":"12:01.734 ","End":"12:05.665","Text":"Now of course, why I say sort of is because that isn\u0027t exactly true."},{"Start":"12:05.665 ","End":"12:07.120","Text":"Because in actual facts,"},{"Start":"12:07.120 ","End":"12:09.400","Text":"if we\u0027re going to be pedantic,"},{"Start":"12:09.400 ","End":"12:11.650","Text":"then this middle term over here actually"},{"Start":"12:11.650 ","End":"12:16.044","Text":"comes from the kinetic energy and comes from the velocity."},{"Start":"12:16.044 ","End":"12:17.875","Text":"It is a bit different."},{"Start":"12:17.875 ","End":"12:23.950","Text":"But we look at it as if it is like that in order to simplify our calculations."},{"Start":"12:23.950 ","End":"12:30.445","Text":"Again, this is our effective potential energy. It\u0027s that."},{"Start":"12:30.445 ","End":"12:34.659","Text":"Now we\u0027ve just spoken about our general equation and let\u0027s speak"},{"Start":"12:34.659 ","End":"12:39.760","Text":"specifically now about dealing with cases of gravity."},{"Start":"12:39.760 ","End":"12:42.310","Text":"Here we have our potential energy,"},{"Start":"12:42.310 ","End":"12:45.189","Text":"which is as a function of r that is equal to"},{"Start":"12:45.189 ","End":"12:49.674","Text":"negative Alpha divided by r when we\u0027re dealing with gravity."},{"Start":"12:49.674 ","End":"12:56.965","Text":"Of course, Alpha is going to be equal to GMm; some constant."},{"Start":"12:56.965 ","End":"13:03.265","Text":"Now that means that given this equation for U as a function of r,"},{"Start":"13:03.265 ","End":"13:07.029","Text":"our U effective by just substituting in this equation over here,"},{"Start":"13:07.029 ","End":"13:11.470","Text":"we\u0027ll get that our U effective when dealing with gravity is equal to L^2"},{"Start":"13:11.470 ","End":"13:15.870","Text":"divided by 2mr^2 the same plus our U as a function of r,"},{"Start":"13:15.870 ","End":"13:18.269","Text":"which is going to be negative Alpha divided by"},{"Start":"13:18.269 ","End":"13:26.769","Text":"r. Now if I draw out the graph for this equation over here,"},{"Start":"13:27.390 ","End":"13:33.265","Text":"I\u0027m drawing the graph for my U effective, this over here."},{"Start":"13:33.265 ","End":"13:39.265","Text":"It\u0027s going according to r and my U effective that I get."},{"Start":"13:39.265 ","End":"13:43.764","Text":"What we\u0027ll see is that my function goes down something like this,"},{"Start":"13:43.764 ","End":"13:45.850","Text":"reaches some minimum point,"},{"Start":"13:45.850 ","End":"13:48.850","Text":"and then we\u0027ll come back up again and slowly,"},{"Start":"13:48.850 ","End":"13:52.075","Text":"slowly as my r increases,"},{"Start":"13:52.075 ","End":"13:56.529","Text":"my function, my U effective will approach 0."},{"Start":"13:56.529 ","End":"14:01.269","Text":"Of course, when my r is approaching 0,"},{"Start":"14:01.269 ","End":"14:07.195","Text":"my U effective will be approaching infinity as we can see from the graph."},{"Start":"14:07.195 ","End":"14:11.725","Text":"Now what I want to do is I want to look at different cases."},{"Start":"14:11.725 ","End":"14:15.310","Text":"What can be and how through this graph,"},{"Start":"14:15.310 ","End":"14:18.790","Text":"we can understand what is happening with our energy."},{"Start":"14:18.790 ","End":"14:22.330","Text":"What we\u0027re going to do is we\u0027re going to look at 3 cases,"},{"Start":"14:22.330 ","End":"14:27.610","Text":"which we saw when we were dealing with our equation for the trajectory."},{"Start":"14:27.610 ","End":"14:31.960","Text":"You have to know how to understand it from the equations of the trajectory."},{"Start":"14:31.960 ","End":"14:35.650","Text":"But you can also understand what our trajectory is going"},{"Start":"14:35.650 ","End":"14:39.580","Text":"to look like based on our energy equation."},{"Start":"14:39.580 ","End":"14:41.965","Text":"That\u0027s what we\u0027re going to do right now."},{"Start":"14:41.965 ","End":"14:46.750","Text":"What do we have on this graph over here is our potential energy."},{"Start":"14:46.750 ","End":"14:50.724","Text":"Now what we\u0027re going to do is we\u0027re going to draw on the same graph,"},{"Start":"14:50.724 ","End":"14:53.650","Text":"our graph for energy."},{"Start":"14:53.650 ","End":"14:57.175","Text":"We\u0027re going to draw it as a function of"},{"Start":"14:57.175 ","End":"15:03.715","Text":"r. As we know from experience and from previous lessons,"},{"Start":"15:03.715 ","End":"15:08.769","Text":"we know that our energy is always going to be a constant."},{"Start":"15:08.769 ","End":"15:13.000","Text":"What that means is even though that kinetic energy"},{"Start":"15:13.000 ","End":"15:16.495","Text":"and our potential energy is dependent on our variable r,"},{"Start":"15:16.495 ","End":"15:23.484","Text":"as we can see, our energy value itself is some constant number."},{"Start":"15:23.484 ","End":"15:26.424","Text":"That means that if I draw it on the graph,"},{"Start":"15:26.424 ","End":"15:30.115","Text":"it\u0027s going to look something like this;"},{"Start":"15:30.115 ","End":"15:34.315","Text":"some straight horizontal line,"},{"Start":"15:34.315 ","End":"15:43.915","Text":"where this value over here is our value for E. Throughout the motion,"},{"Start":"15:43.915 ","End":"15:51.699","Text":"our energy is going to be this constant value of E. I\u0027ve labeled this over here,"},{"Start":"15:51.699 ","End":"15:53.875","Text":"is our E total."},{"Start":"15:53.875 ","End":"15:56.515","Text":"There\u0027s a T over here because it represents"},{"Start":"15:56.515 ","End":"16:00.010","Text":"also kinetic energy and our potential energy over here."},{"Start":"16:00.010 ","End":"16:05.455","Text":"Just like we can have our value for E over here as some positive energy,"},{"Start":"16:05.455 ","End":"16:08.064","Text":"we can have the same thing in the negative."},{"Start":"16:08.064 ","End":"16:12.310","Text":"Then we\u0027ll just again have some constant function."},{"Start":"16:12.310 ","End":"16:17.110","Text":"But now our energy is going to be less than 0."},{"Start":"16:17.110 ","End":"16:18.985","Text":"We have negative energy."},{"Start":"16:18.985 ","End":"16:21.069","Text":"Just like that, we can also have,"},{"Start":"16:21.069 ","End":"16:23.230","Text":"and this is a very interesting point,"},{"Start":"16:23.230 ","End":"16:29.694","Text":"our energy function being constant at this minimum point."},{"Start":"16:29.694 ","End":"16:33.295","Text":"Here we\u0027ll label this as Emin."},{"Start":"16:33.295 ","End":"16:37.449","Text":"This we\u0027ll see is going to be very interesting."},{"Start":"16:37.449 ","End":"16:43.195","Text":"Now, of course, our value for energy cannot be lower than this."},{"Start":"16:43.195 ","End":"16:49.045","Text":"Our energy is always going to be in the range of our U effective graph."},{"Start":"16:49.045 ","End":"16:52.569","Text":"If our U effective graph has some minimum points,"},{"Start":"16:52.569 ","End":"16:58.239","Text":"E min or any value for our E cannot be lower than that point."},{"Start":"16:58.239 ","End":"17:00.024","Text":"Now what we\u0027re going to do,"},{"Start":"17:00.024 ","End":"17:03.250","Text":"it\u0027s going to speak about each one of these cases and what that"},{"Start":"17:03.250 ","End":"17:07.105","Text":"means in terms of our trajectory."},{"Start":"17:07.105 ","End":"17:11.964","Text":"Let\u0027s quickly take a look back at our equation over here."},{"Start":"17:11.964 ","End":"17:15.084","Text":"We\u0027re looking at this equation,"},{"Start":"17:15.084 ","End":"17:18.849","Text":"which we\u0027re saying is like our kinetic energy."},{"Start":"17:18.849 ","End":"17:22.000","Text":"We spoke about it. Why we\u0027re speaking about that?"},{"Start":"17:22.000 ","End":"17:24.100","Text":"It\u0027s around about kinetic energy."},{"Start":"17:24.100 ","End":"17:29.395","Text":"Because we also have this which is dependent on our velocity."},{"Start":"17:29.395 ","End":"17:32.769","Text":"This also comes under the umbrella of kinetic energy."},{"Start":"17:32.769 ","End":"17:34.420","Text":"But we said, never mind."},{"Start":"17:34.420 ","End":"17:36.189","Text":"We\u0027re going to say that this is our kinetic energy"},{"Start":"17:36.189 ","End":"17:38.690","Text":"and that these two terms are potential."},{"Start":"17:38.690 ","End":"17:41.550","Text":"Back to our kinetic energy over here,"},{"Start":"17:41.550 ","End":"17:45.544","Text":"we can see that this term is always going to be positive."},{"Start":"17:45.544 ","End":"17:49.764","Text":"R mass is of course always 1/2 as positive."},{"Start":"17:49.764 ","End":"17:52.684","Text":"R dots because it\u0027s being squared,"},{"Start":"17:52.684 ","End":"17:56.014","Text":"that is also going to be always positive."},{"Start":"17:56.014 ","End":"17:58.264","Text":"So in that case,"},{"Start":"17:58.264 ","End":"18:03.409","Text":"we know that our total energy,"},{"Start":"18:03.409 ","End":"18:10.660","Text":"is always going to be bigger or equal to our effective potential energy."},{"Start":"18:10.660 ","End":"18:14.420","Text":"Why is that? Because kinetic energy over here is"},{"Start":"18:14.420 ","End":"18:18.800","Text":"a positive value so it could also be equal to 0,"},{"Start":"18:18.800 ","End":"18:20.765","Text":"but always bigger or equal to 0."},{"Start":"18:20.765 ","End":"18:25.655","Text":"Which means that our energy is always going to be either U effective"},{"Start":"18:25.655 ","End":"18:31.235","Text":"or U effective plus this positive term over here,"},{"Start":"18:31.235 ","End":"18:33.740","Text":"plus some positive number."},{"Start":"18:33.740 ","End":"18:36.050","Text":"That means that it\u0027s always going to be,"},{"Start":"18:36.050 ","End":"18:38.030","Text":"its minimum will be U effective."},{"Start":"18:38.030 ","End":"18:42.404","Text":"Then it can be bigger depending on what our value for kinetic energy is."},{"Start":"18:42.404 ","End":"18:45.849","Text":"That brings us back to what we said before,"},{"Start":"18:45.849 ","End":"18:50.545","Text":"where our minimum value for our energy is going to be this constant function"},{"Start":"18:50.545 ","End":"18:55.959","Text":"which passes right at our minimum point for our U effective graph."},{"Start":"18:55.959 ","End":"18:58.750","Text":"It\u0027s tangential to that point."},{"Start":"18:58.750 ","End":"19:01.119","Text":"That\u0027s exactly because of this over here,"},{"Start":"19:01.119 ","End":"19:04.179","Text":"because we see that our value for energy must always"},{"Start":"19:04.179 ","End":"19:07.710","Text":"be bigger or equal to our U effective."},{"Start":"19:07.710 ","End":"19:10.239","Text":"That means that it always has to be bigger or"},{"Start":"19:10.239 ","End":"19:13.059","Text":"equal to a minimum value in our U effective?"},{"Start":"19:13.059 ","End":"19:15.085","Text":"Which has this over here."},{"Start":"19:15.085 ","End":"19:18.489","Text":"Specifically our Emin is going to be equal"},{"Start":"19:18.489 ","End":"19:22.630","Text":"to our minimum value of our U effective and as we can see,"},{"Start":"19:22.630 ","End":"19:26.330","Text":"all other options are going to be bigger than that."},{"Start":"19:27.120 ","End":"19:30.604","Text":"Now let\u0027s talk about this section over here,"},{"Start":"19:30.604 ","End":"19:35.750","Text":"where our energy is some negative value gets smaller than 0."},{"Start":"19:35.750 ","End":"19:39.844","Text":"If we take a look at this point over here,"},{"Start":"19:39.844 ","End":"19:45.770","Text":"where energy graph cuts in with our U eff graph."},{"Start":"19:45.770 ","End":"19:52.220","Text":"Also over here. Then we draw these points over here on our x line."},{"Start":"19:52.220 ","End":"19:56.085","Text":"If we call this value our r max,"},{"Start":"19:56.085 ","End":"19:59.730","Text":"because our x-axis is in fact our r axis,"},{"Start":"19:59.730 ","End":"20:03.200","Text":"this will be our r min."},{"Start":"20:03.200 ","End":"20:05.375","Text":"What do these values mean?"},{"Start":"20:05.375 ","End":"20:07.455","Text":"That means that our body,"},{"Start":"20:07.455 ","End":"20:15.050","Text":"or a shape or object is always going to be a distance r away from our origin."},{"Start":"20:15.050 ","End":"20:16.670","Text":"Some value for r,"},{"Start":"20:16.670 ","End":"20:20.595","Text":"which is between our r min and our r max."},{"Start":"20:20.595 ","End":"20:22.990","Text":"Let\u0027s say we have here."},{"Start":"20:22.990 ","End":"20:26.830","Text":"Right here is our center at origin."},{"Start":"20:26.830 ","End":"20:31.195","Text":"Let\u0027s say this is our r min and this is r max."},{"Start":"20:31.195 ","End":"20:36.870","Text":"Our body is going to be somewhere between those two lines."},{"Start":"20:37.850 ","End":"20:42.540","Text":"Let\u0027s take a look at why that is the explanation."},{"Start":"20:42.540 ","End":"20:48.449","Text":"If we take a look, if our objects or our body is located over here at this radius,"},{"Start":"20:48.449 ","End":"20:52.275","Text":"so let\u0027s call it r Tilde, over here."},{"Start":"20:52.275 ","End":"20:54.690","Text":"That means, if we draw a dotted line,"},{"Start":"20:54.690 ","End":"21:01.379","Text":"that means that in the energy it\u0027s located at this point over here. What does that mean?"},{"Start":"21:01.379 ","End":"21:04.694","Text":"As we saw before, our energy function must"},{"Start":"21:04.694 ","End":"21:08.640","Text":"always be bigger or equal to our U effective graph."},{"Start":"21:08.640 ","End":"21:12.465","Text":"If our energy function is this green line over here,"},{"Start":"21:12.465 ","End":"21:14.445","Text":"we can see that right now,"},{"Start":"21:14.445 ","End":"21:16.830","Text":"our U effective graph is this in green."},{"Start":"21:16.830 ","End":"21:24.029","Text":"We can see that our energy graph is smaller than our value for U effective over here."},{"Start":"21:24.029 ","End":"21:27.225","Text":"It\u0027s below this graph, the black graph."},{"Start":"21:27.225 ","End":"21:30.419","Text":"That means that we\u0027re not adhering to this law,"},{"Start":"21:30.419 ","End":"21:32.430","Text":"that we said where this rule,"},{"Start":"21:32.430 ","End":"21:36.040","Text":"which means that our body can\u0027t be located over here."},{"Start":"21:36.440 ","End":"21:44.159","Text":"My energy graph always has to be bigger or equal to my U effective,"},{"Start":"21:44.159 ","End":"21:47.579","Text":"and of course, the same is true on the other side."},{"Start":"21:47.579 ","End":"21:51.116","Text":"If I say that my r is over here,"},{"Start":"21:51.116 ","End":"21:53.309","Text":"of course, because here,"},{"Start":"21:53.309 ","End":"21:58.889","Text":"the black line, my graph for U effective is approaching infinity and of course,"},{"Start":"21:58.889 ","End":"22:01.995","Text":"here my energy and some negative value."},{"Start":"22:01.995 ","End":"22:06.360","Text":"We can see that our radius can also be over here."},{"Start":"22:06.360 ","End":"22:12.910","Text":"It has to be exactly between these two points."},{"Start":"22:13.340 ","End":"22:17.775","Text":"Now let\u0027s talk about if our energy is bigger than 0."},{"Start":"22:17.775 ","End":"22:20.459","Text":"We\u0027re talking about this graph over here."},{"Start":"22:20.459 ","End":"22:22.485","Text":"If our energy is bigger than 0,"},{"Start":"22:22.485 ","End":"22:26.895","Text":"we can see that our only points is over here."},{"Start":"22:26.895 ","End":"22:31.830","Text":"That means that our radius has to be this value over here and then,"},{"Start":"22:31.830 ","End":"22:35.549","Text":"because as our r approaches infinity,"},{"Start":"22:35.549 ","End":"22:40.049","Text":"so we know that our value for U effective approaches 0."},{"Start":"22:40.049 ","End":"22:41.730","Text":"This energy, as we can see,"},{"Start":"22:41.730 ","End":"22:44.190","Text":"is always going to be bigger than 0 because it"},{"Start":"22:44.190 ","End":"22:47.910","Text":"starts off bigger than 0 and it\u0027s a constant function."},{"Start":"22:47.910 ","End":"22:51.930","Text":"This energy is going to go for infinity at"},{"Start":"22:51.930 ","End":"22:56.279","Text":"this value bigger than 0 and our r as it approaches infinity,"},{"Start":"22:56.279 ","End":"22:59.280","Text":"so our U effective will approach 0,"},{"Start":"22:59.280 ","End":"23:03.195","Text":"which means that our energy bar is always going to be higher."},{"Start":"23:03.195 ","End":"23:06.825","Text":"Our energy is always going to be bigger than our U effective."},{"Start":"23:06.825 ","End":"23:10.469","Text":"Which means that our body can be located anywhere from"},{"Start":"23:10.469 ","End":"23:15.525","Text":"this point over here, until infinity."},{"Start":"23:15.525 ","End":"23:22.900","Text":"I can be basically any number as long as it\u0027s bigger or equal to this value over here."},{"Start":"23:23.030 ","End":"23:26.835","Text":"Now, if we\u0027re speaking about our E min,"},{"Start":"23:26.835 ","End":"23:28.615","Text":"this value over here,"},{"Start":"23:28.615 ","End":"23:32.160","Text":"we can see the only point where our graph for"},{"Start":"23:32.160 ","End":"23:37.514","Text":"our energy cuts our graph for our effective potential energy is at this point,"},{"Start":"23:37.514 ","End":"23:40.995","Text":"right over here, just at the minimum point."},{"Start":"23:40.995 ","End":"23:45.060","Text":"Then that means that our radius can only be equal to this,"},{"Start":"23:45.060 ","End":"23:47.530","Text":"which we\u0027ll call r_0."},{"Start":"23:47.930 ","End":"23:53.499","Text":"It can only be r_0 and no other radius."},{"Start":"23:54.380 ","End":"23:58.649","Text":"Now that we\u0027ve discussed the basics of"},{"Start":"23:58.649 ","End":"24:02.475","Text":"where energy graph is and what that means about our radius,"},{"Start":"24:02.475 ","End":"24:06.390","Text":"let\u0027s speak about what that means with our trajectories."},{"Start":"24:06.390 ","End":"24:09.840","Text":"Let\u0027s start over here, where energy is bigger than 0."},{"Start":"24:09.840 ","End":"24:12.180","Text":"We saw that with our energy bigger than 0,"},{"Start":"24:12.180 ","End":"24:18.750","Text":"our r can be located at some minimum for this specific energy when it\u0027s bigger than 0,"},{"Start":"24:18.750 ","End":"24:24.390","Text":"and we can use any r from then on until infinity."},{"Start":"24:24.390 ","End":"24:29.820","Text":"Because we don\u0027t have some range that our radius has to be,"},{"Start":"24:29.820 ","End":"24:33.074","Text":"so that really describes hyperbolic motion."},{"Start":"24:33.074 ","End":"24:38.955","Text":"With hyperbolic motion, we\u0027re going to have our planet over here, let\u0027s say our earth."},{"Start":"24:38.955 ","End":"24:41.129","Text":"We\u0027re going to have some meteor,"},{"Start":"24:41.129 ","End":"24:48.165","Text":"or whatever it is coming close going around and then going back to infinity."},{"Start":"24:48.165 ","End":"24:49.829","Text":"That\u0027s exactly this."},{"Start":"24:49.829 ","End":"24:53.039","Text":"This is the r min over"},{"Start":"24:53.039 ","End":"24:57.134","Text":"here that we showed on the graph here and now with this black line,"},{"Start":"24:57.134 ","End":"25:01.079","Text":"and then our r can go to infinity,"},{"Start":"25:01.079 ","End":"25:06.345","Text":"and that\u0027s when it comes from infinity and goes to infinity afterwards,"},{"Start":"25:06.345 ","End":"25:08.010","Text":"so that\u0027s hyperbolic motion."},{"Start":"25:08.010 ","End":"25:13.284","Text":"When our E, a value for E is bigger or equal to 0."},{"Start":"25:13.284 ","End":"25:18.380","Text":"If our E is exactly equal to 0 as resin over here,"},{"Start":"25:18.380 ","End":"25:19.939","Text":"then we have parabolic,"},{"Start":"25:19.939 ","End":"25:22.260","Text":"but it\u0027s very similar."},{"Start":"25:23.110 ","End":"25:27.695","Text":"Now let\u0027s speak about when our E is smaller than 0."},{"Start":"25:27.695 ","End":"25:33.599","Text":"As we can see, our body can move when it\u0027s distance from"},{"Start":"25:33.599 ","End":"25:40.905","Text":"the origin is in this range between r_min over here and r_max over here."},{"Start":"25:40.905 ","End":"25:43.485","Text":"Let\u0027s take a look what that means."},{"Start":"25:43.485 ","End":"25:46.755","Text":"Over here, if we have our planet and here is our origin,"},{"Start":"25:46.755 ","End":"25:53.190","Text":"so our body can be some distance r_min away from the origin."},{"Start":"25:53.190 ","End":"25:55.635","Text":"This is our r_min."},{"Start":"25:55.635 ","End":"26:00.390","Text":"Then it can reach a distance of r_max away."},{"Start":"26:00.390 ","End":"26:06.225","Text":"This would be our r_max and then we can see"},{"Start":"26:06.225 ","End":"26:12.150","Text":"that because our range of radiuses is always between this,"},{"Start":"26:12.150 ","End":"26:16.498","Text":"because our, radius isn\u0027t approaching infinity,"},{"Start":"26:16.498 ","End":"26:21.840","Text":"it\u0027s not like when our E was bigger than 0 and then we had this huge range of infinity."},{"Start":"26:21.840 ","End":"26:24.960","Text":"Because we\u0027re always stuck in this range,"},{"Start":"26:24.960 ","End":"26:30.330","Text":"we know that we have some kind of closed circuit that our object has to be in."},{"Start":"26:30.330 ","End":"26:32.759","Text":"That means some circular circuit,"},{"Start":"26:32.759 ","End":"26:34.439","Text":"but because our radius is changing,"},{"Start":"26:34.439 ","End":"26:37.150","Text":"it\u0027s going to be an ellipse."},{"Start":"26:37.370 ","End":"26:40.724","Text":"When our E is smaller than 0,"},{"Start":"26:40.724 ","End":"26:43.920","Text":"we\u0027re always going to be located within a range of"},{"Start":"26:43.920 ","End":"26:50.549","Text":"radiuses that doesn\u0027t include an infinite range."},{"Start":"26:50.549 ","End":"26:55.769","Text":"Some discrete range, which means that we have motion and an ellipse,"},{"Start":"26:55.769 ","End":"26:58.560","Text":"and it will look something like this."},{"Start":"26:58.560 ","End":"27:03.299","Text":"Then our final option is when our energy is exactly"},{"Start":"27:03.299 ","End":"27:07.620","Text":"equal to our minimum point for our U effective."},{"Start":"27:07.620 ","End":"27:10.695","Text":"Our U effective minimum over here."},{"Start":"27:10.695 ","End":"27:11.985","Text":"Now, what does that mean?"},{"Start":"27:11.985 ","End":"27:14.084","Text":"It means that throughout the motion,"},{"Start":"27:14.084 ","End":"27:18.870","Text":"our radius can only be this value, our r_0."},{"Start":"27:18.870 ","End":"27:24.225","Text":"Of course, because our radius doesn\u0027t approach infinity,"},{"Start":"27:24.225 ","End":"27:29.295","Text":"over here, that means that we have a closed circuit."},{"Start":"27:29.295 ","End":"27:33.374","Text":"If we have a closed circuit where our radius can only be 1 value,"},{"Start":"27:33.374 ","End":"27:38.820","Text":"that means that our body is orbiting in a perfect circle."},{"Start":"27:38.820 ","End":"27:42.705","Text":"Again, here we have our planet and here is our origin,"},{"Start":"27:42.705 ","End":"27:48.870","Text":"so that means that our body is always going in a perfect circle around,"},{"Start":"27:48.870 ","End":"27:55.769","Text":"always at a distance of r_0 away from the origin of course."},{"Start":"27:55.769 ","End":"28:04.720","Text":"This is only when our energy is exactly equal to our U effective minimum points."},{"Start":"28:06.120 ","End":"28:11.544","Text":"A little note about this graph that I drew of the ellipse over here."},{"Start":"28:11.544 ","End":"28:15.775","Text":"I know that this distance looks smaller than my value for r_min."},{"Start":"28:15.775 ","End":"28:18.879","Text":"It\u0027s not meant to be, its just my drawing skills."},{"Start":"28:18.879 ","End":"28:24.414","Text":"Imagine that the ellipse goes more out like this."},{"Start":"28:24.414 ","End":"28:29.650","Text":"That this length over here is really the shortest length that we have."},{"Start":"28:29.650 ","End":"28:35.359","Text":"Now let\u0027s speak about our kinetic energy and then I\u0027m going to rub out these diagrams."},{"Start":"28:35.430 ","End":"28:38.724","Text":"Now we\u0027re speaking about our kinetic energy."},{"Start":"28:38.724 ","End":"28:44.500","Text":"Again, I\u0027m going to put this around to show that this isn\u0027t exactly our kinetic energy,"},{"Start":"28:44.500 ","End":"28:49.675","Text":"but we\u0027re just calling it that for the sake of convention and for ease."},{"Start":"28:49.675 ","End":"28:56.364","Text":"We know that that\u0027s going to be equal to our 1/2m r dot squared."},{"Start":"28:56.364 ","End":"28:58.480","Text":"Now let\u0027s take this,"},{"Start":"28:58.480 ","End":"29:05.200","Text":"our kinetic energy and see how we can work out what\u0027s going on with our body."},{"Start":"29:05.200 ","End":"29:08.860","Text":"What we can see from this equation over here is"},{"Start":"29:08.860 ","End":"29:12.070","Text":"that if we isolate out our kinetic energy,"},{"Start":"29:12.070 ","End":"29:14.830","Text":"we isolate out this term over here."},{"Start":"29:14.830 ","End":"29:18.564","Text":"I get that our kinetic energy is equal to the difference"},{"Start":"29:18.564 ","End":"29:23.839","Text":"between our total energy and our U effective."},{"Start":"29:23.910 ","End":"29:26.080","Text":"Let\u0027s take a look at the graph,"},{"Start":"29:26.080 ","End":"29:27.969","Text":"let\u0027s see what that means."},{"Start":"29:27.969 ","End":"29:33.025","Text":"Let\u0027s take our case where we\u0027re speaking about the motion of an ellipse."},{"Start":"29:33.025 ","End":"29:36.879","Text":"That means when our E is smaller than 0,"},{"Start":"29:36.879 ","End":"29:40.900","Text":"so that means that my radius is going to be somewhere in this region."},{"Start":"29:40.900 ","End":"29:44.930","Text":"Let\u0027s say that my radius is right over here."},{"Start":"29:45.990 ","End":"29:50.605","Text":"My radius is going to be here on my graph for my energy."},{"Start":"29:50.605 ","End":"29:57.775","Text":"As we can see, it\u0027s going to cut somewhere over here on my graph from my U effective."},{"Start":"29:57.775 ","End":"30:03.895","Text":"Then if I draw an arrow between these 2 points."},{"Start":"30:03.895 ","End":"30:06.805","Text":"This distance over here,"},{"Start":"30:06.805 ","End":"30:11.570","Text":"so this is my kinetic energy."},{"Start":"30:12.270 ","End":"30:16.120","Text":"My kinetic energy is the difference between"},{"Start":"30:16.120 ","End":"30:20.305","Text":"my total energy and my value for my U effective."},{"Start":"30:20.305 ","End":"30:23.034","Text":"We can see obviously right at the ends,"},{"Start":"30:23.034 ","End":"30:27.190","Text":"at the ends of my region where my body can be in motion."},{"Start":"30:27.190 ","End":"30:30.594","Text":"We can see that my kinetic energy will be equal to 0."},{"Start":"30:30.594 ","End":"30:37.435","Text":"Because the difference between my energy and my U effective is going to be equal to 0."},{"Start":"30:37.435 ","End":"30:40.720","Text":"These points on the edge over here where I\u0027m putting"},{"Start":"30:40.720 ","End":"30:45.444","Text":"my red dots are special points and we can calculate them."},{"Start":"30:45.444 ","End":"30:47.994","Text":"Now the way I can calculate these points,"},{"Start":"30:47.994 ","End":"30:50.335","Text":"these points are r _min and r_max,"},{"Start":"30:50.335 ","End":"31:00.265","Text":"is that I can say that my energy at these points is equal to my U effective at r_min."},{"Start":"31:00.265 ","End":"31:06.910","Text":"It\u0027s also equal to my U effective at r_max."},{"Start":"31:06.910 ","End":"31:10.585","Text":"At these points, and my points r_min and r_max,"},{"Start":"31:10.585 ","End":"31:14.770","Text":"my r dot over here is equal to 0,"},{"Start":"31:14.770 ","End":"31:21.475","Text":"which means that my total energy is simply equal to this, my U effective."},{"Start":"31:21.475 ","End":"31:24.250","Text":"This is another way that we can work out"},{"Start":"31:24.250 ","End":"31:26.650","Text":"what are our r_min and what are our r_max values are."},{"Start":"31:26.650 ","End":"31:32.695","Text":"I can say that if I know my value for my energy,"},{"Start":"31:32.695 ","End":"31:38.590","Text":"it can just equate it to my U effective and find these values."},{"Start":"31:38.590 ","End":"31:43.284","Text":"When we\u0027re dealing with some kind of question which is dealing with gravity."},{"Start":"31:43.284 ","End":"31:48.054","Text":"My U effective will be equal to l^2 divided by"},{"Start":"31:48.054 ","End":"31:53.349","Text":"2mr^2 minus Alpha divided by"},{"Start":"31:53.349 ","End":"31:55.749","Text":"r. We can see that every single thing in"},{"Start":"31:55.749 ","End":"32:00.260","Text":"this equation is a constant aside from my radius."},{"Start":"32:00.840 ","End":"32:03.309","Text":"In order to solve this,"},{"Start":"32:03.309 ","End":"32:06.790","Text":"I just have to rearrange and find out what my values for r are."},{"Start":"32:06.790 ","End":"32:11.424","Text":"Those will be corresponding to my r_min and r_max values."},{"Start":"32:11.424 ","End":"32:18.130","Text":"How I do this is that I\u0027m going to multiply everything by r^2."},{"Start":"32:18.130 ","End":"32:24.295","Text":"Then we\u0027ll get Er^2 is equal to"},{"Start":"32:24.295 ","End":"32:31.405","Text":"negative Alpha r plus l^2 divided by 2m."},{"Start":"32:31.405 ","End":"32:35.424","Text":"Then we have just a simple quadratic equation."},{"Start":"32:35.424 ","End":"32:40.225","Text":"I can move these 2 terms over to the other side of the equals sign."},{"Start":"32:40.225 ","End":"32:43.825","Text":"Then I can use my formula for solving a quadratic equation."},{"Start":"32:43.825 ","End":"32:47.320","Text":"Well, get my solutions r1 and r2,"},{"Start":"32:47.320 ","End":"32:53.125","Text":"where r1 and r2 will obviously correspond to my r_min and my r_max."},{"Start":"32:53.125 ","End":"32:59.095","Text":"Now the last thing we\u0027re going to speak about during this lesson is our r_0 over here."},{"Start":"32:59.095 ","End":"33:03.295","Text":"That\u0027s our radius when we\u0027re dealing with circular motion."},{"Start":"33:03.295 ","End":"33:06.025","Text":"When our body is moving in a perfect circle."},{"Start":"33:06.025 ","End":"33:15.445","Text":"Our r_0 is located at the minimum point of our potential effect of energy."},{"Start":"33:15.445 ","End":"33:16.795","Text":"How do we find it?"},{"Start":"33:16.795 ","End":"33:22.975","Text":"Because it\u0027s at the exact minimum points of our U effective graph."},{"Start":"33:22.975 ","End":"33:30.054","Text":"The way that we have to find it is by taking the derivative of our U effective."},{"Start":"33:30.054 ","End":"33:35.620","Text":"M as a function of r. Then we set it equal to 0."},{"Start":"33:35.620 ","End":"33:37.794","Text":"This will find our extremes."},{"Start":"33:37.794 ","End":"33:41.125","Text":"Here specifically our extreme will be our minimum points."},{"Start":"33:41.125 ","End":"33:47.124","Text":"Then what we\u0027ll find is that when we set this to 0 and we rearrange,"},{"Start":"33:47.124 ","End":"33:50.950","Text":"so we will find our value for r_0."},{"Start":"33:50.950 ","End":"33:56.589","Text":"Well, r_0 corresponds to circular motion."},{"Start":"33:56.589 ","End":"34:02.005","Text":"Now an important note here is to differentiate between the case of our r_0,"},{"Start":"34:02.005 ","End":"34:07.554","Text":"which is at minimum point of the U effective graph."},{"Start":"34:07.554 ","End":"34:09.565","Text":"The minimum point."},{"Start":"34:09.565 ","End":"34:12.940","Text":"Our value for r_min,"},{"Start":"34:12.940 ","End":"34:20.380","Text":"which is the minimum value that our radius can be in range of."},{"Start":"34:20.380 ","End":"34:22.617","Text":"What does that mean?"},{"Start":"34:22.617 ","End":"34:26.650","Text":"When our E is smaller than 0 and we\u0027re moving in an ellipse."},{"Start":"34:26.650 ","End":"34:31.750","Text":"So our radius can be in a certain range between some value,"},{"Start":"34:31.750 ","End":"34:36.160","Text":"which is our r_min, and some value which is our r_maximum."},{"Start":"34:36.160 ","End":"34:44.380","Text":"The smallest radius that can be in elliptical motion is going to be called our r_min."},{"Start":"34:44.380 ","End":"34:47.684","Text":"That\u0027s here right at the edge,"},{"Start":"34:47.684 ","End":"34:48.970","Text":"where the red dot is."},{"Start":"34:48.970 ","End":"34:52.149","Text":"You have to make a difference in your mind between"},{"Start":"34:52.149 ","End":"34:56.080","Text":"that value our r_min our value for r_0,"},{"Start":"34:56.080 ","End":"35:01.300","Text":"which is at the minimum point of energy of our U effective graph."},{"Start":"35:01.300 ","End":"35:03.880","Text":"That\u0027s that green point over here."},{"Start":"35:03.880 ","End":"35:08.035","Text":"That corresponds to circular motion,"},{"Start":"35:08.035 ","End":"35:10.944","Text":"when our motion isn\u0027t a perfect circle."},{"Start":"35:10.944 ","End":"35:14.740","Text":"Now for a mini conclusion of this lesson,"},{"Start":"35:14.740 ","End":"35:19.720","Text":"the first thing that we did is we express the energy of our system,"},{"Start":"35:19.720 ","End":"35:22.945","Text":"which is rotating about some large body."},{"Start":"35:22.945 ","End":"35:28.315","Text":"We expressed it with respect to simply one variable or one unknown,"},{"Start":"35:28.315 ","End":"35:33.114","Text":"which in this case was equal to r. Now the way that we did it is that we"},{"Start":"35:33.114 ","End":"35:38.095","Text":"wrote our velocity of the body in terms of its polar coordinates."},{"Start":"35:38.095 ","End":"35:44.229","Text":"We wrote that our value for Theta dot was in terms of our angular momentum."},{"Start":"35:44.229 ","End":"35:46.900","Text":"So then what we did is we plugged in all of"},{"Start":"35:46.900 ","End":"35:50.950","Text":"our values into our equation for energy over here."},{"Start":"35:50.950 ","End":"35:53.230","Text":"When we were done rearranging everything,"},{"Start":"35:53.230 ","End":"35:55.809","Text":"we were left with this equation over here,"},{"Start":"35:55.809 ","End":"35:59.289","Text":"which is the exact same equation for energy except only"},{"Start":"35:59.289 ","End":"36:03.490","Text":"expressed with our variable r and its first derivative."},{"Start":"36:03.490 ","End":"36:08.860","Text":"This expression, we said that this section over here was as if it\u0027s the kinetic energy,"},{"Start":"36:08.860 ","End":"36:12.430","Text":"which we said that we\u0027re putting these speech marks"},{"Start":"36:12.430 ","End":"36:13.629","Text":"around it because this isn\u0027t"},{"Start":"36:13.629 ","End":"36:16.855","Text":"exactly kinetic energy as it\u0027s to do with the potential energy."},{"Start":"36:16.855 ","End":"36:20.935","Text":"But for the sake of making this equation slightly easier."},{"Start":"36:20.935 ","End":"36:23.949","Text":"Then we said that the remaining part of"},{"Start":"36:23.949 ","End":"36:27.944","Text":"our energy equation was going to be our U effective."},{"Start":"36:27.944 ","End":"36:30.579","Text":"Then we looked at the case where we\u0027re speaking"},{"Start":"36:30.579 ","End":"36:34.944","Text":"about gravity and we\u0027re trying to find the energy,"},{"Start":"36:34.944 ","End":"36:40.435","Text":"and the shape of a rotating system when we\u0027re dealing with gravitational force."},{"Start":"36:40.435 ","End":"36:45.009","Text":"So we wrote our U effective for this case and we drew a graph,"},{"Start":"36:45.009 ","End":"36:49.885","Text":"showing our graph for our U effective and our corresponding energies,"},{"Start":"36:49.885 ","End":"36:52.434","Text":"which was this value over here."},{"Start":"36:52.434 ","End":"36:56.770","Text":"We had when our energy was greater than 0, less than 0,"},{"Start":"36:56.770 ","End":"37:00.690","Text":"and at the minimum point of our U effective graph,"},{"Start":"37:00.690 ","End":"37:02.984","Text":"tangential to the minimum points."},{"Start":"37:02.984 ","End":"37:07.020","Text":"Then we saw what these corresponding energy is"},{"Start":"37:07.020 ","End":"37:11.830","Text":"meant in terms of the shape of the orbital trajectory."},{"Start":"37:11.830 ","End":"37:14.800","Text":"That\u0027s what we spoke about So that through energy,"},{"Start":"37:14.800 ","End":"37:20.905","Text":"we could find out the shape that our body was orbiting in, the trajectory."},{"Start":"37:20.905 ","End":"37:25.749","Text":"Also, we discussed how we could find our values in the case of an ellipse,"},{"Start":"37:25.749 ","End":"37:29.275","Text":"for instance, for our r_min and our r_max."},{"Start":"37:29.275 ","End":"37:34.520","Text":"So we can really tell a lot of information by our energy equation."},{"Start":"37:34.520 ","End":"37:39.570","Text":"Lastly, we spoke about when we\u0027re dealing with perfect circular motion,"},{"Start":"37:39.570 ","End":"37:43.754","Text":"that we have a single radius which is r_0 and how to find it."},{"Start":"37:43.754 ","End":"37:46.410","Text":"That\u0027s by taking the first derivative of our U"},{"Start":"37:46.410 ","End":"37:50.384","Text":"effective of this and setting it equal to 0."},{"Start":"37:50.384 ","End":"37:53.070","Text":"Then we\u0027ll find out what our r_0 is."},{"Start":"37:53.070 ","End":"37:55.630","Text":"That\u0027s the end of our lesson."}],"ID":9461},{"Watched":false,"Name":"Effective Potential Question","Duration":"9m 49s","ChapterTopicVideoID":9192,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.840","Text":"Hello. In this question,"},{"Start":"00:01.840 ","End":"00:05.050","Text":"we\u0027re being told that a body of mass m moves in"},{"Start":"00:05.050 ","End":"00:08.830","Text":"a circular motion with the potential of U as a function of r,"},{"Start":"00:08.830 ","End":"00:13.690","Text":"which is equal to a negative A divided by the square root of r,"},{"Start":"00:13.690 ","End":"00:16.315","Text":"where A is some known constant."},{"Start":"00:16.315 ","End":"00:18.760","Text":"We\u0027re also being told that the angular momentum of"},{"Start":"00:18.760 ","End":"00:22.175","Text":"the body is given by L. In the first question,"},{"Start":"00:22.175 ","End":"00:24.790","Text":"we\u0027re being asked, what is the radius of the circle?"},{"Start":"00:24.790 ","End":"00:26.395","Text":"Then in the second question,"},{"Start":"00:26.395 ","End":"00:28.690","Text":"what is the velocity of the body?"},{"Start":"00:28.690 ","End":"00:32.200","Text":"Let\u0027s start off with the first question."},{"Start":"00:32.200 ","End":"00:37.390","Text":"The first thing that we\u0027re going to use is our idea of energy."},{"Start":"00:37.390 ","End":"00:40.330","Text":"Let\u0027s write out our energy equation."},{"Start":"00:40.330 ","End":"00:43.650","Text":"It\u0027s going to be equal to our kinetic energy,"},{"Start":"00:43.650 ","End":"00:46.785","Text":"which is 1/2 m multiplied by v^2."},{"Start":"00:46.785 ","End":"00:51.315","Text":"Now v^2 because we\u0027re given r and we\u0027re being told to find r,"},{"Start":"00:51.315 ","End":"00:52.710","Text":"in fact, that\u0027s our unknown,"},{"Start":"00:52.710 ","End":"00:54.480","Text":"so instead of writing v^2,"},{"Start":"00:54.480 ","End":"00:56.670","Text":"I\u0027ll write r.^2,"},{"Start":"00:56.670 ","End":"00:59.480","Text":"which as we know is the first derivative of r,"},{"Start":"00:59.480 ","End":"01:03.820","Text":"which is its velocity, and then plus our potential energy."},{"Start":"01:03.820 ","End":"01:05.630","Text":"Now our potential energy,"},{"Start":"01:05.630 ","End":"01:09.245","Text":"we\u0027re going to be using our effective potential energy."},{"Start":"01:09.245 ","End":"01:11.420","Text":"Because when we\u0027re dealing with circular motion,"},{"Start":"01:11.420 ","End":"01:16.018","Text":"if you remember in the second look at our equation for it,"},{"Start":"01:16.018 ","End":"01:19.505","Text":"but it also uses the idea of our radius."},{"Start":"01:19.505 ","End":"01:23.210","Text":"Our effective potential energy is a function of"},{"Start":"01:23.210 ","End":"01:29.690","Text":"r. Now let\u0027s take a look at what our U effective is equal to."},{"Start":"01:29.690 ","End":"01:33.500","Text":"It\u0027s the function of r. It\u0027s going to be equal to"},{"Start":"01:33.500 ","End":"01:39.800","Text":"our angular momentum squared divided by 2mr^2,"},{"Start":"01:39.800 ","End":"01:43.760","Text":"and then plus our potential energy over here,"},{"Start":"01:43.760 ","End":"01:46.985","Text":"so that\u0027s going to be our Ur."},{"Start":"01:46.985 ","End":"01:57.675","Text":"Then the whole thing is going to be equal to L^2 divided by 2mr^2 minus"},{"Start":"01:57.675 ","End":"02:04.595","Text":"A divided by the square root of r. Then we can just simplify it a little"},{"Start":"02:04.595 ","End":"02:12.075","Text":"bit by writing L^2 divided by 2m and moving the r from the denominator to over here."},{"Start":"02:12.075 ","End":"02:15.285","Text":"It\u0027d be the r^negative 2,"},{"Start":"02:15.285 ","End":"02:17.607","Text":"and then negative A,"},{"Start":"02:17.607 ","End":"02:25.160","Text":"and then instead of having divided by the square root of r, wrote r^negative 1/2."},{"Start":"02:25.160 ","End":"02:28.860","Text":"It\u0027s the exact same thing just written slightly differently."},{"Start":"02:29.390 ","End":"02:34.080","Text":"The idea, just like we saw in 1 of our previous lessons,"},{"Start":"02:34.080 ","End":"02:38.225","Text":"if a body is moving in circular motion,"},{"Start":"02:38.225 ","End":"02:42.235","Text":"then that means that its radius is a constant."},{"Start":"02:42.235 ","End":"02:47.200","Text":"It\u0027s located at the minimum of our effective potential."},{"Start":"02:47.200 ","End":"02:49.100","Text":"How do we find that?"},{"Start":"02:49.100 ","End":"02:52.940","Text":"We know from calculus that if I just take the derivative of"},{"Start":"02:52.940 ","End":"02:59.006","Text":"my effective potential and then I set that equal to 0,"},{"Start":"02:59.006 ","End":"03:01.930","Text":"then I can find that minimum point."},{"Start":"03:01.930 ","End":"03:06.810","Text":"Let\u0027s do that. Our effective potential,"},{"Start":"03:06.810 ","End":"03:08.765","Text":"so when we take the derivative of it,"},{"Start":"03:08.765 ","End":"03:15.205","Text":"it\u0027s going to be equal to L^2 divided by 2m as a constant."},{"Start":"03:15.205 ","End":"03:20.155","Text":"But then when we take the derivative of r^negative 2,"},{"Start":"03:20.155 ","End":"03:22.725","Text":"our 2 will cancel out."},{"Start":"03:22.725 ","End":"03:27.485","Text":"We\u0027ll be left here with a minimum of m and the r^negative 3."},{"Start":"03:27.485 ","End":"03:31.565","Text":"Then we have our Ar^negative 1/2."},{"Start":"03:31.565 ","End":"03:35.960","Text":"When we take the derivative of r,"},{"Start":"03:35.960 ","End":"03:37.130","Text":"we\u0027ll have a minus and a minus,"},{"Start":"03:37.130 ","End":"03:39.920","Text":"which is a plus, multiplied by our A,"},{"Start":"03:39.920 ","End":"03:43.640","Text":"and then we\u0027ll have r^negative 3 over 2."},{"Start":"03:43.640 ","End":"03:48.705","Text":"Then we will have that divided by 2 over here."},{"Start":"03:48.705 ","End":"03:50.690","Text":"Then as we said, in order to find the minimum,"},{"Start":"03:50.690 ","End":"03:52.990","Text":"we set this equal to 0."},{"Start":"03:52.990 ","End":"03:55.250","Text":"Now by simple algebra,"},{"Start":"03:55.250 ","End":"03:57.290","Text":"we can rearrange this equation,"},{"Start":"03:57.290 ","End":"04:03.705","Text":"and what we\u0027ll get is that A divided by 2r^negative 3 over"},{"Start":"04:03.705 ","End":"04:11.460","Text":"2 is going to be equal to L^2 divided by m r^negative 3."},{"Start":"04:11.460 ","End":"04:18.215","Text":"Then when we multiply both sides by 2 divided by A multiplied by r^3,"},{"Start":"04:18.215 ","End":"04:27.570","Text":"we will get that our r^3 over 2 is equal to 2L^2 divided by mA."},{"Start":"04:27.570 ","End":"04:33.310","Text":"Then we can just get rid of this to the power of 3 over 2,"},{"Start":"04:33.310 ","End":"04:39.770","Text":"so that means squaring both sides and taking the cube root of both sides."},{"Start":"04:39.770 ","End":"04:41.150","Text":"Then we\u0027ll get that our r,"},{"Start":"04:41.150 ","End":"04:44.375","Text":"that we\u0027re trying to find the radius of the circle, so let\u0027s call r_0,"},{"Start":"04:44.375 ","End":"04:47.325","Text":"to show that, because this r is an unknown,"},{"Start":"04:47.325 ","End":"04:50.315","Text":"but r_0 is what we\u0027re trying to find."},{"Start":"04:50.315 ","End":"04:55.490","Text":"It\u0027s going to simply be equal to 2L^2 divided by mA,"},{"Start":"04:55.490 ","End":"05:01.070","Text":"and then to the power of 2 divided by 3."},{"Start":"05:01.070 ","End":"05:04.650","Text":"Now just before we go on to answering Question number 2,"},{"Start":"05:04.650 ","End":"05:08.135","Text":"the idea was that a body was moving in a circular motion,"},{"Start":"05:08.135 ","End":"05:10.685","Text":"which means that in order to find the radius,"},{"Start":"05:10.685 ","End":"05:15.115","Text":"we have to find the minimum value for our effective potential."},{"Start":"05:15.115 ","End":"05:17.570","Text":"How we do that is we take the derivative of"},{"Start":"05:17.570 ","End":"05:20.750","Text":"our effective potential and set that equal to 0 and then do"},{"Start":"05:20.750 ","End":"05:26.105","Text":"some basic algebraic processes in order to inside out r,"},{"Start":"05:26.105 ","End":"05:28.180","Text":"which is what we\u0027re trying to find."},{"Start":"05:28.180 ","End":"05:36.370","Text":"Now briefly right here what that exactly means about our effective potential?"},{"Start":"05:36.370 ","End":"05:41.880","Text":"If we have some kind of function and this is"},{"Start":"05:41.880 ","End":"05:48.105","Text":"our U effective as a function of r. This is some kind of function."},{"Start":"05:48.105 ","End":"05:52.300","Text":"Then here we have our axis."},{"Start":"05:52.430 ","End":"06:02.130","Text":"That means that our body is, lets say,"},{"Start":"06:02.130 ","End":"06:04.170","Text":"located something like this,"},{"Start":"06:04.170 ","End":"06:10.375","Text":"the energy of the body is higher than the minimum energy,"},{"Start":"06:10.375 ","End":"06:12.125","Text":"as we can see over here."},{"Start":"06:12.125 ","End":"06:14.025","Text":"In this case,"},{"Start":"06:14.025 ","End":"06:16.994","Text":"then our body can carry on moving."},{"Start":"06:16.994 ","End":"06:21.050","Text":"Only when the energy is located at"},{"Start":"06:21.050 ","End":"06:27.300","Text":"this minimum point then our body will be moving in circular motion."},{"Start":"06:27.300 ","End":"06:36.260","Text":"Only when our energy is equal to our potential energy at the minimum point,"},{"Start":"06:36.260 ","End":"06:39.430","Text":"then our object will be moving in circular motion."},{"Start":"06:39.430 ","End":"06:42.560","Text":"Or in other words, like in this specific question,"},{"Start":"06:42.560 ","End":"06:46.025","Text":"we were being told that it\u0027s moving in a circular motion,"},{"Start":"06:46.025 ","End":"06:50.210","Text":"which means that our energy has to be"},{"Start":"06:50.210 ","End":"06:54.935","Text":"at this minimum point in order for it to be moving in a circular motion."},{"Start":"06:54.935 ","End":"06:56.945","Text":"We know that it\u0027s this minimum point,"},{"Start":"06:56.945 ","End":"06:58.955","Text":"we just have to find that minimum point."},{"Start":"06:58.955 ","End":"07:04.950","Text":"How do we find it? By taking the derivative of our U effective and setting it equal to 0."},{"Start":"07:04.950 ","End":"07:07.740","Text":"Now let\u0027s go on to Question number 2."},{"Start":"07:07.740 ","End":"07:11.450","Text":"We\u0027re being told to find what is the velocity of the body,"},{"Start":"07:11.450 ","End":"07:14.938","Text":"and we know what the angular momentum of the body is,"},{"Start":"07:14.938 ","End":"07:18.815","Text":"and it\u0027s equal to L. Using our equation for angular momentum,"},{"Start":"07:18.815 ","End":"07:22.925","Text":"I have that my angular momentum is equal to and then simply the equation,"},{"Start":"07:22.925 ","End":"07:26.405","Text":"the mass multiplied by the velocity multiplied by the radius,"},{"Start":"07:26.405 ","End":"07:28.615","Text":"which we found in Question 1."},{"Start":"07:28.615 ","End":"07:31.040","Text":"Now, just as a reminder,"},{"Start":"07:31.040 ","End":"07:36.590","Text":"when we\u0027re dealing with angular momentum when working in a circular motion,"},{"Start":"07:36.590 ","End":"07:38.630","Text":"imagine this is a perfect circle,"},{"Start":"07:38.630 ","End":"07:41.150","Text":"we know that if we have a radius of the circle,"},{"Start":"07:41.150 ","End":"07:47.580","Text":"our velocity vector is going to be at exactly 90 degrees to our radius."},{"Start":"07:47.580 ","End":"07:51.110","Text":"In actual facts, if you remember the equation for angular momentum,"},{"Start":"07:51.110 ","End":"07:54.380","Text":"it\u0027s mass times velocity times radius multiplied by"},{"Start":"07:54.380 ","End":"07:57.995","Text":"its sine of the angle between the velocity and the radius vectors."},{"Start":"07:57.995 ","End":"08:01.310","Text":"Now here because the angle in a circle is always"},{"Start":"08:01.310 ","End":"08:04.580","Text":"going to be exactly 90 degrees between r and v,"},{"Start":"08:04.580 ","End":"08:07.655","Text":"so sine of 90 is simply equal to 1."},{"Start":"08:07.655 ","End":"08:10.495","Text":"That\u0027s where we get that from."},{"Start":"08:10.495 ","End":"08:13.055","Text":"Then all we have to do is we have to"},{"Start":"08:13.055 ","End":"08:15.860","Text":"isolate out our v because that\u0027s what we want to find."},{"Start":"08:15.860 ","End":"08:20.725","Text":"Have that our v is equal to our L divided by our mr_0."},{"Start":"08:20.725 ","End":"08:23.420","Text":"Then we know what r_0 is."},{"Start":"08:23.420 ","End":"08:26.705","Text":"It\u0027s going to be L divided by m,"},{"Start":"08:26.705 ","End":"08:34.290","Text":"and then our r_0 is 2L^2 divided by mA^2/3."},{"Start":"08:34.430 ","End":"08:37.035","Text":"That\u0027s the end of the question,"},{"Start":"08:37.035 ","End":"08:41.380","Text":"but another important thing that is useful to know is what if we want to"},{"Start":"08:41.380 ","End":"08:46.550","Text":"find the force acting on the body."},{"Start":"08:46.550 ","End":"08:55.205","Text":"Our force is going to be equal to negative the gradient of our potential energy."},{"Start":"08:55.205 ","End":"08:57.920","Text":"Now in this case specifically,"},{"Start":"08:57.920 ","End":"09:02.155","Text":"because our potential is only as a function of r,"},{"Start":"09:02.155 ","End":"09:09.500","Text":"this negative brad of U will simply be equal to negative d by"},{"Start":"09:09.500 ","End":"09:17.095","Text":"dr of our U function just because specifically here it\u0027s only dependent on r,"},{"Start":"09:17.095 ","End":"09:21.675","Text":"and of course, this would be going in the radial direction."},{"Start":"09:21.675 ","End":"09:27.332","Text":"Then if we take the derivative of this with respect to r,,"},{"Start":"09:27.332 ","End":"09:35.110","Text":"we\u0027ll get that our force acting will be negative Ar^negative 1/2."},{"Start":"09:35.110 ","End":"09:39.650","Text":"This will be the force acting on the body in the radial direction."},{"Start":"09:39.650 ","End":"09:42.875","Text":"What this means, I\u0027m not going to explain it right now,"},{"Start":"09:42.875 ","End":"09:47.420","Text":"but it\u0027s useful to know just in case you have a Question that is also asking this."},{"Start":"09:47.420 ","End":"09:50.640","Text":"That\u0027s the end of this lesson."}],"ID":9462},{"Watched":false,"Name":"Position As A Function Of Time And Period","Duration":"10m 24s","ChapterTopicVideoID":9193,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.935","Text":"Hello. In this lesson,"},{"Start":"00:01.935 ","End":"00:04.874","Text":"we\u0027re going to be speaking about the relationship between r,"},{"Start":"00:04.874 ","End":"00:06.645","Text":"our radius, and time."},{"Start":"00:06.645 ","End":"00:11.442","Text":"We\u0027ll also be speaking about how to find the time for 1 period,"},{"Start":"00:11.442 ","End":"00:13.620","Text":"specifically in the case of an ellipse,"},{"Start":"00:13.620 ","End":"00:16.410","Text":"but also in a more general case."},{"Start":"00:16.410 ","End":"00:19.575","Text":"Just as a reminder, up until now,"},{"Start":"00:19.575 ","End":"00:23.704","Text":"we\u0027ve been working out our radius as a function of Theta,"},{"Start":"00:23.704 ","End":"00:26.735","Text":"if you remember this equation from before."},{"Start":"00:26.735 ","End":"00:31.850","Text":"What we\u0027re going to do now is we\u0027re going to work in a more general fashion."},{"Start":"00:31.850 ","End":"00:35.255","Text":"We\u0027re going to start with our energy equation."},{"Start":"00:35.255 ","End":"00:44.585","Text":"As we know, it\u0027s going to be equal to 1/2 mr dot squared plus our u effective,"},{"Start":"00:44.585 ","End":"00:46.850","Text":"which is as a function of r,"},{"Start":"00:46.850 ","End":"00:51.860","Text":"like we saw in our lesson about r effective potential energy."},{"Start":"00:51.860 ","End":"00:55.430","Text":"Where our u effective as a function of"},{"Start":"00:55.430 ","End":"01:00.050","Text":"r was equal to when we\u0027re dealing with a central force,"},{"Start":"01:00.050 ","End":"01:05.089","Text":"L^2 divided by 2mr^2,"},{"Start":"01:05.089 ","End":"01:08.045","Text":"and this comes from the kinetic energy plus"},{"Start":"01:08.045 ","End":"01:12.479","Text":"some potential energy if it\u0027s due to gravity or whatever it might be."},{"Start":"01:12.479 ","End":"01:17.075","Text":"This ur is still some general potential energy."},{"Start":"01:17.075 ","End":"01:19.850","Text":"Usually in these types of questions we\u0027re going to be dealing"},{"Start":"01:19.850 ","End":"01:24.860","Text":"with our potential energy from a central force such as gravity."},{"Start":"01:24.860 ","End":"01:29.220","Text":"But really this refers to any central energy that we might have."},{"Start":"01:29.220 ","End":"01:33.230","Text":"As I said, the goal of the lesson was to find the relationship of r"},{"Start":"01:33.230 ","End":"01:36.995","Text":"as a function of time rather than as a function of Theta,"},{"Start":"01:36.995 ","End":"01:39.305","Text":"which is what we\u0027ve seen up until now."},{"Start":"01:39.305 ","End":"01:44.660","Text":"What I\u0027m going to do therefore is I\u0027m going to try and isolate out my r over here."},{"Start":"01:44.660 ","End":"01:49.980","Text":"From simple algebra, I\u0027ll get that my r dot is equal"},{"Start":"01:49.980 ","End":"01:56.165","Text":"to and then we\u0027ll have our energy minus our u effective,"},{"Start":"01:56.165 ","End":"02:01.250","Text":"which is as a function of r. Then this is going to"},{"Start":"02:01.250 ","End":"02:06.635","Text":"be multiplied by 2 divided by m from here and then,"},{"Start":"02:06.635 ","End":"02:08.645","Text":"because we have r dot squared,"},{"Start":"02:08.645 ","End":"02:13.075","Text":"so our final thing will be to take the square root of all of that."},{"Start":"02:13.075 ","End":"02:18.780","Text":"From our simple algebra of isolating out our r dot,"},{"Start":"02:18.780 ","End":"02:21.650","Text":"plus minus square root of E minus our u"},{"Start":"02:21.650 ","End":"02:27.440","Text":"effective multiplied by 2 divided by m square root of all of that."},{"Start":"02:27.440 ","End":"02:29.735","Text":"Now, we have our r dot over here."},{"Start":"02:29.735 ","End":"02:33.175","Text":"What I\u0027m actually going to do is I\u0027m going to try and solve this."},{"Start":"02:33.175 ","End":"02:35.990","Text":"This is done in the same way as solving"},{"Start":"02:35.990 ","End":"02:40.885","Text":"a differential equation because this is in fact just that differential equation."},{"Start":"02:40.885 ","End":"02:43.125","Text":"R dot, as we know,"},{"Start":"02:43.125 ","End":"02:48.750","Text":"is our dr by dt and that\u0027s going to equal this."},{"Start":"02:48.750 ","End":"02:53.045","Text":"Plus minus the square root of 2 divided by m,"},{"Start":"02:53.045 ","End":"02:56.720","Text":"E minus u effective,"},{"Start":"02:56.720 ","End":"03:01.050","Text":"which is a function of r, square root."},{"Start":"03:01.550 ","End":"03:06.200","Text":"Now, what we\u0027re going to do is we\u0027re going to separate this out."},{"Start":"03:06.200 ","End":"03:09.770","Text":"We\u0027re going to multiply both sides by our dt and"},{"Start":"03:09.770 ","End":"03:13.595","Text":"divide both sides by what\u0027s going on over here,"},{"Start":"03:13.595 ","End":"03:16.195","Text":"and then the square root."},{"Start":"03:16.195 ","End":"03:20.270","Text":"What we\u0027re going to be left with is plus or"},{"Start":"03:20.270 ","End":"03:25.540","Text":"minus and then we\u0027re going to have 2 divided by m,"},{"Start":"03:25.540 ","End":"03:36.245","Text":"E minus u effective as a function of r. This is going to be the power of negative 1/2,"},{"Start":"03:36.245 ","End":"03:38.960","Text":"because here the expression is to the power of a half."},{"Start":"03:38.960 ","End":"03:40.985","Text":"When we divide both sides,"},{"Start":"03:40.985 ","End":"03:42.855","Text":"that it\u0027s like negative 1/2,"},{"Start":"03:42.855 ","End":"03:49.750","Text":"this is going to be multiplied by dr and this is going to be equal to dt."},{"Start":"03:50.150 ","End":"03:57.994","Text":"Now, we\u0027re going to add in our integration sign and our borders for integration."},{"Start":"03:57.994 ","End":"04:00.750","Text":"So r is going to be going from our r_0"},{"Start":"04:00.750 ","End":"04:06.805","Text":"to r as a function of time and I\u0027m reminding you this is what we\u0027re trying to find."},{"Start":"04:06.805 ","End":"04:11.670","Text":"Rt is going to be going from some t_0 until a"},{"Start":"04:11.670 ","End":"04:16.070","Text":"general t. Usually when we do this integration,"},{"Start":"04:16.070 ","End":"04:18.170","Text":"our t_0, our starting time,"},{"Start":"04:18.170 ","End":"04:19.700","Text":"is usually going to be 0."},{"Start":"04:19.700 ","End":"04:25.270","Text":"Our r_0 corresponds to a radius at time t equals 0."},{"Start":"04:25.270 ","End":"04:30.020","Text":"Then we\u0027re going until some general t and finding the radius at that general time"},{"Start":"04:30.020 ","End":"04:36.950","Text":"t. This is the general integral that we will get for any type of central force."},{"Start":"04:36.950 ","End":"04:39.350","Text":"Now, if we\u0027re specifically going to be dealing"},{"Start":"04:39.350 ","End":"04:42.380","Text":"with gravitational forces as our central for us,"},{"Start":"04:42.380 ","End":"04:46.050","Text":"then we know that for gravity,"},{"Start":"04:46.050 ","End":"04:53.820","Text":"our u effective as a function of r is going to be equal to L^2"},{"Start":"04:53.820 ","End":"05:02.180","Text":"divided by 2mr^2 negative Alpha divided by r. Now,"},{"Start":"05:02.180 ","End":"05:06.095","Text":"what we can do is we can substitute this into our equation."},{"Start":"05:06.095 ","End":"05:11.135","Text":"We\u0027re going to have our integral sign plus minus."},{"Start":"05:11.135 ","End":"05:17.265","Text":"Then we have 2 divided by m multiplied by E negative our u effective."},{"Start":"05:17.265 ","End":"05:24.620","Text":"Negative L^2 divided by 2mr^2 and then negative and negative,"},{"Start":"05:24.620 ","End":"05:25.985","Text":"so that\u0027s a positive,"},{"Start":"05:25.985 ","End":"05:29.210","Text":"Alpha divided by r. Then to the power of"},{"Start":"05:29.210 ","End":"05:37.793","Text":"negative 1/2 dr and then we\u0027re going to have our integral dt as well."},{"Start":"05:37.793 ","End":"05:42.995","Text":"Then once we set in our borders of r_0 and rt over here,"},{"Start":"05:42.995 ","End":"05:49.055","Text":"and from t_0 is most commonly equal to 0 until some time t,"},{"Start":"05:49.055 ","End":"05:51.320","Text":"we can find a value for rt."},{"Start":"05:51.320 ","End":"05:54.280","Text":"We just have to integrate and then we\u0027ll find that value."},{"Start":"05:54.280 ","End":"05:55.970","Text":"If you want to practice that,"},{"Start":"05:55.970 ","End":"05:58.670","Text":"you can do that right now in a piece of paper."},{"Start":"05:58.670 ","End":"06:00.545","Text":"What I do want to show you, however,"},{"Start":"06:00.545 ","End":"06:02.960","Text":"is not how to solve this integral,"},{"Start":"06:02.960 ","End":"06:05.240","Text":"and that\u0027s how we will find rt,"},{"Start":"06:05.240 ","End":"06:09.835","Text":"but rather how to find the time taken for 1 period."},{"Start":"06:09.835 ","End":"06:13.105","Text":"When we\u0027re dealing with our u effective,"},{"Start":"06:13.105 ","End":"06:15.935","Text":"which has relating specifically to gravity,"},{"Start":"06:15.935 ","End":"06:20.285","Text":"then we know that our orbit is going to be some ellipse."},{"Start":"06:20.285 ","End":"06:26.855","Text":"That means that the range that our radiuses or radii will be,"},{"Start":"06:26.855 ","End":"06:35.370","Text":"will be between some value for r_min and a value for r_max."},{"Start":"06:35.420 ","End":"06:43.260","Text":"Then what I meant to get once I do that integral is 1/2 period."},{"Start":"06:43.670 ","End":"06:49.125","Text":"We\u0027ll get some value from 0 until t divided by 2,"},{"Start":"06:49.125 ","End":"06:52.700","Text":"which t is the period and we\u0027ll have,"},{"Start":"06:52.700 ","End":"06:57.955","Text":"at the end, this integral will be equal to t divided by 2."},{"Start":"06:57.955 ","End":"07:00.270","Text":"Let\u0027s see why this is."},{"Start":"07:00.270 ","End":"07:03.725","Text":"I\u0027m reminding you that when we\u0027re dealing with gravity,"},{"Start":"07:03.725 ","End":"07:06.245","Text":"then that means that we\u0027re moving in an ellipse."},{"Start":"07:06.245 ","End":"07:10.670","Text":"When we\u0027re moving in the trajectory of an ellipse,"},{"Start":"07:10.670 ","End":"07:13.655","Text":"that means that our energy is some negative value,"},{"Start":"07:13.655 ","End":"07:15.740","Text":"it\u0027s smaller than 0."},{"Start":"07:15.740 ","End":"07:22.736","Text":"As we said, we\u0027re integrating between a range of r_min and r_max."},{"Start":"07:22.736 ","End":"07:26.590","Text":"This has meant to be r_min and this r_max."},{"Start":"07:26.590 ","End":"07:30.320","Text":"I don\u0027t know if you can see the subscript over here, but it\u0027s opposite."},{"Start":"07:30.320 ","End":"07:32.960","Text":"If this is an r_min and we\u0027re"},{"Start":"07:32.960 ","End":"07:36.740","Text":"integrating all along this until we reach this point over here,"},{"Start":"07:36.740 ","End":"07:39.072","Text":"which is our point of r_max,"},{"Start":"07:39.072 ","End":"07:42.695","Text":"as we can see, we\u0027ve only done 1/2 of a rotation."},{"Start":"07:42.695 ","End":"07:46.040","Text":"We haven\u0027t integrated along this section,"},{"Start":"07:46.040 ","End":"07:48.530","Text":"the bottom half, so to speak,"},{"Start":"07:48.530 ","End":"07:51.988","Text":"which means that we\u0027ve done exactly 1/2 of a period."},{"Start":"07:51.988 ","End":"07:58.580","Text":"I\u0027m reminding you that one period is from our starting point all the way back to here."},{"Start":"07:58.580 ","End":"08:01.410","Text":"Here we\u0027ve only done halfway."},{"Start":"08:01.690 ","End":"08:07.020","Text":"I hear you asking, then why am I integrating from r_min"},{"Start":"08:07.020 ","End":"08:11.620","Text":"until r_max and not doing a fuller integration like this?"},{"Start":"08:11.620 ","End":"08:16.510","Text":"The reason I\u0027m doing that is to get rid of this plus minus over here."},{"Start":"08:16.510 ","End":"08:22.520","Text":"I know that my integration from my r_min until my r_max,"},{"Start":"08:22.520 ","End":"08:28.318","Text":"my value for r dot over here is some positive value"},{"Start":"08:28.318 ","End":"08:31.310","Text":"because this whole integration"},{"Start":"08:31.310 ","End":"08:35.300","Text":"comes from r dot and we can see that it has a value of plus or minus."},{"Start":"08:35.300 ","End":"08:38.510","Text":"I know that from my r_min until my r_max,"},{"Start":"08:38.510 ","End":"08:42.320","Text":"my r dot is just this positive value."},{"Start":"08:42.320 ","End":"08:46.280","Text":"That means that when I integrate along that I can get"},{"Start":"08:46.280 ","End":"08:50.735","Text":"rid of this minus and only focus on this plus over here."},{"Start":"08:50.735 ","End":"08:56.360","Text":"If you wanted to instead find just t instead of t divided by 2,"},{"Start":"08:56.360 ","End":"09:00.380","Text":"you\u0027d have to integrate from r_min until r_max on the"},{"Start":"09:00.380 ","End":"09:07.870","Text":"positive and then from r_max until r_min on the negative."},{"Start":"09:08.230 ","End":"09:16.040","Text":"Both integrals will work out to the same value but then instead of getting 1/2 a period,"},{"Start":"09:16.040 ","End":"09:19.725","Text":"you\u0027ll just get your full period t over here."},{"Start":"09:19.725 ","End":"09:22.354","Text":"I\u0027m not going to work out this integral."},{"Start":"09:22.354 ","End":"09:25.250","Text":"It\u0027s also slightly complicated."},{"Start":"09:25.250 ","End":"09:28.940","Text":"Usually if you\u0027ll get this type of question in an exam,"},{"Start":"09:28.940 ","End":"09:34.280","Text":"you will have a slightly easier function to integrate on easier than this."},{"Start":"09:34.280 ","End":"09:36.965","Text":"For instance, instead of having Alpha divided by r,"},{"Start":"09:36.965 ","End":"09:42.030","Text":"you\u0027ll maybe have something like Alpha divided by r^2."},{"Start":"09:42.730 ","End":"09:45.070","Text":"That\u0027s the end of this lesson."},{"Start":"09:45.070 ","End":"09:51.367","Text":"The general idea was to show you how you can find your radius as a function of time,"},{"Start":"09:51.367 ","End":"09:58.135","Text":"which meant just simply isolating out our r dot and then solving a differential equation,"},{"Start":"09:58.135 ","End":"10:03.505","Text":"getting a result with your rt somewhere in the equation and then just isolating your rt."},{"Start":"10:03.505 ","End":"10:05.650","Text":"You can also substitute in,"},{"Start":"10:05.650 ","End":"10:08.724","Text":"in order to find your period or half-year period."},{"Start":"10:08.724 ","End":"10:10.720","Text":"The borders for your radius,"},{"Start":"10:10.720 ","End":"10:13.217","Text":"for something that you know is 1/2 of your periods,"},{"Start":"10:13.217 ","End":"10:15.875","Text":"so here from r_min until r_max,"},{"Start":"10:15.875 ","End":"10:18.855","Text":"1/2 a circle or an ellipse."},{"Start":"10:18.855 ","End":"10:22.290","Text":"Then you integrate and you\u0027ll find 1/2 the period."},{"Start":"10:22.290 ","End":"10:25.660","Text":"That\u0027s the end of this lesson."}],"ID":9463},{"Watched":false,"Name":"Time Period","Duration":"16m 23s","ChapterTopicVideoID":9194,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"Hello. In this question,"},{"Start":"00:01.920 ","End":"00:07.080","Text":"we\u0027re being told that a body of mass m moves on a straight line with a potential of"},{"Start":"00:07.080 ","End":"00:12.820","Text":"U_x being equal to B multiplied by the absolute value of x,"},{"Start":"00:12.820 ","End":"00:15.065","Text":"where B is some constant."},{"Start":"00:15.065 ","End":"00:19.430","Text":"We\u0027re being told that the greatest distance reached by the body is A."},{"Start":"00:19.430 ","End":"00:23.650","Text":"If we\u0027re moving along some straight line."},{"Start":"00:23.650 ","End":"00:25.565","Text":"Imagine that this is straight."},{"Start":"00:25.565 ","End":"00:27.530","Text":"If we start at some position,"},{"Start":"00:27.530 ","End":"00:29.390","Text":"let\u0027s say at x=0,"},{"Start":"00:29.390 ","End":"00:35.230","Text":"the greatest distance that our body will reach is this point here, where x=A."},{"Start":"00:35.230 ","End":"00:38.795","Text":"Now our first question is to find the general value"},{"Start":"00:38.795 ","End":"00:42.350","Text":"for the energy of the body and our second question is,"},{"Start":"00:42.350 ","End":"00:44.885","Text":"what is the period?"},{"Start":"00:44.885 ","End":"00:48.105","Text":"Before we start to answer,"},{"Start":"00:48.105 ","End":"00:50.810","Text":"if I can reach a maximum distance of A,"},{"Start":"00:50.810 ","End":"00:56.765","Text":"that means I can reach maximum distance of A either side of my x=0."},{"Start":"00:56.765 ","End":"01:01.170","Text":"I can also reach a value of negative A over here."},{"Start":"01:01.250 ","End":"01:05.490","Text":"Let\u0027s start off with Question number 1."},{"Start":"01:05.490 ","End":"01:07.850","Text":"The first thing that I\u0027m going to do is I\u0027m going to"},{"Start":"01:07.850 ","End":"01:10.735","Text":"write out the energy equation for the body."},{"Start":"01:10.735 ","End":"01:17.105","Text":"My energy is going to be equal to 1/2 mv^2 for the kinetic energy."},{"Start":"01:17.105 ","End":"01:20.795","Text":"Instead of v^2, I\u0027m going to take the first derivative may x^2,"},{"Start":"01:20.795 ","End":"01:22.645","Text":"as we know, this is for velocity."},{"Start":"01:22.645 ","End":"01:25.505","Text":"Then plus my potential energy,"},{"Start":"01:25.505 ","End":"01:27.695","Text":"where my potential energy is this."},{"Start":"01:27.695 ","End":"01:31.910","Text":"It\u0027s B multiplied by the absolute value of x."},{"Start":"01:31.910 ","End":"01:37.175","Text":"Now, because we\u0027re dealing with our body moving simply in 1 dimension,"},{"Start":"01:37.175 ","End":"01:40.495","Text":"this is going to be much easier."},{"Start":"01:40.495 ","End":"01:46.235","Text":"Now what I\u0027m going to do is I\u0027m going to draw the graph for our energy."},{"Start":"01:46.235 ","End":"01:49.140","Text":"Some general graph."},{"Start":"01:49.640 ","End":"01:52.820","Text":"Over here I have my x."},{"Start":"01:52.820 ","End":"01:55.655","Text":"Now because I know that my potential energy"},{"Start":"01:55.655 ","End":"02:00.140","Text":"is some constant multiplied by the absolute value of x."},{"Start":"02:00.140 ","End":"02:06.675","Text":"The graph for that is going to look something along these lines."},{"Start":"02:06.675 ","End":"02:12.169","Text":"Now our energy we know is always going to be some constant value,"},{"Start":"02:12.169 ","End":"02:18.685","Text":"which translates when we\u0027re drawing it as a graph to a constant graph."},{"Start":"02:18.685 ","End":"02:23.435","Text":"Just going to be this straight line over here."},{"Start":"02:23.435 ","End":"02:27.080","Text":"Now, the area that our body can be found in,"},{"Start":"02:27.080 ","End":"02:29.930","Text":"as we know it is according to the rule that"},{"Start":"02:29.930 ","End":"02:37.040","Text":"our general energy E has to always be larger than our potential energy."},{"Start":"02:37.040 ","End":"02:40.430","Text":"That means that we have to find a section in this graph"},{"Start":"02:40.430 ","End":"02:44.975","Text":"where our energy is bigger than our potential energy."},{"Start":"02:44.975 ","End":"02:51.395","Text":"That means somewhere in this area over here,"},{"Start":"02:51.395 ","End":"02:53.150","Text":"in this triangle over here,"},{"Start":"02:53.150 ","End":"02:57.695","Text":"because we can see that we\u0027re located above our potential energy,"},{"Start":"02:57.695 ","End":"03:01.060","Text":"but below our E value."},{"Start":"03:01.060 ","End":"03:04.520","Text":"Now the next thing that we have to note is that we were told that"},{"Start":"03:04.520 ","End":"03:08.455","Text":"the greatest distance reached by the body is A."},{"Start":"03:08.455 ","End":"03:16.850","Text":"That\u0027s going to be at the intersection between our graph for E and our graph for our U."},{"Start":"03:16.850 ","End":"03:20.770","Text":"That\u0027s going to be at this point over here."},{"Start":"03:20.770 ","End":"03:30.670","Text":"If we draw a dotted line down that will correspond to this point being x=A."},{"Start":"03:30.670 ","End":"03:35.690","Text":"Now what I want to do is I want to find this point over here."},{"Start":"03:35.690 ","End":"03:43.445","Text":"Now what I have to do is I have to say that my value for E is equal to my value for U."},{"Start":"03:43.445 ","End":"03:48.170","Text":"Or alternatively, I can say that my x dots over"},{"Start":"03:48.170 ","End":"03:54.090","Text":"here is equal to 0 at this point because I\u0027m changing direction."},{"Start":"03:54.200 ","End":"04:00.030","Text":"The velocity is equal to 0, the x dot=0."},{"Start":"04:00.260 ","End":"04:05.600","Text":"Then I can say that my energy is therefore equal 2."},{"Start":"04:05.600 ","End":"04:09.635","Text":"We said that my x dot over here at this point is equal to 0."},{"Start":"04:09.635 ","End":"04:18.050","Text":"It\u0027s going to be 0 plus my B constant multiplied by the absolute value of my x."},{"Start":"04:18.050 ","End":"04:21.905","Text":"My absolute value over here is equal to A."},{"Start":"04:21.905 ","End":"04:27.190","Text":"Therefore, I can know that my energy is equal to B_A,"},{"Start":"04:27.190 ","End":"04:30.490","Text":"where they\u0027re both constants given in my question."},{"Start":"04:30.490 ","End":"04:34.910","Text":"Now of course this isn\u0027t just the value of energy at this specific point,"},{"Start":"04:34.910 ","End":"04:38.705","Text":"because we know that our energy is some constant function."},{"Start":"04:38.705 ","End":"04:45.925","Text":"My energy is always going to be equal to B multiplied by A for any value."},{"Start":"04:45.925 ","End":"04:49.010","Text":"However, this is very easy way to work it out."},{"Start":"04:49.010 ","End":"04:54.140","Text":"Because if we had to work out our value for energy at some random point,"},{"Start":"04:54.140 ","End":"04:55.593","Text":"let\u0027s say over here,"},{"Start":"04:55.593 ","End":"04:59.660","Text":"it will be a lot harder and bordering on impossible to find it out."},{"Start":"04:59.660 ","End":"05:03.650","Text":"But when we take this extreme point over here,"},{"Start":"05:03.650 ","End":"05:08.280","Text":"as we can see, it\u0027s really easy to work out the value."},{"Start":"05:08.870 ","End":"05:12.045","Text":"That\u0027s our answer for Question number 1."},{"Start":"05:12.045 ","End":"05:15.675","Text":"Now, let\u0027s move on to Question number 2."},{"Start":"05:15.675 ","End":"05:19.535","Text":"In Question number 2, we\u0027re trying to find the period or the time period."},{"Start":"05:19.535 ","End":"05:23.630","Text":"Now, we spoke about how to do this in the previous lesson."},{"Start":"05:23.630 ","End":"05:27.470","Text":"The only difference is that there we were using a variable of R,"},{"Start":"05:27.470 ","End":"05:30.110","Text":"and here we\u0027re using a variable of x."},{"Start":"05:30.110 ","End":"05:33.200","Text":"Now it means the exact same thing."},{"Start":"05:33.200 ","End":"05:38.880","Text":"There\u0027s no difference, it\u0027s just our variables are represented by different symbols."},{"Start":"05:39.740 ","End":"05:41.750","Text":"In the previous lesson,"},{"Start":"05:41.750 ","End":"05:46.355","Text":"we saw that what do we have to do to answer this is we have to isolate out."},{"Start":"05:46.355 ","End":"05:51.680","Text":"There we isolated out our r. and here we have to isolate out our x dot."},{"Start":"05:51.680 ","End":"05:54.630","Text":"Let\u0027s just do that really quickly."},{"Start":"05:54.630 ","End":"06:01.115","Text":"We\u0027ll see that our x dot is equal to plus or minus the square root of 2 divided by"},{"Start":"06:01.115 ","End":"06:08.780","Text":"m multiplied by E minus our value for our potential energy,"},{"Start":"06:08.780 ","End":"06:11.240","Text":"which here is going to be what?"},{"Start":"06:11.240 ","End":"06:13.310","Text":"Let\u0027s just substitute that in a second."},{"Start":"06:13.310 ","End":"06:17.000","Text":"Our value for a potential energy and then the square root of this."},{"Start":"06:17.000 ","End":"06:23.965","Text":"Now we remember that our x dot is equal to dx by dt."},{"Start":"06:23.965 ","End":"06:29.720","Text":"Now what we want to do is we want to multiply both sides by dt"},{"Start":"06:29.720 ","End":"06:35.270","Text":"and divide both sides by this expression over here with the square root."},{"Start":"06:35.270 ","End":"06:39.495","Text":"What we\u0027re going to be left with once we do that is plus minus,"},{"Start":"06:39.495 ","End":"06:43.160","Text":"then we\u0027re going to have the square root of"},{"Start":"06:43.160 ","End":"06:47.750","Text":"2 divided by m. Actually we\u0027ll do this without the square root,"},{"Start":"06:47.750 ","End":"06:53.410","Text":"2 divided by m E minus our U."},{"Start":"06:53.410 ","End":"06:58.880","Text":"Then this is going to be to the power of negative 1/2 dx,"},{"Start":"06:58.880 ","End":"07:02.400","Text":"and that\u0027s going to be equal to dt."},{"Start":"07:03.050 ","End":"07:07.610","Text":"Before we start putting in our integral sign"},{"Start":"07:07.610 ","End":"07:12.695","Text":"and putting in our borders for integrating our bounds,"},{"Start":"07:12.695 ","End":"07:18.120","Text":"let\u0027s take a look at what it means 1 period in this system."},{"Start":"07:18.120 ","End":"07:23.440","Text":"Now this graph is obviously meant to be symmetrical about the y-axis over here."},{"Start":"07:23.440 ","End":"07:27.790","Text":"1 period means that if, for instance,"},{"Start":"07:27.790 ","End":"07:30.865","Text":"our body starts from here,"},{"Start":"07:30.865 ","End":"07:39.505","Text":"that means that its starting position at t=0=0."},{"Start":"07:39.505 ","End":"07:45.010","Text":"1 period would be from here all the way to the end,"},{"Start":"07:45.010 ","End":"07:47.710","Text":"back past the origin,"},{"Start":"07:47.710 ","End":"07:49.315","Text":"to the other end,"},{"Start":"07:49.315 ","End":"07:51.370","Text":"and then back to the origin."},{"Start":"07:51.370 ","End":"07:57.260","Text":"We\u0027ll go right ways till our A,"},{"Start":"07:57.260 ","End":"08:00.050","Text":"then left all the way to negative A,"},{"Start":"08:00.050 ","End":"08:03.890","Text":"then back to the origin."},{"Start":"08:03.890 ","End":"08:08.920","Text":"All of that is going to be considered 1 period."},{"Start":"08:08.920 ","End":"08:13.070","Text":"In order to make my integral a little bit simpler,"},{"Start":"08:13.070 ","End":"08:16.420","Text":"I want to integrate from my t=0."},{"Start":"08:16.420 ","End":"08:17.750","Text":"For my starting point,"},{"Start":"08:17.750 ","End":"08:22.965","Text":"my origin until I get just to my A value."},{"Start":"08:22.965 ","End":"08:25.290","Text":"If we count for 1 period,"},{"Start":"08:25.290 ","End":"08:27.390","Text":"we have 1,"},{"Start":"08:27.390 ","End":"08:29.713","Text":"2, 3, 4."},{"Start":"08:29.713 ","End":"08:32.560","Text":"That means if I\u0027m going just from the origin"},{"Start":"08:32.560 ","End":"08:35.769","Text":"until my A and not carrying on the rest of my period,"},{"Start":"08:35.769 ","End":"08:43.120","Text":"so that means that my time is going to be that of a quarter of a period."},{"Start":"08:43.120 ","End":"08:45.685","Text":"From the origin until this point,"},{"Start":"08:45.685 ","End":"08:49.315","Text":"A over here is a 1/4 of the period."},{"Start":"08:49.315 ","End":"08:57.680","Text":"I want to integrate between 0 and t divided by 4."},{"Start":"08:58.140 ","End":"09:01.360","Text":"From the symmetry, we\u0027ll know that it will"},{"Start":"09:01.360 ","End":"09:04.105","Text":"take the same amount of time to get from the origin to A,"},{"Start":"09:04.105 ","End":"09:05.860","Text":"from A back to the origin,"},{"Start":"09:05.860 ","End":"09:11.480","Text":"from the origin to negative A and from negative A back to the origin."},{"Start":"09:11.850 ","End":"09:15.820","Text":"Why do I want to specifically integrate a"},{"Start":"09:15.820 ","End":"09:20.155","Text":"1/4 of my period and not just integrate along the whole period?"},{"Start":"09:20.155 ","End":"09:25.900","Text":"From t from 0 until capital T until my period."},{"Start":"09:25.900 ","End":"09:27.475","Text":"Why do I want to do that?"},{"Start":"09:27.475 ","End":"09:29.290","Text":"As we can see over here,"},{"Start":"09:29.290 ","End":"09:33.295","Text":"my x dot value can get a positive and a negative,"},{"Start":"09:33.295 ","End":"09:37.540","Text":"which will mean that I have to do lots of separate integrals when I can"},{"Start":"09:37.540 ","End":"09:42.805","Text":"just do one integral and then multiply it by 4 in order to get the time period."},{"Start":"09:42.805 ","End":"09:49.490","Text":"What I want to do is I want to integrate along where my x is positive."},{"Start":"09:49.800 ","End":"09:52.090","Text":"My x is bigger than 0,"},{"Start":"09:52.090 ","End":"09:55.420","Text":"which we can see from 0 until my A,"},{"Start":"09:55.420 ","End":"10:01.855","Text":"my x is going to be positive and we can see that my slope is a positive slope."},{"Start":"10:01.855 ","End":"10:04.015","Text":"We\u0027re at an incline."},{"Start":"10:04.015 ","End":"10:09.475","Text":"That means that my x dot is also bigger than 0."},{"Start":"10:09.475 ","End":"10:12.355","Text":"That means that when I\u0027m integrating,"},{"Start":"10:12.355 ","End":"10:15.100","Text":"I don\u0027t have to take into account this negative and I"},{"Start":"10:15.100 ","End":"10:18.475","Text":"can just integrate along with this value for the positive."},{"Start":"10:18.475 ","End":"10:21.580","Text":"That\u0027s solves our problem with our x dot,"},{"Start":"10:21.580 ","End":"10:23.980","Text":"so we can cross out the negative over here."},{"Start":"10:23.980 ","End":"10:27.370","Text":"Then the other thing is that my potential energy"},{"Start":"10:27.370 ","End":"10:31.285","Text":"is given by B multiplied by the absolute value of x."},{"Start":"10:31.285 ","End":"10:34.780","Text":"Now we don\u0027t really know how to deal with that when we\u0027re integrating,"},{"Start":"10:34.780 ","End":"10:38.830","Text":"unless we take the case when x is just positive or just negative."},{"Start":"10:38.830 ","End":"10:41.800","Text":"Specifically in this area,"},{"Start":"10:41.800 ","End":"10:44.860","Text":"if we\u0027re integrating along over here so"},{"Start":"10:44.860 ","End":"10:48.685","Text":"specifically here we can see that our x is always going to be positive."},{"Start":"10:48.685 ","End":"10:52.135","Text":"Which means that in our integration,"},{"Start":"10:52.135 ","End":"10:55.299","Text":"we don\u0027t have to include these lines representing"},{"Start":"10:55.299 ","End":"10:59.600","Text":"the absolute value and then we\u0027ve solved that problem as well."},{"Start":"10:59.790 ","End":"11:02.965","Text":"Now let\u0027s write out our integrals."},{"Start":"11:02.965 ","End":"11:07.120","Text":"We know that we\u0027re going to integrate along the positive value for this."},{"Start":"11:07.120 ","End":"11:14.620","Text":"We\u0027re going to have 2 divided by m multiplied by E minus and then"},{"Start":"11:14.620 ","End":"11:18.790","Text":"our U is going to be B multiplied by"},{"Start":"11:18.790 ","End":"11:23.620","Text":"x which we know is always positive so just multiplied by x."},{"Start":"11:23.620 ","End":"11:32.680","Text":"Then this all is going to be to the power of negative 1/2 dx."},{"Start":"11:32.680 ","End":"11:36.980","Text":"This is going to be equal to dt."},{"Start":"11:37.230 ","End":"11:42.310","Text":"Now we can substitute in our integral signs and not"},{"Start":"11:42.310 ","End":"11:46.855","Text":"forgetting our value for E we worked out in Question 1."},{"Start":"11:46.855 ","End":"11:51.130","Text":"We know that it equals BA over here."},{"Start":"11:51.130 ","End":"11:53.890","Text":"Now let\u0027s put in our bounds."},{"Start":"11:53.890 ","End":"11:56.740","Text":"We know that our time period,"},{"Start":"11:56.740 ","End":"12:03.520","Text":"we\u0027re integrating from 0 until 1/4 of the time period as we saw over here."},{"Start":"12:03.520 ","End":"12:09.265","Text":"Because we saw one full period is this back here and back."},{"Start":"12:09.265 ","End":"12:14.140","Text":"We\u0027re just going from this until our A over here,"},{"Start":"12:14.140 ","End":"12:16.570","Text":"so it\u0027s 1/4 of a time period."},{"Start":"12:16.570 ","End":"12:19.720","Text":"Then we\u0027re integrating from our x value,"},{"Start":"12:19.720 ","End":"12:23.605","Text":"from our x=0 until our x=A."},{"Start":"12:23.605 ","End":"12:25.330","Text":"This distance over here,"},{"Start":"12:25.330 ","End":"12:28.910","Text":"so 0 until A."},{"Start":"12:29.280 ","End":"12:32.290","Text":"Let\u0027s integrate this."},{"Start":"12:32.290 ","End":"12:35.320","Text":"Let\u0027s rearrange this like so."},{"Start":"12:35.320 ","End":"12:38.275","Text":"We have our bounds from 0 until A."},{"Start":"12:38.275 ","End":"12:41.380","Text":"Then let\u0027s take out our constants."},{"Start":"12:41.380 ","End":"12:47.590","Text":"We\u0027ll have 2B divided by m. Then that is going"},{"Start":"12:47.590 ","End":"12:53.670","Text":"to be to the power of negative 1/2 over here."},{"Start":"12:53.670 ","End":"12:58.550","Text":"Then it\u0027s going to be multiplied by A minus x,"},{"Start":"12:58.550 ","End":"13:03.340","Text":"which is also to the power of negative 1/2 dx."},{"Start":"13:03.340 ","End":"13:05.755","Text":"Then we can already just integrate this."},{"Start":"13:05.755 ","End":"13:08.110","Text":"The integral of this,"},{"Start":"13:08.110 ","End":"13:11.750","Text":"the answer is going to be T divided by 4."},{"Start":"13:12.660 ","End":"13:17.125","Text":"Now what we can do is we can just integrate this already."},{"Start":"13:17.125 ","End":"13:18.820","Text":"Because these are constants,"},{"Start":"13:18.820 ","End":"13:23.190","Text":"we\u0027re going to have 2B divided by m to the"},{"Start":"13:23.190 ","End":"13:27.780","Text":"negative 1/2 and then we\u0027re going to integrate these brackets so it\u0027s"},{"Start":"13:27.780 ","End":"13:38.410","Text":"going to be A minus x to the power of 1/2 multiplied by negative 2."},{"Start":"13:38.570 ","End":"13:44.275","Text":"Then we\u0027re going to see what this is between 0 and A."},{"Start":"13:44.275 ","End":"13:49.180","Text":"Of course, again, it\u0027s equal to T divided by 4."},{"Start":"13:49.180 ","End":"13:52.240","Text":"When we substitute in our values,"},{"Start":"13:52.240 ","End":"14:00.355","Text":"so we\u0027re going to have negative 2 multiplied by the square root of."},{"Start":"14:00.355 ","End":"14:02.920","Text":"Because we have a negative over here,"},{"Start":"14:02.920 ","End":"14:06.475","Text":"so we can flip our fraction over here upside"},{"Start":"14:06.475 ","End":"14:11.860","Text":"down and then just do the square root because the negative means 1 over."},{"Start":"14:11.860 ","End":"14:15.190","Text":"Then we\u0027re just going to have m divided by"},{"Start":"14:15.190 ","End":"14:19.510","Text":"2B instead of 2B divided by m and of course the square root."},{"Start":"14:19.510 ","End":"14:23.080","Text":"Then we\u0027re going to have this by"},{"Start":"14:23.080 ","End":"14:33.860","Text":"A minus A minus and then A minus 0."},{"Start":"14:34.110 ","End":"14:36.280","Text":"This is, of course,"},{"Start":"14:36.280 ","End":"14:38.515","Text":"equal to T divided by 4."},{"Start":"14:38.515 ","End":"14:40.960","Text":"A minus A is obviously going to be equal to"},{"Start":"14:40.960 ","End":"14:45.080","Text":"0 and then we\u0027re just going to be left with this."},{"Start":"14:47.280 ","End":"14:52.360","Text":"Now notice this negative and this negative cross out."},{"Start":"14:52.360 ","End":"14:58.645","Text":"We\u0027re going to have 2 multiply by the square root of m divided"},{"Start":"14:58.645 ","End":"15:04.959","Text":"2B multiplied"},{"Start":"15:04.959 ","End":"15:10.555","Text":"by A=T divided by 4."},{"Start":"15:10.555 ","End":"15:13.885","Text":"Now, let\u0027s quickly go back to the question, we\u0027re being asked."},{"Start":"15:13.885 ","End":"15:15.430","Text":"What is the period?"},{"Start":"15:15.430 ","End":"15:18.865","Text":"Right now we have an answer for a 1/4 of the period."},{"Start":"15:18.865 ","End":"15:21.700","Text":"What we want to do now is multiply both sides by 4"},{"Start":"15:21.700 ","End":"15:24.715","Text":"and then we\u0027ll get our full time period."},{"Start":"15:24.715 ","End":"15:30.085","Text":"We\u0027ll have that T=8 multiplied by the square root of m"},{"Start":"15:30.085 ","End":"15:36.880","Text":"divided by 2B square root multiplied by A."},{"Start":"15:36.880 ","End":"15:39.130","Text":"This is our final answer."},{"Start":"15:39.130 ","End":"15:43.210","Text":"Now let\u0027s 1 second go back over here."},{"Start":"15:43.210 ","End":"15:46.570","Text":"Now remember we said that our body starts at the origin,"},{"Start":"15:46.570 ","End":"15:50.260","Text":"and then that\u0027s what we used in order to calculate the period."},{"Start":"15:50.260 ","End":"15:51.940","Text":"Now when we\u0027re calculating the period,"},{"Start":"15:51.940 ","End":"15:55.210","Text":"it doesn\u0027t matter if we say that our body starts here or"},{"Start":"15:55.210 ","End":"15:58.570","Text":"here or here anywhere because our period is going to be"},{"Start":"15:58.570 ","End":"16:05.560","Text":"the same whichever place is our initial starting position, so it doesn\u0027t matter."},{"Start":"16:05.560 ","End":"16:11.155","Text":"If however, we were asked to find our position of the body as a function of time,"},{"Start":"16:11.155 ","End":"16:18.460","Text":"so x is a function of T then our initial starting position will be important to know."},{"Start":"16:18.460 ","End":"16:21.505","Text":"But for finding the time period, it doesn\u0027t matter."},{"Start":"16:21.505 ","End":"16:24.710","Text":"That\u0027s the end of this lesson."}],"ID":9464},{"Watched":false,"Name":"Gravitational Force Inside A Full Sphere","Duration":"8m 21s","ChapterTopicVideoID":9195,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"Hello. In this lesson,"},{"Start":"00:01.755 ","End":"00:05.880","Text":"I wanted to speak of a case where we have a mass m,"},{"Start":"00:05.880 ","End":"00:07.515","Text":"which is a small mass,"},{"Start":"00:07.515 ","End":"00:15.150","Text":"and it\u0027s located inside some larger body of mass M. Here we have a small mass m,"},{"Start":"00:15.150 ","End":"00:19.020","Text":"which is inside a full sphere with uniform density and of mass"},{"Start":"00:19.020 ","End":"00:24.600","Text":"M. What will happen is that our gravitational force from"},{"Start":"00:24.600 ","End":"00:29.295","Text":"this large body with another small body of mass m inside will be"},{"Start":"00:29.295 ","End":"00:34.635","Text":"slightly different and the mass that we will use in order to solve this,"},{"Start":"00:34.635 ","End":"00:37.965","Text":"not the whole mass of everything together and"},{"Start":"00:37.965 ","End":"00:41.990","Text":"not to do with this M. But rather it will be"},{"Start":"00:41.990 ","End":"00:51.230","Text":"some function which is dependent on where this mass is inside the large sphere."},{"Start":"00:51.230 ","End":"00:55.455","Text":"We\u0027ll write our equation for our force,"},{"Start":"00:55.455 ","End":"01:03.290","Text":"and this is going to be equal to negative GM as a function of r multiplied by m,"},{"Start":"01:03.290 ","End":"01:04.805","Text":"which is our small mass,"},{"Start":"01:04.805 ","End":"01:09.715","Text":"divided by r^2 in the radial direction."},{"Start":"01:09.715 ","End":"01:14.180","Text":"The mass, which the force is dependent on,"},{"Start":"01:14.180 ","End":"01:19.515","Text":"is dependent on the position of the small mass."},{"Start":"01:19.515 ","End":"01:24.800","Text":"The idea is that when the mass is located inside the sphere,"},{"Start":"01:24.800 ","End":"01:29.837","Text":"is that anything that will be in the vicinity of this small mass,"},{"Start":"01:29.837 ","End":"01:33.335","Text":"so let\u0027s say here we have some object,"},{"Start":"01:33.335 ","End":"01:40.310","Text":"it will experience a force to here rather than to the center of the sphere."},{"Start":"01:40.310 ","End":"01:43.820","Text":"Now, when we have a uniform density,"},{"Start":"01:43.820 ","End":"01:47.945","Text":"then all of the pieces that we\u0027ll have in here together,"},{"Start":"01:47.945 ","End":"01:51.870","Text":"so this and this, will cancel out."},{"Start":"01:52.220 ","End":"01:58.130","Text":"All the objects which are located outside the radius that the small mass is"},{"Start":"01:58.130 ","End":"02:05.030","Text":"located at from the center will cancel each other out their forces."},{"Start":"02:05.030 ","End":"02:11.405","Text":"Then the mass which I need to calculate is going to be the mass inside"},{"Start":"02:11.405 ","End":"02:17.720","Text":"the sphere of the same radius as that of my small mass."},{"Start":"02:17.720 ","End":"02:21.590","Text":"As we can see from the center of the circle to my location of the small mass I have"},{"Start":"02:21.590 ","End":"02:29.224","Text":"radius r. This entire circle of radius r of a sphere of radius r over here,"},{"Start":"02:29.224 ","End":"02:33.175","Text":"so that\u0027s the mass that I need to calculate."},{"Start":"02:33.175 ","End":"02:37.835","Text":"When I\u0027m calculating questions of this type,"},{"Start":"02:37.835 ","End":"02:44.490","Text":"so all of the mass which is located outside in this whitespace isn\u0027t interesting to me."},{"Start":"02:44.490 ","End":"02:48.860","Text":"All I have to do is I have to calculate the mass of this sphere of"},{"Start":"02:48.860 ","End":"02:54.950","Text":"radius r which is the distance that my small mass is from the center of the ball,"},{"Start":"02:54.950 ","End":"03:04.535","Text":"and that will be the mass which is affecting my mass m. The only mass that"},{"Start":"03:04.535 ","End":"03:14.345","Text":"is going to have some effect on this mass m over here is all the masses inside this area."},{"Start":"03:14.345 ","End":"03:20.060","Text":"Which means that all the points which are closer to the center of the sphere,"},{"Start":"03:20.060 ","End":"03:23.570","Text":"then this mass over here."},{"Start":"03:23.570 ","End":"03:27.470","Text":"Now, I\u0027m not going to prove to you why all the masses which are"},{"Start":"03:27.470 ","End":"03:32.105","Text":"located outside of the smaller sphere cancel each other out,"},{"Start":"03:32.105 ","End":"03:35.810","Text":"but we\u0027re just going to accept it as a fact."},{"Start":"03:35.810 ","End":"03:40.490","Text":"But what we are going to work out and what is maybe more important right"},{"Start":"03:40.490 ","End":"03:45.950","Text":"now is how to find out what this mass is over here."},{"Start":"03:45.950 ","End":"03:56.675","Text":"That we denoted as our M as a function of r. If I have a mass of m,"},{"Start":"03:56.675 ","End":"04:01.814","Text":"which is located at some radius inside my sphere,"},{"Start":"04:01.814 ","End":"04:05.630","Text":"what mass do I have to calculate in order"},{"Start":"04:05.630 ","End":"04:10.060","Text":"to substitute in to my force equation over here?"},{"Start":"04:10.060 ","End":"04:14.255","Text":"The mass which is located over here,"},{"Start":"04:14.255 ","End":"04:18.295","Text":"so that\u0027s equal to M(r),"},{"Start":"04:18.295 ","End":"04:22.045","Text":"is equal to the density, so that\u0027s Rho,"},{"Start":"04:22.045 ","End":"04:26.885","Text":"multiplied by the volume of this small sphere."},{"Start":"04:26.885 ","End":"04:32.415","Text":"The volume of a sphere is 4/3 Pir^3."},{"Start":"04:32.415 ","End":"04:35.190","Text":"I\u0027m reminding you it\u0027s a r because that\u0027s"},{"Start":"04:35.190 ","End":"04:39.420","Text":"the radius from the center until where mass is located."},{"Start":"04:39.420 ","End":"04:44.495","Text":"Here specifically, the density is uniform."},{"Start":"04:44.495 ","End":"04:45.965","Text":"If it wasn\u0027t uniform,"},{"Start":"04:45.965 ","End":"04:48.380","Text":"then we\u0027d have to do some form of"},{"Start":"04:48.380 ","End":"04:53.070","Text":"integration over here in order to get our value for our density."},{"Start":"04:53.960 ","End":"04:58.735","Text":"Now what I want to find out is what is my density?"},{"Start":"04:58.735 ","End":"05:03.665","Text":"My density I can find out from my entire sphere."},{"Start":"05:03.665 ","End":"05:07.130","Text":"Density is mass divided by volume."},{"Start":"05:07.130 ","End":"05:12.800","Text":"We\u0027re being told that the mass of my full sphere is this M over here."},{"Start":"05:12.800 ","End":"05:19.860","Text":"The mass of the full sphere is M divided by the volume of the full sphere,"},{"Start":"05:19.860 ","End":"05:22.440","Text":"which is 4/3 Pi,"},{"Start":"05:22.440 ","End":"05:24.930","Text":"and here it\u0027s R^3,"},{"Start":"05:24.930 ","End":"05:31.459","Text":"because the radius of the full sphere is R. That is going to be my density,"},{"Start":"05:31.459 ","End":"05:33.140","Text":"and the density is uniform."},{"Start":"05:33.140 ","End":"05:37.040","Text":"The density of the entire sphere over here is going to be the same as"},{"Start":"05:37.040 ","End":"05:42.780","Text":"the density in here in my imaginary central sphere."},{"Start":"05:43.130 ","End":"05:49.135","Text":"Now what we\u0027re going to do is we\u0027re going to substitute in our Rho into here."},{"Start":"05:49.135 ","End":"05:51.665","Text":"Then by doing some simple algebra,"},{"Start":"05:51.665 ","End":"05:57.695","Text":"we will get that our mass as a function of the radius."},{"Start":"05:57.695 ","End":"06:03.935","Text":"This mass over here is going to"},{"Start":"06:03.935 ","End":"06:10.920","Text":"be equal to the mass of the entire sphere multiplied by (r divided by R)^3."},{"Start":"06:13.250 ","End":"06:17.435","Text":"Now what we\u0027re going to do is we\u0027re going to substitute"},{"Start":"06:17.435 ","End":"06:22.070","Text":"in this value into our equation for force."},{"Start":"06:22.070 ","End":"06:26.475","Text":"Then we\u0027ll get that our force is equal to,"},{"Start":"06:26.475 ","End":"06:30.260","Text":"and then by simple algebra, canceling everything out,"},{"Start":"06:30.260 ","End":"06:40.450","Text":"we\u0027ll get that it\u0027s equal to GMmr in the radial direction divided by R^3,"},{"Start":"06:40.450 ","End":"06:42.435","Text":"where R^3 over here,"},{"Start":"06:42.435 ","End":"06:46.485","Text":"R^3 is the radius of the whole sphere."},{"Start":"06:46.485 ","End":"06:49.025","Text":"Now, another way to write this,"},{"Start":"06:49.025 ","End":"06:57.810","Text":"because I\u0027ll remind you that our r hat is equal to this r. We can also write"},{"Start":"06:57.810 ","End":"07:07.410","Text":"it as GMmr divided by R^3."},{"Start":"07:07.410 ","End":"07:09.100","Text":"These are the same thing,"},{"Start":"07:09.100 ","End":"07:11.170","Text":"just different ways of writing it."},{"Start":"07:11.170 ","End":"07:15.940","Text":"Because this is r in the direction of our unit vector r,"},{"Start":"07:15.940 ","End":"07:17.890","Text":"and this is our r vector."},{"Start":"07:17.890 ","End":"07:23.895","Text":"This is useful if we want to write this equation in the Cartesian coordinates."},{"Start":"07:23.895 ","End":"07:29.665","Text":"Then we\u0027ll have that this is equal to GMm divided by R^3,"},{"Start":"07:29.665 ","End":"07:31.510","Text":"these are constants,"},{"Start":"07:31.510 ","End":"07:40.790","Text":"multiplied by x in the x-direction plus a y in the y-direction."},{"Start":"07:41.360 ","End":"07:45.235","Text":"It\u0027s very important to remember this."},{"Start":"07:45.235 ","End":"07:50.020","Text":"That means that if we have some body of some mass,"},{"Start":"07:50.020 ","End":"07:51.665","Text":"and inside that body,"},{"Start":"07:51.665 ","End":"07:53.599","Text":"we have another mass."},{"Start":"07:53.599 ","End":"07:59.480","Text":"Then our equation for our gravitational force is going to be different."},{"Start":"07:59.480 ","End":"08:03.800","Text":"It\u0027s no longer going to be GMm divided by R^2,"},{"Start":"08:03.800 ","End":"08:09.980","Text":"but it\u0027s going to be GM as a function of the radius multiplied by the"},{"Start":"08:09.980 ","End":"08:13.670","Text":"m. That means that we\u0027re going to have"},{"Start":"08:13.670 ","End":"08:19.425","Text":"to utilize the position of our smaller mass inside."},{"Start":"08:19.425 ","End":"08:22.210","Text":"That\u0027s the end of our lesson."}],"ID":9465},{"Watched":false,"Name":"Body Moves Through Tunne","Duration":"10m 14s","ChapterTopicVideoID":9196,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"Hello. In this question,"},{"Start":"00:01.920 ","End":"00:05.640","Text":"we\u0027re being told that a body travels through a tunnel located"},{"Start":"00:05.640 ","End":"00:09.360","Text":"at R divided by 2 from the center of a sphere of"},{"Start":"00:09.360 ","End":"00:17.565","Text":"mass M. We have some full sphere of mass M and a radius capital R,"},{"Start":"00:17.565 ","End":"00:20.340","Text":"and we\u0027re told halfway through the radius that we have"},{"Start":"00:20.340 ","End":"00:23.520","Text":"some tunnel that a body is moving through."},{"Start":"00:23.520 ","End":"00:26.700","Text":"The body begins its motion from rest at one end"},{"Start":"00:26.700 ","End":"00:30.045","Text":"of the tunnel and there is no frictional force acting."},{"Start":"00:30.045 ","End":"00:36.070","Text":"Now the goal is to find the position of the body as a function of time."},{"Start":"00:36.070 ","End":"00:43.070","Text":"The axis that I\u0027m going to work with is my x-axis because I\u0027m going to try and"},{"Start":"00:43.070 ","End":"00:45.770","Text":"find the position as a function of time so I\u0027m going to try and"},{"Start":"00:45.770 ","End":"00:49.900","Text":"find my x as a function of time."},{"Start":"00:49.900 ","End":"00:51.680","Text":"That\u0027s what I\u0027m going to do."},{"Start":"00:51.680 ","End":"00:57.070","Text":"My y-axis doesn\u0027t really interests me because I have no motion in the y direction."},{"Start":"00:57.070 ","End":"01:02.675","Text":"What I want to find out now is which forces are acting on this body."},{"Start":"01:02.675 ","End":"01:04.370","Text":"Let\u0027s see what this is."},{"Start":"01:04.370 ","End":"01:10.220","Text":"We already know that the gravitational force from the last lesson it\u0027s going to be equal"},{"Start":"01:10.220 ","End":"01:16.755","Text":"to negative GMm divided by R^3,"},{"Start":"01:16.755 ","End":"01:20.030","Text":"R being the radius of the sphere,"},{"Start":"01:20.030 ","End":"01:23.075","Text":"multiplied by our R vector."},{"Start":"01:23.075 ","End":"01:30.230","Text":"We showed in the past lesson that the closer that the body gets to the center of"},{"Start":"01:30.230 ","End":"01:38.880","Text":"the sphere so the force acting on the body will become smaller."},{"Start":"01:38.890 ","End":"01:45.630","Text":"In the end, our gravitational force is going to be in this form."},{"Start":"01:46.280 ","End":"01:52.730","Text":"Now what I want to do is I want to work with my Cartesian coordinate,"},{"Start":"01:52.730 ","End":"01:54.245","Text":"so that\u0027s x and y."},{"Start":"01:54.245 ","End":"01:55.745","Text":"Let\u0027s rewrite this."},{"Start":"01:55.745 ","End":"02:01.070","Text":"I\u0027ll have negative GMm divided by R^3,"},{"Start":"02:01.070 ","End":"02:02.960","Text":"and then in my Cartesian coordinates,"},{"Start":"02:02.960 ","End":"02:09.300","Text":"it\u0027s going to be x in the x direction plus y in the y-direction."},{"Start":"02:09.310 ","End":"02:14.930","Text":"As we said, the only thing that interests me is the x values,"},{"Start":"02:14.930 ","End":"02:17.435","Text":"because we don\u0027t really have much movement,"},{"Start":"02:17.435 ","End":"02:22.595","Text":"if any in the y-direction so let\u0027s rewrite this only in the x-direction."},{"Start":"02:22.595 ","End":"02:28.760","Text":"We can say that the sum of all of the forces in the x-direction it\u0027s going to be equal"},{"Start":"02:28.760 ","End":"02:37.330","Text":"to negative GMm divided by R^3 in the x-direction."},{"Start":"02:37.330 ","End":"02:40.670","Text":"I don\u0027t need to write this x hat because it\u0027s"},{"Start":"02:40.670 ","End":"02:43.895","Text":"clear from this subscript that I\u0027m in the x-direction,"},{"Start":"02:43.895 ","End":"02:48.755","Text":"and that is going to be equal to my mass multiplied by my acceleration,"},{"Start":"02:48.755 ","End":"02:51.715","Text":"also in the x-direction."},{"Start":"02:51.715 ","End":"02:55.890","Text":"Now I can cancel out my small ms,"},{"Start":"02:55.890 ","End":"03:00.560","Text":"and I can rewrite this by writing negative GM divided"},{"Start":"03:00.560 ","End":"03:08.060","Text":"by R^3 multiplied by x is equal to my acceleration in the x-direction."},{"Start":"03:08.060 ","End":"03:09.335","Text":"Now how else can I write this?"},{"Start":"03:09.335 ","End":"03:11.825","Text":"I can write this as x double-dot,"},{"Start":"03:11.825 ","End":"03:13.555","Text":"these are the same things."},{"Start":"03:13.555 ","End":"03:19.790","Text":"Now we can see that what we\u0027re left with is an equation for harmonic motion."},{"Start":"03:19.790 ","End":"03:21.950","Text":"We have our negative, which we always have in"},{"Start":"03:21.950 ","End":"03:26.375","Text":"harmonic motion and then we have the coefficient of our x,"},{"Start":"03:26.375 ","End":"03:28.605","Text":"which is like our k,"},{"Start":"03:28.605 ","End":"03:31.995","Text":"and then our coefficient of our x double dots is our mass."},{"Start":"03:31.995 ","End":"03:37.610","Text":"Here our coefficient is 1 so our mass is going to be equal to 1."},{"Start":"03:37.610 ","End":"03:40.640","Text":"Then we can write our Omega,"},{"Start":"03:40.640 ","End":"03:49.495","Text":"which is going to be equal to the square root of GM divided by R^3."},{"Start":"03:49.495 ","End":"03:53.450","Text":"Remember that our Omega is equal to the square root of"},{"Start":"03:53.450 ","End":"03:57.170","Text":"k divided by m so here because our m=1,"},{"Start":"03:57.170 ","End":"04:04.715","Text":"it\u0027s just going to be the square root of k. Now once we get some harmonic equation,"},{"Start":"04:04.715 ","End":"04:07.340","Text":"we have to know immediately and automatically,"},{"Start":"04:07.340 ","End":"04:10.070","Text":"if someone wakes you up in the night and asks you"},{"Start":"04:10.070 ","End":"04:13.475","Text":"for the general solution for harmonic equation,"},{"Start":"04:13.475 ","End":"04:15.590","Text":"you\u0027re meant to know it off by heart."},{"Start":"04:15.590 ","End":"04:20.840","Text":"That is that our x as a function of t is going to be equal to"},{"Start":"04:20.840 ","End":"04:27.790","Text":"our A cosine of Omega t plus Phi."},{"Start":"04:27.800 ","End":"04:31.395","Text":"Omega, we already have, it\u0027s this over here,"},{"Start":"04:31.395 ","End":"04:34.280","Text":"and now what we want to do is we want to find our values for A and"},{"Start":"04:34.280 ","End":"04:38.215","Text":"Phi by using our initial conditions."},{"Start":"04:38.215 ","End":"04:42.155","Text":"My initial conditions as given in the question,"},{"Start":"04:42.155 ","End":"04:49.760","Text":"is that the body begins its motion from rest and begins at one end of the tunnel."},{"Start":"04:49.760 ","End":"04:56.360","Text":"Let\u0027s begin from our position. Let\u0027s see."},{"Start":"04:56.360 ","End":"05:01.550","Text":"Our x as a function of time when t=0."},{"Start":"05:01.550 ","End":"05:03.830","Text":"This is our starting position."},{"Start":"05:03.830 ","End":"05:07.455","Text":"If we say that this is our origin,"},{"Start":"05:07.455 ","End":"05:09.315","Text":"right in the center of the sphere,"},{"Start":"05:09.315 ","End":"05:11.525","Text":"and this up until here,"},{"Start":"05:11.525 ","End":"05:14.930","Text":"is going to be our radius R. We know that the radius of"},{"Start":"05:14.930 ","End":"05:19.145","Text":"the larger sphere is R. This whole length is R,"},{"Start":"05:19.145 ","End":"05:22.160","Text":"and this length is R divided by 2."},{"Start":"05:22.160 ","End":"05:27.555","Text":"Now what we want to find is this distance over here."},{"Start":"05:27.555 ","End":"05:31.940","Text":"Through Pythagoras, we know that this length over here,"},{"Start":"05:31.940 ","End":"05:36.505","Text":"so that\u0027s R divided by 2^2,"},{"Start":"05:36.505 ","End":"05:38.285","Text":"plus this length over here,"},{"Start":"05:38.285 ","End":"05:39.485","Text":"let\u0027s call it x,"},{"Start":"05:39.485 ","End":"05:44.600","Text":"so plus x^2 is going to be equal to our hypotenuse squared."},{"Start":"05:44.600 ","End":"05:46.505","Text":"That\u0027s our R^2."},{"Start":"05:46.505 ","End":"05:50.335","Text":"What we want do is we want to isolate out this x^2."},{"Start":"05:50.335 ","End":"05:56.090","Text":"Therefore, we\u0027ll get that our x is going to be equal to plus or minus"},{"Start":"05:56.090 ","End":"06:04.560","Text":"the square root of R^2 minus R divided by 2^2."},{"Start":"06:05.030 ","End":"06:07.760","Text":"We have our plus or minus over here,"},{"Start":"06:07.760 ","End":"06:09.200","Text":"now because we said that this is"},{"Start":"06:09.200 ","End":"06:12.980","Text":"the origin and we want to find our body right at the start,"},{"Start":"06:12.980 ","End":"06:15.875","Text":"where it\u0027s to the left of the origin,"},{"Start":"06:15.875 ","End":"06:22.830","Text":"so we\u0027re just going to take the negative because it\u0027s in the negative x values."},{"Start":"06:22.830 ","End":"06:26.900","Text":"That means that we can just substitute that in over here so we\u0027re going to have"},{"Start":"06:26.900 ","End":"06:33.960","Text":"negative the square root of R^2 minus R divided by 2^2."},{"Start":"06:34.370 ","End":"06:37.665","Text":"Then we can simplify that."},{"Start":"06:37.665 ","End":"06:44.140","Text":"That is going to be equal to negative root 3 over 2R."},{"Start":"06:44.140 ","End":"06:48.350","Text":"Then we have to set it equal to our x as a function of t,"},{"Start":"06:48.350 ","End":"06:50.780","Text":"where we set that t=0."},{"Start":"06:50.780 ","End":"06:52.805","Text":"We look at this over here,"},{"Start":"06:52.805 ","End":"06:58.440","Text":"so it\u0027s going to be equal to A cosine of Omega multiplied by t,"},{"Start":"06:58.440 ","End":"07:00.020","Text":"but our t=0,"},{"Start":"07:00.020 ","End":"07:04.170","Text":"so it\u0027s going to be equal to A cosine of Phi."},{"Start":"07:04.460 ","End":"07:07.920","Text":"This is our first equation,"},{"Start":"07:07.920 ","End":"07:10.550","Text":"and now we have to find our second equation and that\u0027s going to"},{"Start":"07:10.550 ","End":"07:14.745","Text":"come from our velocity equation."},{"Start":"07:14.745 ","End":"07:18.410","Text":"How do we do that? We have to take the first derivative of our x"},{"Start":"07:18.410 ","End":"07:23.030","Text":"because we know that the first derivative of our position is going to be our velocity."},{"Start":"07:23.030 ","End":"07:26.825","Text":"We\u0027re going to find x dot of t,"},{"Start":"07:26.825 ","End":"07:30.590","Text":"and that\u0027s going to be equal to v(t),"},{"Start":"07:30.590 ","End":"07:32.075","Text":"and then the derivative of this,"},{"Start":"07:32.075 ","End":"07:36.245","Text":"so we\u0027re going to have A multiplied by the derivative of cosine."},{"Start":"07:36.245 ","End":"07:40.650","Text":"It\u0027s going to be negative sine of what\u0027s inside,"},{"Start":"07:40.650 ","End":"07:44.740","Text":"so Omega t plus Phi."},{"Start":"07:45.530 ","End":"07:47.669","Text":"That all."},{"Start":"07:47.669 ","End":"07:50.970","Text":"Then multiplied by Omega over here."},{"Start":"07:51.790 ","End":"07:56.960","Text":"Our Omega is the inner derivative coming from this over here."},{"Start":"07:56.960 ","End":"07:59.960","Text":"As we said, the motion,"},{"Start":"07:59.960 ","End":"08:02.195","Text":"so it begins from rest,"},{"Start":"08:02.195 ","End":"08:07.520","Text":"our body, so that means that I have to set my velocity at t=0,"},{"Start":"08:07.520 ","End":"08:10.670","Text":"so my velocity at the start,"},{"Start":"08:10.670 ","End":"08:13.982","Text":"the beginning of the motion is equal to 0,"},{"Start":"08:13.982 ","End":"08:19.650","Text":"and that is equal to negative A Omega sine of,"},{"Start":"08:19.650 ","End":"08:21.620","Text":"now because our t=0,"},{"Start":"08:21.620 ","End":"08:26.365","Text":"Omega times 0 is 0 so it\u0027s going to be Omega sine Phi."},{"Start":"08:26.365 ","End":"08:30.570","Text":"Now this is our second equation."},{"Start":"08:30.570 ","End":"08:33.200","Text":"Let\u0027s take a look. We have to have that"},{"Start":"08:33.200 ","End":"08:36.605","Text":"this expression over here is going to be equal to 0."},{"Start":"08:36.605 ","End":"08:38.885","Text":"Now we know that our A can be equal to 0,"},{"Start":"08:38.885 ","End":"08:40.955","Text":"we know that our Omega is this,"},{"Start":"08:40.955 ","End":"08:42.770","Text":"and this is definitely not equal to 0,"},{"Start":"08:42.770 ","End":"08:47.030","Text":"so that means that our sine Phi has to be equal to 0."},{"Start":"08:47.030 ","End":"08:50.735","Text":"For which value of Phi will our sine Phi equal to 0?"},{"Start":"08:50.735 ","End":"08:56.235","Text":"When our Phi=0,"},{"Start":"08:56.235 ","End":"08:58.070","Text":"our sine Phi=0,"},{"Start":"08:58.070 ","End":"09:01.555","Text":"and then our velocity at t=0 is 0."},{"Start":"09:01.555 ","End":"09:05.840","Text":"Perfect. Now we know that this is equal to 0,"},{"Start":"09:05.840 ","End":"09:08.480","Text":"and now we can find what our A is equal to."},{"Start":"09:08.480 ","End":"09:13.855","Text":"When Phi=0, our cosine of 0=1."},{"Start":"09:13.855 ","End":"09:19.875","Text":"That means that our negative root 3 divided by 2R=A."},{"Start":"09:19.875 ","End":"09:21.975","Text":"Now we found what our A is."},{"Start":"09:21.975 ","End":"09:28.990","Text":"Our A is going to equal to negative root 3 divided by 2R."},{"Start":"09:28.990 ","End":"09:32.910","Text":"Now we have our final equation,"},{"Start":"09:32.910 ","End":"09:34.665","Text":"let\u0027s write this out."},{"Start":"09:34.665 ","End":"09:38.075","Text":"We\u0027ll have that our x as a function of time,"},{"Start":"09:38.075 ","End":"09:41.390","Text":"so our position as a function of time is going to be A,"},{"Start":"09:41.390 ","End":"09:49.265","Text":"which is negative root 3 divided by 2R multiplied by R cosine."},{"Start":"09:49.265 ","End":"09:52.190","Text":"Then in our brackets we have our Omega,"},{"Start":"09:52.190 ","End":"09:56.420","Text":"which is going to be the square root of GM divided by"},{"Start":"09:56.420 ","End":"10:04.285","Text":"R^3 multiplied by t plus our Phi, and our Phi=0."},{"Start":"10:04.285 ","End":"10:06.695","Text":"This is our final answer."},{"Start":"10:06.695 ","End":"10:12.370","Text":"This is our position of the body as it moves through the tunnel as a function of time,"},{"Start":"10:12.370 ","End":"10:14.990","Text":"and that\u0027s the end of this lesson."}],"ID":9466},{"Watched":false,"Name":"Deriving Trajectory Equation","Duration":"19m 6s","ChapterTopicVideoID":9197,"CourseChapterTopicPlaylistID":5361,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.995","Text":"Hello. In this lesson,"},{"Start":"00:01.995 ","End":"00:07.995","Text":"I\u0027ll be going over how we get to the equation for the trajectory,"},{"Start":"00:07.995 ","End":"00:10.620","Text":"which we were speaking about with the central force"},{"Start":"00:10.620 ","End":"00:14.250","Text":"and specifically when we\u0027re dealing with gravity."},{"Start":"00:14.250 ","End":"00:18.915","Text":"In fact, the equation that I want to reach is this equation over here."},{"Start":"00:18.915 ","End":"00:24.785","Text":"If this reminds you of one of the beginning lessons that we had in this unit."},{"Start":"00:24.785 ","End":"00:30.915","Text":"We will show that our Theta_0 over here is in fact equal to 0."},{"Start":"00:30.915 ","End":"00:35.175","Text":"We\u0027ll be left with an equation that our r as a function of Theta,"},{"Start":"00:35.175 ","End":"00:43.150","Text":"is going to be equal to r_0 divided by 1 plus Epsilon multiplied by cosine of Theta."},{"Start":"00:43.400 ","End":"00:47.575","Text":"Where r_0 and our Epsilon are constants,"},{"Start":"00:47.575 ","End":"00:48.970","Text":"which are these over here,"},{"Start":"00:48.970 ","End":"00:52.255","Text":"and our Alpha, which features in both constants,"},{"Start":"00:52.255 ","End":"00:55.480","Text":"is equal to GMm."},{"Start":"00:55.480 ","End":"01:01.404","Text":"Now, this is going to be less useful for you in answering questions."},{"Start":"01:01.404 ","End":"01:04.523","Text":"However, it\u0027s good to know where your equations come from,"},{"Start":"01:04.523 ","End":"01:07.420","Text":"and depending where you\u0027re studying and what course you\u0027re doing,"},{"Start":"01:07.420 ","End":"01:10.940","Text":"sometimes you will be required to know this."},{"Start":"01:11.450 ","End":"01:14.725","Text":"The way we\u0027re going to start is by using"},{"Start":"01:14.725 ","End":"01:19.995","Text":"our energy equation in its polar coordinate formats,"},{"Start":"01:19.995 ","End":"01:28.440","Text":"so that\u0027s going to be 1/2 mr dot^2 plus L^2 divided by"},{"Start":"01:28.440 ","End":"01:34.725","Text":"2mr^2 minus Alpha divided by"},{"Start":"01:34.725 ","End":"01:41.999","Text":"r. We spoke about this in our lesson about effective potential energy,"},{"Start":"01:41.999 ","End":"01:44.690","Text":"and you\u0027ll recognize over here that this is"},{"Start":"01:44.690 ","End":"01:48.870","Text":"specifically referring to the gravitational pull."},{"Start":"01:49.250 ","End":"01:53.549","Text":"In the lesson about our u effective,"},{"Start":"01:53.549 ","End":"01:57.830","Text":"I showed that my 1/2 mv^2 was broken up when I"},{"Start":"01:57.830 ","End":"02:04.490","Text":"change into polar coordinates with just one variable being r in it\u0027s first derivative."},{"Start":"02:04.490 ","End":"02:09.960","Text":"I saw that the kinetic energy breaks up into these 2 parts."},{"Start":"02:10.400 ","End":"02:18.840","Text":"This came from my radial velocity and this came from my velocity in the Theta direction."},{"Start":"02:19.340 ","End":"02:24.855","Text":"My Alpha over here is this GMm."},{"Start":"02:24.855 ","End":"02:31.090","Text":"Now what I want to do from this point is I want to isolate out my r dot."},{"Start":"02:31.090 ","End":"02:33.230","Text":"Let\u0027s isolate it out."},{"Start":"02:33.230 ","End":"02:34.865","Text":"It\u0027s really simple algebra,"},{"Start":"02:34.865 ","End":"02:40.085","Text":"so I\u0027ll have that my r dot is going to be equal to plus or minus."},{"Start":"02:40.085 ","End":"02:48.770","Text":"Then we\u0027re going to have 2 divided by m multiplied by E minus L^2 divided by"},{"Start":"02:48.770 ","End":"02:54.060","Text":"2mr^2 plus Alpha divided by"},{"Start":"02:54.060 ","End":"03:00.425","Text":"r. Then all of this is going to be to the power of negative 1/2."},{"Start":"03:00.425 ","End":"03:06.040","Text":"We did this again in our lesson about our u effective."},{"Start":"03:06.040 ","End":"03:10.910","Text":"Sorry, this is to the power of a 1/2 not negative 1/2."},{"Start":"03:10.910 ","End":"03:12.395","Text":"We\u0027re taking the square root,"},{"Start":"03:12.395 ","End":"03:15.090","Text":"not 1 divided by the square root."},{"Start":"03:15.170 ","End":"03:20.015","Text":"Now, instead of having our variable over here of r dot,"},{"Start":"03:20.015 ","End":"03:22.025","Text":"so what is our r dot?"},{"Start":"03:22.025 ","End":"03:26.165","Text":"Our r dot is dr by dt."},{"Start":"03:26.165 ","End":"03:32.945","Text":"What I want to do now is I want to make this an expression in terms of Theta."},{"Start":"03:32.945 ","End":"03:36.800","Text":"Because I want to get this equation over here where I have a function in terms"},{"Start":"03:36.800 ","End":"03:40.950","Text":"of Theta. What can I say?"},{"Start":"03:40.950 ","End":"03:46.180","Text":"I can say that my dr by dt=dr"},{"Start":"03:46.180 ","End":"03:52.950","Text":"by d Theta multiplied by d Theta by dt."},{"Start":"03:54.500 ","End":"03:58.285","Text":"This is your chain rule right over here."},{"Start":"03:58.285 ","End":"04:02.955","Text":"Now, I can rewrite this as dr by d Theta,"},{"Start":"04:02.955 ","End":"04:07.095","Text":"and d Theta by dt is simply Theta dot."},{"Start":"04:07.095 ","End":"04:11.780","Text":"Now what I want to do is I want to get rid of my Theta dot over here."},{"Start":"04:11.780 ","End":"04:16.465","Text":"Now what I\u0027m going to do is I\u0027m going to use the same trick using"},{"Start":"04:16.465 ","End":"04:21.130","Text":"angular momentum in order to get rid of this Theta dot over here."},{"Start":"04:21.130 ","End":"04:24.880","Text":"This is the exact same trick that I did in order to get"},{"Start":"04:24.880 ","End":"04:29.740","Text":"my energy equation in terms of just this variable here r,"},{"Start":"04:29.740 ","End":"04:31.255","Text":"so it\u0027s the same trick."},{"Start":"04:31.255 ","End":"04:36.765","Text":"Again, we can go back to our potential energy lesson,"},{"Start":"04:36.765 ","End":"04:41.670","Text":"our u effective lesson and there it goes into more detail."},{"Start":"04:42.460 ","End":"04:46.265","Text":"With a trick, I can say that my angular momentum,"},{"Start":"04:46.265 ","End":"04:50.510","Text":"when we\u0027re dealing with gravity is only in the z direction,"},{"Start":"04:50.510 ","End":"04:52.580","Text":"and that is going to be equal to,"},{"Start":"04:52.580 ","End":"04:54.050","Text":"when we\u0027re dealing with gravity,"},{"Start":"04:54.050 ","End":"04:58.895","Text":"it\u0027s going to be mr^2 multiplied by Theta dot."},{"Start":"04:58.895 ","End":"05:02.990","Text":"Then we can isolate out our Theta dot and we can say that it\u0027s"},{"Start":"05:02.990 ","End":"05:05.530","Text":"equal to L divided by"},{"Start":"05:05.530 ","End":"05:19.140","Text":"mr^2."},{"Start":"05:19.140 ","End":"05:22.305","Text":"How do I get to this equation?"},{"Start":"05:22.305 ","End":"05:30.110","Text":"I can say that my angular momentum is going to be equal to my r vector,"},{"Start":"05:30.110 ","End":"05:32.255","Text":"cross multiplied by my mass,"},{"Start":"05:32.255 ","End":"05:34.955","Text":"multiplied by my velocity vectors."},{"Start":"05:34.955 ","End":"05:38.045","Text":"Then when I write this out in polar coordinates,"},{"Start":"05:38.045 ","End":"05:41.850","Text":"I will be left with this equation over here."},{"Start":"05:43.550 ","End":"05:47.700","Text":"I\u0027ll have this expression and then I isolate out my Theta,"},{"Start":"05:47.700 ","End":"05:50.555","Text":"and here of course I\u0027m just going to write my L because"},{"Start":"05:50.555 ","End":"05:53.855","Text":"even though my angular momentum is just in the z-axis,"},{"Start":"05:53.855 ","End":"05:58.610","Text":"that\u0027s also my total angular momentum because there\u0027s none in the x or y-axis,"},{"Start":"05:58.610 ","End":"06:00.950","Text":"so I don\u0027t have to keep on writing my L,"},{"Start":"06:00.950 ","End":"06:04.710","Text":"and with a z subscript just like this."},{"Start":"06:05.260 ","End":"06:10.595","Text":"Now, of course, my angular momentum is some constant,"},{"Start":"06:10.595 ","End":"06:14.150","Text":"just like my energy over here is some constant as well."},{"Start":"06:14.150 ","End":"06:17.600","Text":"What I do is I work out my angular momentum at some specific point,"},{"Start":"06:17.600 ","End":"06:20.015","Text":"and because we have conservation of angular momentum,"},{"Start":"06:20.015 ","End":"06:25.320","Text":"that will be the value for my angular momentum throughout the motion."},{"Start":"06:25.460 ","End":"06:32.945","Text":"Now what I have to do is I have to substitute in my Theta dot into here."},{"Start":"06:32.945 ","End":"06:39.420","Text":"Then this entire expression is going to switch this over here."},{"Start":"06:40.040 ","End":"06:42.150","Text":"Now rewriting this,"},{"Start":"06:42.150 ","End":"06:43.590","Text":"so let\u0027s put this in."},{"Start":"06:43.590 ","End":"06:52.430","Text":"Our r dot is going to be equal to our dr by d Theta multiplied by our Theta dot,"},{"Start":"06:52.430 ","End":"06:53.885","Text":"which is this over here,"},{"Start":"06:53.885 ","End":"06:58.445","Text":"multiplied by l divided by mr^2."},{"Start":"06:58.445 ","End":"07:06.185","Text":"That is going to be equal to plus or minus r2 divided by m multiplied by"},{"Start":"07:06.185 ","End":"07:15.300","Text":"E minus L^2 divided by 2mr^2 plus Alpha divided by r,"},{"Start":"07:15.300 ","End":"07:19.050","Text":"and all of this to the power of 1/2."},{"Start":"07:19.050 ","End":"07:23.085","Text":"Now I have some differential equation,"},{"Start":"07:23.085 ","End":"07:24.765","Text":"so I want to solve this."},{"Start":"07:24.765 ","End":"07:31.350","Text":"What I\u0027m going to do is I\u0027m going to multiply both sides by my d Theta,"},{"Start":"07:31.350 ","End":"07:38.930","Text":"and I\u0027m also going to divide both sides by this expression over here."},{"Start":"07:38.930 ","End":"07:44.665","Text":"Let\u0027s call this entire thing A,"},{"Start":"07:44.665 ","End":"07:48.455","Text":"so I\u0027m going to divide both sides by my A over here,"},{"Start":"07:48.455 ","End":"07:51.840","Text":"which is including this power sign."},{"Start":"07:52.370 ","End":"07:54.985","Text":"Let\u0027s rewrite this."},{"Start":"07:54.985 ","End":"07:59.625","Text":"We\u0027re going to have our Ldr from here,"},{"Start":"07:59.625 ","End":"08:07.950","Text":"and then it\u0027s going to be divided by mr^2 multiplied by 2 divided by"},{"Start":"08:07.950 ","End":"08:13.200","Text":"m E negative L^2 divided"},{"Start":"08:13.200 ","End":"08:19.095","Text":"by 2mr^2 plus Alpha r,"},{"Start":"08:19.095 ","End":"08:23.165","Text":"and this is going to be to the power of a 1/2."},{"Start":"08:23.165 ","End":"08:27.005","Text":"Then that\u0027s going to be equal to plus or minus,"},{"Start":"08:27.005 ","End":"08:31.210","Text":"our plus or minus is from here, d Theta."},{"Start":"08:32.120 ","End":"08:36.235","Text":"Now what\u0027s left to do is to integrate both sides,"},{"Start":"08:36.235 ","End":"08:41.330","Text":"and the way we\u0027re going to do that is by integration via substitution."},{"Start":"08:41.330 ","End":"08:46.225","Text":"The new variable, which we\u0027re going to substitute in is going to be our u,"},{"Start":"08:46.225 ","End":"08:49.750","Text":"which is equal to 1 divided by r,"},{"Start":"08:49.750 ","End":"08:53.305","Text":"and then when we take the derivative of our u,"},{"Start":"08:53.305 ","End":"08:59.950","Text":"we\u0027ll get that our du is going to be equal to negative 1 divided by"},{"Start":"08:59.950 ","End":"09:09.180","Text":"r^2 dr. Now let\u0027s substitute this into our equation over here."},{"Start":"09:09.180 ","End":"09:13.830","Text":"Here we can see that we have dr divided by r^2."},{"Start":"09:13.830 ","End":"09:23.640","Text":"We\u0027ll have L, dr divided by r^2 is exactly our negative du, so negative du."},{"Start":"09:23.640 ","End":"09:31.620","Text":"Then we\u0027ll have that divided by our m over here."},{"Start":"09:31.620 ","End":"09:36.855","Text":"Then it\u0027s going to be multiplied by 2 over m. Then"},{"Start":"09:36.855 ","End":"09:42.735","Text":"we\u0027ll have our E minus L^2 divided by 2m."},{"Start":"09:42.735 ","End":"09:47.430","Text":"Then here we have 1 over r^2,"},{"Start":"09:47.430 ","End":"09:51.105","Text":"which is exactly a u^2."},{"Start":"09:51.105 ","End":"09:54.250","Text":"Here we\u0027ll have u^2."},{"Start":"09:54.500 ","End":"10:01.425","Text":"Then we have plus r Alpha divided by r,"},{"Start":"10:01.425 ","End":"10:05.580","Text":"which is like Alpha times 1 divided by I,"},{"Start":"10:05.580 ","End":"10:09.675","Text":"which is just Alpha multiplied by u."},{"Start":"10:09.675 ","End":"10:14.325","Text":"Then all of this to the power of 1/2,"},{"Start":"10:14.325 ","End":"10:20.640","Text":"and that is of course going to be equal to d Theta over here."},{"Start":"10:20.640 ","End":"10:25.200","Text":"Of course, we also have a plus minus over here."},{"Start":"10:25.200 ","End":"10:29.910","Text":"Now we can see that we have all of these ms and whatnot over here."},{"Start":"10:29.910 ","End":"10:31.575","Text":"Let\u0027s just rearrange this."},{"Start":"10:31.575 ","End":"10:37.020","Text":"I have negative L du divided by,"},{"Start":"10:37.020 ","End":"10:38.850","Text":"then when we just sort it out,"},{"Start":"10:38.850 ","End":"10:40.065","Text":"these m\u0027s over here,"},{"Start":"10:40.065 ","End":"10:47.775","Text":"we\u0027ll have 2m E negative L^2 u^2"},{"Start":"10:47.775 ","End":"10:56.175","Text":"plus 2m Alpha multiplied by u."},{"Start":"10:56.175 ","End":"11:00.915","Text":"All of this is going to be to the power of 1/2."},{"Start":"11:00.915 ","End":"11:05.130","Text":"This is going to be equal to a plus minus d Theta."},{"Start":"11:05.130 ","End":"11:09.135","Text":"Now in order to get rid of this plus minus over here,"},{"Start":"11:09.135 ","End":"11:11.610","Text":"because I can integrate like this."},{"Start":"11:11.610 ","End":"11:15.075","Text":"What I\u0027m going to do is I\u0027m going to integrate along the area"},{"Start":"11:15.075 ","End":"11:20.235","Text":"where r dot is some positive value."},{"Start":"11:20.235 ","End":"11:25.450","Text":"Then I will just multiply my integral by 2."},{"Start":"11:25.850 ","End":"11:29.204","Text":"That means if we\u0027re dealing with some ellipse,"},{"Start":"11:29.204 ","End":"11:32.955","Text":"so the area where my r dot is going to be positive."},{"Start":"11:32.955 ","End":"11:41.140","Text":"We\u0027ll have a d Theta over here and we can put in our integration signs."},{"Start":"11:41.630 ","End":"11:44.730","Text":"Just to explain this point a little bit better,"},{"Start":"11:44.730 ","End":"11:51.915","Text":"if I have my ellipse over here and my body is orbiting around this x over here,"},{"Start":"11:51.915 ","End":"11:53.699","Text":"and we\u0027re moving in this direction."},{"Start":"11:53.699 ","End":"11:57.645","Text":"I know that in this direction my radius is constantly growing."},{"Start":"11:57.645 ","End":"12:01.840","Text":"My I dot is always going to be positive."},{"Start":"12:02.330 ","End":"12:11.355","Text":"All I have to do is I have to integrate from 0 until Pi."},{"Start":"12:11.355 ","End":"12:16.290","Text":"Then the exact same thing happens when my I dot is negative."},{"Start":"12:16.290 ","End":"12:19.695","Text":"From my Pi to 2Pi,"},{"Start":"12:19.695 ","End":"12:22.695","Text":"however, in the opposite way and the negative."},{"Start":"12:22.695 ","End":"12:24.270","Text":"I\u0027m doing the exact same thing,"},{"Start":"12:24.270 ","End":"12:28.120","Text":"but I just have to do it twice, multiply by 2."},{"Start":"12:29.120 ","End":"12:34.665","Text":"In order to solve this integral because it\u0027s a little bit complicated."},{"Start":"12:34.665 ","End":"12:38.505","Text":"I\u0027m not going to do it completely through right now."},{"Start":"12:38.505 ","End":"12:44.190","Text":"But what I am going to give you is a general equation to solve integrals of this type."},{"Start":"12:44.190 ","End":"12:47.115","Text":"You should write this really in your equation sheet."},{"Start":"12:47.115 ","End":"12:54.690","Text":"If you have an integral of du divided by the square root of a plus"},{"Start":"12:54.690 ","End":"13:04.382","Text":"bu minus cu^2,"},{"Start":"13:04.382 ","End":"13:09.210","Text":"then the integral is going to be equal to 1 divided by"},{"Start":"13:09.210 ","End":"13:16.110","Text":"the square root of c multiplied by cosine to the negative 1 or I cos,"},{"Start":"13:16.110 ","End":"13:17.505","Text":"however you want to call it,"},{"Start":"13:17.505 ","End":"13:28.740","Text":"of to 2cu minus b divided by the square root of b^2 plus 4ac."},{"Start":"13:30.770 ","End":"13:35.430","Text":"This is a general equation and it\u0027s very useful."},{"Start":"13:35.430 ","End":"13:42.075","Text":"Now we can take a look and we can equate this integral to what we have over here."},{"Start":"13:42.075 ","End":"13:47.655","Text":"We can see that our a is some constant,"},{"Start":"13:47.655 ","End":"13:49.215","Text":"which isn\u0027t to do with u."},{"Start":"13:49.215 ","End":"13:50.910","Text":"Over here in our example,"},{"Start":"13:50.910 ","End":"13:53.420","Text":"that\u0027s going to be our 2mE."},{"Start":"13:53.420 ","End":"13:59.240","Text":"Then our b is the coefficient of what\u0027s multiplying our u,"},{"Start":"13:59.240 ","End":"14:01.930","Text":"our linear value, which is this over here."},{"Start":"14:01.930 ","End":"14:04.080","Text":"It\u0027s 2m Alpha."},{"Start":"14:04.080 ","End":"14:09.000","Text":"Then c is our coefficient which is multiplying our u^2."},{"Start":"14:09.000 ","End":"14:12.195","Text":"Notice that there\u0027s a minus over here and a minus over here."},{"Start":"14:12.195 ","End":"14:17.110","Text":"That means that our c is simply going to be L^2."},{"Start":"14:17.450 ","End":"14:20.625","Text":"Now in order to solve some time,"},{"Start":"14:20.625 ","End":"14:22.470","Text":"I\u0027m not going to solve this integral,"},{"Start":"14:22.470 ","End":"14:24.810","Text":"but you can see that you\u0027re going to get this and then you just have to"},{"Start":"14:24.810 ","End":"14:28.110","Text":"substitute in your bounds."},{"Start":"14:28.110 ","End":"14:32.955","Text":"The same with your Theta integrated on Theta and substitute in the bounds."},{"Start":"14:32.955 ","End":"14:38.505","Text":"Once you\u0027ve done that and you integrate according to the formula over here,"},{"Start":"14:38.505 ","End":"14:41.535","Text":"you\u0027re going to get an answer that looks something like this."},{"Start":"14:41.535 ","End":"14:51.465","Text":"R=L^2 divided by mass times Alpha divided by 1 plus the square root"},{"Start":"14:51.465 ","End":"14:58.185","Text":"of 1 plus 2EL^2 divided by"},{"Start":"14:58.185 ","End":"15:07.005","Text":"m Alpha^2 multiplied by cosine of Theta minus Theta_0."},{"Start":"15:07.005 ","End":"15:12.510","Text":"This Theta_0 is our constant from integrating."},{"Start":"15:12.510 ","End":"15:14.565","Text":"Remember that every time we integrate,"},{"Start":"15:14.565 ","End":"15:16.335","Text":"when we integrate without borders,"},{"Start":"15:16.335 ","End":"15:19.800","Text":"we have to add in a plus some constant."},{"Start":"15:19.800 ","End":"15:22.230","Text":"This is that over here."},{"Start":"15:22.230 ","End":"15:28.110","Text":"Now we can see that we have some complicated expression over here."},{"Start":"15:28.110 ","End":"15:32.925","Text":"In order to simplify this and make it look a little bit neater,"},{"Start":"15:32.925 ","End":"15:39.060","Text":"we can say that what is inside this dotted circle is called r_0."},{"Start":"15:39.060 ","End":"15:43.890","Text":"Then we can say that this expression over here"},{"Start":"15:43.890 ","End":"15:49.350","Text":"with the square roots is called our Epsilon."},{"Start":"15:49.350 ","End":"15:51.945","Text":"Then once we rewrite this,"},{"Start":"15:51.945 ","End":"15:57.372","Text":"we\u0027ll have that this is equal to r_0 divided by 1 plus Epsilon"},{"Start":"15:57.372 ","End":"16:04.530","Text":"cosine of Theta minus Theta_0."},{"Start":"16:04.530 ","End":"16:10.050","Text":"This is exactly the expression that we wanted to get initially."},{"Start":"16:10.050 ","End":"16:16.860","Text":"When we scroll back up and we look at what our r_0 and our Epsilon are equal to,"},{"Start":"16:16.860 ","End":"16:20.895","Text":"we can look over here,"},{"Start":"16:20.895 ","End":"16:26.230","Text":"r_0 and Epsilon is equal to exactly that."},{"Start":"16:27.020 ","End":"16:30.195","Text":"We receive the equation that we were looking for."},{"Start":"16:30.195 ","End":"16:38.505","Text":"Now let\u0027s take a look at this Theta_0 and show that it is in fact equal to 0."},{"Start":"16:38.505 ","End":"16:48.570","Text":"This Theta_0 is actually something depending on how we draw our system with our diagram."},{"Start":"16:48.570 ","End":"16:52.440","Text":"Let\u0027s say we have our ellipse over here."},{"Start":"16:52.440 ","End":"16:56.745","Text":"Then we say that our origin is over here."},{"Start":"16:56.745 ","End":"16:59.160","Text":"This is the direction of our x-axis,"},{"Start":"16:59.160 ","End":"17:02.580","Text":"and this is the direction of our y-axis."},{"Start":"17:02.580 ","End":"17:05.535","Text":"Then at our point over here,"},{"Start":"17:05.535 ","End":"17:08.080","Text":"where we have our r_min."},{"Start":"17:12.560 ","End":"17:18.900","Text":"Then I\u0027m saying that when my Theta is equal 0,"},{"Start":"17:18.900 ","End":"17:23.025","Text":"I need my r over here to be minimal."},{"Start":"17:23.025 ","End":"17:26.110","Text":"At some minimum point."},{"Start":"17:26.180 ","End":"17:29.145","Text":"That means that I have to do this."},{"Start":"17:29.145 ","End":"17:31.125","Text":"I have to say that my r,"},{"Start":"17:31.125 ","End":"17:34.455","Text":"when my Theta=0,"},{"Start":"17:34.455 ","End":"17:38.265","Text":"has to be equal to my r_min."},{"Start":"17:38.265 ","End":"17:44.100","Text":"What does that mean? It means that when I substitute in Theta\u0027s equal to 0 over here,"},{"Start":"17:44.100 ","End":"17:49.710","Text":"I have to get that my denominator over here is the largest possible."},{"Start":"17:49.710 ","End":"17:55.780","Text":"Then my whole expression will be the smallest it can possibly be."},{"Start":"17:59.300 ","End":"18:04.110","Text":"The only thing that is a variable here in order to make my denominator as"},{"Start":"18:04.110 ","End":"18:08.790","Text":"large as possible is my cosine of Theta_0."},{"Start":"18:08.790 ","End":"18:14.490","Text":"I want my cosine of Theta_0 to be the largest possible value."},{"Start":"18:16.330 ","End":"18:23.040","Text":"The largest value that cosine of anything can be is positive 1."},{"Start":"18:23.040 ","End":"18:27.525","Text":"Then if my cosine of Theta_0=1,"},{"Start":"18:27.525 ","End":"18:32.445","Text":"that means that my Theta_0 must be equal to 0."},{"Start":"18:32.445 ","End":"18:35.186","Text":"Then we get that exactly."},{"Start":"18:35.186 ","End":"18:38.990","Text":"Now we\u0027ve shown how we got to"},{"Start":"18:38.990 ","End":"18:43.790","Text":"this equation and we\u0027ve also shown that our Theta_0 is in fact equal to 0."},{"Start":"18:43.790 ","End":"18:46.730","Text":"Now we can write this in where we substitute in"},{"Start":"18:46.730 ","End":"18:52.100","Text":"our Theta_0=0 and we\u0027ll get that our equation for our position as"},{"Start":"18:52.100 ","End":"19:00.850","Text":"a function of Theta=r_0 divided by 1 plus Epsilon multiplied by cosine of Theta."},{"Start":"19:00.850 ","End":"19:04.430","Text":"That is exactly the expression that we wanted to get."},{"Start":"19:04.430 ","End":"19:06.810","Text":"That\u0027s the end of this lesson."}],"ID":9467}],"Thumbnail":null,"ID":5361},{"Name":"Keplers Laws","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Kepler\u0026#39;s First And Second Laws","Duration":"12m 57s","ChapterTopicVideoID":9198,"CourseChapterTopicPlaylistID":5362,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Hello. In this lesson,"},{"Start":"00:01.830 ","End":"00:04.169","Text":"we\u0027ll be speaking about Kepler\u0027s laws."},{"Start":"00:04.169 ","End":"00:07.650","Text":"In this video, we\u0027ll be speaking about the first and second laws,"},{"Start":"00:07.650 ","End":"00:10.590","Text":"and in the next video we\u0027ll speak about the third law."},{"Start":"00:10.590 ","End":"00:15.165","Text":"The first law says that the orbit of every planet around the sun is"},{"Start":"00:15.165 ","End":"00:19.617","Text":"an ellipse where the sun is located at one of the foci."},{"Start":"00:19.617 ","End":"00:22.350","Text":"This is something that we already know,"},{"Start":"00:22.350 ","End":"00:25.245","Text":"we\u0027ve already seen this in our equations."},{"Start":"00:25.245 ","End":"00:27.960","Text":"But what\u0027s important to know is that Kepler knew"},{"Start":"00:27.960 ","End":"00:31.665","Text":"this long before we develop the equations."},{"Start":"00:31.665 ","End":"00:35.049","Text":"Kepler knew it from observation."},{"Start":"00:35.049 ","End":"00:39.870","Text":"We already know that all of the planets orbit the sun in some kind of ellipse."},{"Start":"00:39.870 ","End":"00:43.685","Text":"Here\u0027s our Earth also orbiting in an ellipse."},{"Start":"00:43.685 ","End":"00:45.800","Text":"What\u0027s important to know is that with"},{"Start":"00:45.800 ","End":"00:49.850","Text":"any big body which is located at some point in space,"},{"Start":"00:49.850 ","End":"00:53.180","Text":"and if another smaller body is orbiting about"},{"Start":"00:53.180 ","End":"00:58.675","Text":"that larger body then the trajectory will always be elliptical."},{"Start":"00:58.675 ","End":"01:04.550","Text":"The second law is that the line which connects the planet to the sun,"},{"Start":"01:04.550 ","End":"01:07.355","Text":"that\u0027s this line over here, the radius."},{"Start":"01:07.355 ","End":"01:15.245","Text":"The position vector covers equal areas at equal time intervals. What does that mean?"},{"Start":"01:15.245 ","End":"01:18.680","Text":"If we have our Earth over here at this position vector"},{"Start":"01:18.680 ","End":"01:23.325","Text":"here and then with some time interval,"},{"Start":"01:23.325 ","End":"01:25.180","Text":"let\u0027s call it Delta t_1."},{"Start":"01:25.180 ","End":"01:29.395","Text":"Our Earth continues in the orbit until it reaches this point over here."},{"Start":"01:29.395 ","End":"01:36.300","Text":"If we calculate the area between its starting point and its end point,"},{"Start":"01:36.300 ","End":"01:41.180","Text":"connecting back to our focal point number 1 where our sun is."},{"Start":"01:41.450 ","End":"01:44.733","Text":"That area, let\u0027s call it S_1."},{"Start":"01:44.733 ","End":"01:49.085","Text":"If our Earth carries on in its orbit and then it\u0027s located over here."},{"Start":"01:49.085 ","End":"01:52.610","Text":"Again, this is its position vector,"},{"Start":"01:52.610 ","End":"01:57.940","Text":"and it carries on in its orbit until this point over here in a time of Delta t_2."},{"Start":"01:57.940 ","End":"02:00.515","Text":"Then we calculate again this area,"},{"Start":"02:00.515 ","End":"02:02.405","Text":"the area of this segment."},{"Start":"02:02.405 ","End":"02:11.755","Text":"What Kepler\u0027s second law says is that if our Delta t_1 is equal to our Delta t_2,"},{"Start":"02:11.755 ","End":"02:15.320","Text":"then that means that our area of the segments,"},{"Start":"02:15.320 ","End":"02:20.240","Text":"so our S_1 will be equal to our S_2."},{"Start":"02:20.240 ","End":"02:26.030","Text":"Now, what if our Delta t_1 does not equal to our Delta t_2?"},{"Start":"02:26.030 ","End":"02:28.510","Text":"They\u0027re not equal."},{"Start":"02:28.510 ","End":"02:31.625","Text":"That means over here written,"},{"Start":"02:31.625 ","End":"02:33.905","Text":"if the time intervals are not equal,"},{"Start":"02:33.905 ","End":"02:39.215","Text":"the ratio between the area covered and the time interval will be fixed."},{"Start":"02:39.215 ","End":"02:43.020","Text":"That means that the area covered with Delta t_1,"},{"Start":"02:43.020 ","End":"02:47.715","Text":"so that\u0027s going to be S_1 divided by Delta t_1."},{"Start":"02:47.715 ","End":"02:51.255","Text":"The ratio is going to be constant."},{"Start":"02:51.255 ","End":"02:58.140","Text":"S_1 divided by Delta t_1 is going to be the same as S_2 divided by Delta t_2."},{"Start":"02:58.140 ","End":"03:01.340","Text":"In fact, the total area inside the ellipse,"},{"Start":"03:01.340 ","End":"03:06.630","Text":"so S total divided by the period."},{"Start":"03:06.630 ","End":"03:12.510","Text":"That\u0027s the time taken to complete 1 full orbit is also going to be equal."},{"Start":"03:12.510 ","End":"03:15.710","Text":"These are 2 very useful equations."},{"Start":"03:15.710 ","End":"03:18.890","Text":"Now another useful equation if we\u0027re already speaking about an ellipse,"},{"Start":"03:18.890 ","End":"03:27.385","Text":"is that the total area of an ellipse is going to be equal to Pi times a times b."},{"Start":"03:27.385 ","End":"03:30.650","Text":"What I\u0027m going to show now is how we can get to"},{"Start":"03:30.650 ","End":"03:35.008","Text":"Kepler\u0027s second law from the idea of angular momentum."},{"Start":"03:35.008 ","End":"03:40.205","Text":"I\u0027m going to start by working out an infinitesimal area,"},{"Start":"03:40.205 ","End":"03:44.315","Text":"dS, which our radius will cover."},{"Start":"03:44.315 ","End":"03:48.740","Text":"Let\u0027s imagine that our Earth is over here,"},{"Start":"03:48.740 ","End":"03:51.320","Text":"and this is its radius."},{"Start":"03:51.320 ","End":"03:55.145","Text":"Then after a short period of time, Delta t,"},{"Start":"03:55.145 ","End":"04:01.380","Text":"my Earth has orbited and reached this new point over here."},{"Start":"04:01.380 ","End":"04:07.205","Text":"We can see that the angle covered over here from this point to here,"},{"Start":"04:07.205 ","End":"04:12.545","Text":"we can call this d Theta to represent a very small angle."},{"Start":"04:12.545 ","End":"04:14.150","Text":"In fact over here,"},{"Start":"04:14.150 ","End":"04:18.500","Text":"we can call this time dt to show that it\u0027s a very,"},{"Start":"04:18.500 ","End":"04:20.005","Text":"very small amount of time."},{"Start":"04:20.005 ","End":"04:24.534","Text":"The area that I have covered is this area over here,"},{"Start":"04:24.534 ","End":"04:27.149","Text":"and we can call this area dS."},{"Start":"04:27.149 ","End":"04:30.064","Text":"That is what I want to calculate."},{"Start":"04:30.064 ","End":"04:31.580","Text":"How much is this area?"},{"Start":"04:31.580 ","End":"04:32.660","Text":"This is a very,"},{"Start":"04:32.660 ","End":"04:36.260","Text":"very useful calculation to know how to do."},{"Start":"04:36.260 ","End":"04:39.020","Text":"When I want to calculate my area dS,"},{"Start":"04:39.020 ","End":"04:45.020","Text":"I have to integrate on my infinitesimal area,"},{"Start":"04:45.020 ","End":"04:50.890","Text":"which in polar coordinates is going to be r dr d Theta."},{"Start":"04:50.890 ","End":"04:54.080","Text":"What I\u0027m integrating on over here is"},{"Start":"04:54.080 ","End":"04:58.160","Text":"simply my r because my r is what is in fact changing."},{"Start":"04:58.160 ","End":"05:03.740","Text":"My Theta, my angle over here is going to be so small that what I could do is"},{"Start":"05:03.740 ","End":"05:09.500","Text":"I can add in another integration sign and have my bounds from 0 until d Theta."},{"Start":"05:09.500 ","End":"05:13.130","Text":"Then when I do that, I\u0027ll just be left with again, my d Theta."},{"Start":"05:13.130 ","End":"05:19.420","Text":"We can just leave it like that and integrate via r. Let\u0027s add in our bounds for r."},{"Start":"05:19.420 ","End":"05:25.897","Text":"We\u0027re integrating from our radius is equal to 0 until we reach some value for r,"},{"Start":"05:25.897 ","End":"05:28.850","Text":"which as we can see as a function of Theta."},{"Start":"05:28.850 ","End":"05:31.490","Text":"Depending on the angle that we\u0027re at r,"},{"Start":"05:31.490 ","End":"05:34.205","Text":"radius will be slightly bigger."},{"Start":"05:34.205 ","End":"05:37.055","Text":"Just as an example, when our Theta is equal to 0,"},{"Start":"05:37.055 ","End":"05:39.890","Text":"we can see that our r is at some minimum."},{"Start":"05:39.890 ","End":"05:41.945","Text":"When our Theta is equal to Pi,"},{"Start":"05:41.945 ","End":"05:44.500","Text":"we can see that our r will be at a maximum."},{"Start":"05:44.500 ","End":"05:46.970","Text":"Now when I integrate,"},{"Start":"05:46.970 ","End":"05:56.005","Text":"I\u0027m going to have my r as a function of Theta^2 divided by 2 and multiplied by d Theta."},{"Start":"05:56.005 ","End":"05:59.010","Text":"This is what my dS is equal to,"},{"Start":"05:59.010 ","End":"06:03.755","Text":"some small area that I cover."},{"Start":"06:03.755 ","End":"06:11.040","Text":"Another way of getting to this answer is to say that because we move our dt,"},{"Start":"06:11.040 ","End":"06:14.450","Text":"is so such a small value that our d Theta is so small."},{"Start":"06:14.450 ","End":"06:16.340","Text":"Then we can say that at an estimate,"},{"Start":"06:16.340 ","End":"06:19.395","Text":"this is like some kind of triangle."},{"Start":"06:19.395 ","End":"06:24.625","Text":"Then our dS is simply going to be the area of said triangle."},{"Start":"06:24.625 ","End":"06:28.310","Text":"Then we\u0027ll get to the exact same answer"},{"Start":"06:28.310 ","End":"06:31.760","Text":"because we\u0027ll just have a 1/2 times the base times the height,"},{"Start":"06:31.760 ","End":"06:33.985","Text":"and we\u0027ll get this value again."},{"Start":"06:33.985 ","End":"06:36.935","Text":"What we found over here is very important."},{"Start":"06:36.935 ","End":"06:39.695","Text":"I\u0027ll just square this in red."},{"Start":"06:39.695 ","End":"06:45.250","Text":"This is very, very useful to know and it will come in handy in all calculations."},{"Start":"06:45.250 ","End":"06:54.210","Text":"The next thing that I want to know is what my value of dS is relative to dt."},{"Start":"06:54.210 ","End":"06:56.845","Text":"My area is dependent on time."},{"Start":"06:56.845 ","End":"06:59.900","Text":"What I want to show in the end is that"},{"Start":"06:59.900 ","End":"07:04.999","Text":"my area is dependent on time but it has a fixed dependency,"},{"Start":"07:04.999 ","End":"07:06.785","Text":"it\u0027s some constant,"},{"Start":"07:06.785 ","End":"07:09.160","Text":"and the value is never changing."},{"Start":"07:09.160 ","End":"07:11.085","Text":"Let\u0027s rewrite this."},{"Start":"07:11.085 ","End":"07:16.545","Text":"I want to find my dS as some relation to dt."},{"Start":"07:16.545 ","End":"07:22.760","Text":"My dS by dt is going to be equal to this also divided by dt."},{"Start":"07:22.760 ","End":"07:27.210","Text":"So I\u0027ll have my r as a function of Theta^2 divided by 2,"},{"Start":"07:27.210 ","End":"07:31.610","Text":"d Theta by dt."},{"Start":"07:31.610 ","End":"07:35.180","Text":"Now what I\u0027m going to do is I\u0027m going to multiply both sides by dt."},{"Start":"07:35.180 ","End":"07:37.805","Text":"I\u0027ll be left over here with my dS,"},{"Start":"07:37.805 ","End":"07:42.010","Text":"and then I have my dt over here."},{"Start":"07:42.010 ","End":"07:47.170","Text":"Now we know that d Theta by dt is equal to Theta dot."},{"Start":"07:47.170 ","End":"07:49.250","Text":"We can rewrite this as r,"},{"Start":"07:49.250 ","End":"07:56.545","Text":"as a function of Theta^2 divide by 2 multiplied by Theta dot dt."},{"Start":"07:56.545 ","End":"08:01.290","Text":"Now what I want to do is I want to find what my Theta dot is equal to,"},{"Start":"08:01.290 ","End":"08:05.089","Text":"and I\u0027m going to do that by working out my angular momentum."},{"Start":"08:05.089 ","End":"08:13.205","Text":"I\u0027m going to say that my angular momentum is equal to mr^2 multiplied by Theta dot."},{"Start":"08:13.205 ","End":"08:17.150","Text":"I\u0027ve shown how we got to this equation a few times."},{"Start":"08:17.150 ","End":"08:20.875","Text":"If you can\u0027t remember, go back to some previous lessons."},{"Start":"08:20.875 ","End":"08:26.765","Text":"Just really quickly, our angular momentum comes from the equation of r cross mv."},{"Start":"08:26.765 ","End":"08:31.370","Text":"Then this is always the answer that will get when we\u0027re dealing with central force."},{"Start":"08:31.370 ","End":"08:33.665","Text":"Of course, as previously discussed,"},{"Start":"08:33.665 ","End":"08:39.350","Text":"our angular momentum over here is always going to be only in the z-direction."},{"Start":"08:39.350 ","End":"08:43.715","Text":"But for the sake of foster writing and easier to read,"},{"Start":"08:43.715 ","End":"08:46.160","Text":"we just write it as L because we anyway,"},{"Start":"08:46.160 ","End":"08:49.785","Text":"have no angular momentum in the x or y-axis."},{"Start":"08:49.785 ","End":"08:52.445","Text":"Just to make this a little bit clear,"},{"Start":"08:52.445 ","End":"08:59.750","Text":"our angular momentum is equal to our r vector cross our mv vector,"},{"Start":"08:59.750 ","End":"09:04.835","Text":"which in polar coordinates is going to be r in the r direction plus"},{"Start":"09:04.835 ","End":"09:14.070","Text":"m multiplied by r dot in the r direction plus r Theta dot in the Theta direction."},{"Start":"09:14.070 ","End":"09:16.500","Text":"This is cross multiplied over here. Then when"},{"Start":"09:16.500 ","End":"09:24.525","Text":"we have our r cross our r, that\u0027s going to have a 0."},{"Start":"09:24.525 ","End":"09:27.050","Text":"Then r cross this over here,"},{"Start":"09:27.050 ","End":"09:29.840","Text":"we will get our angular momentum just in"},{"Start":"09:29.840 ","End":"09:33.965","Text":"the z direction and it will be equal to this equation over here."},{"Start":"09:33.965 ","End":"09:37.670","Text":"Now what we do is we want to isolate out our Theta dot."},{"Start":"09:37.670 ","End":"09:43.505","Text":"I have that our Theta dot is equal to l divided by mr^2."},{"Start":"09:43.505 ","End":"09:47.464","Text":"Then we substitute in this Theta dot in over here."},{"Start":"09:47.464 ","End":"09:53.045","Text":"Then we\u0027ll see that our r^2 over here will cross out with this r^2."},{"Start":"09:53.045 ","End":"10:03.715","Text":"What will be left with is that our dS is equal to l divided by m multiplied by dt."},{"Start":"10:03.715 ","End":"10:07.005","Text":"Once we substitute in this Theta dot over here."},{"Start":"10:07.005 ","End":"10:08.795","Text":"I forgot a 2 over here."},{"Start":"10:08.795 ","End":"10:13.460","Text":"Now we can see that my angular momentum is constant,"},{"Start":"10:13.460 ","End":"10:16.644","Text":"my m is constant and my 2 is constant."},{"Start":"10:16.644 ","End":"10:20.315","Text":"Then when I divide both sides by dt,"},{"Start":"10:20.315 ","End":"10:27.410","Text":"I\u0027ll get my dS by dt and I\u0027ll be left with l divided by 2m,"},{"Start":"10:27.410 ","End":"10:30.850","Text":"which is of course a constant value."},{"Start":"10:30.850 ","End":"10:33.560","Text":"This is exactly what I wanted to show,"},{"Start":"10:33.560 ","End":"10:38.300","Text":"that the relationship or the ratio between my area and my time,"},{"Start":"10:38.300 ","End":"10:41.780","Text":"no matter which area and time I\u0027m substituting in,"},{"Start":"10:41.780 ","End":"10:45.665","Text":"is always going to be a constant value,"},{"Start":"10:45.665 ","End":"10:50.270","Text":"which is exactly what we said in Kepler\u0027s second law."},{"Start":"10:50.270 ","End":"10:54.770","Text":"Even if our dt_1 is different to my dt_2,"},{"Start":"10:54.770 ","End":"11:00.900","Text":"the area covered in my Delta t_1 divided by"},{"Start":"11:00.900 ","End":"11:07.665","Text":"my Delta t_1 is going to be the same constant value as my area S_2,"},{"Start":"11:07.665 ","End":"11:09.720","Text":"which was covered in Delta t_2,"},{"Start":"11:09.720 ","End":"11:11.990","Text":"and that\u0027s going to be the same as"},{"Start":"11:11.990 ","End":"11:16.985","Text":"the total area of the ellipse divided by 1 time period."},{"Start":"11:16.985 ","End":"11:23.100","Text":"Now another way of looking at it is if I decide to integrate at this stage over here,"},{"Start":"11:23.380 ","End":"11:26.395","Text":"can put in my integration."},{"Start":"11:26.395 ","End":"11:28.760","Text":"I will see that once I\u0027ve integrated,"},{"Start":"11:28.760 ","End":"11:36.530","Text":"I\u0027ll have that my S is equal to l divided by 2m multiplied by this time interval."},{"Start":"11:36.530 ","End":"11:39.545","Text":"Let\u0027s call it some Delta t. Then again,"},{"Start":"11:39.545 ","End":"11:44.720","Text":"when I divide my S by this Delta t and left with some constant value."},{"Start":"11:44.720 ","End":"11:50.064","Text":"No matter what my S and my Delta T is for any general values."},{"Start":"11:50.064 ","End":"11:53.000","Text":"Just to make what I said slightly clearer,"},{"Start":"11:53.000 ","End":"11:57.140","Text":"the S has to correspond to my Delta t. This is"},{"Start":"11:57.140 ","End":"12:01.795","Text":"specifically the area covered in this time interval."},{"Start":"12:01.795 ","End":"12:03.770","Text":"That\u0027s what that means."},{"Start":"12:03.770 ","End":"12:08.210","Text":"That\u0027s the end of our lesson for Kepler\u0027s first and second law."},{"Start":"12:08.210 ","End":"12:10.805","Text":"in the next video, we\u0027ll be speaking about the third law."},{"Start":"12:10.805 ","End":"12:12.290","Text":"What we learned over here,"},{"Start":"12:12.290 ","End":"12:17.935","Text":"the most important thing is that for any elliptical orbits,"},{"Start":"12:17.935 ","End":"12:20.765","Text":"in some kind of time interval,"},{"Start":"12:20.765 ","End":"12:24.725","Text":"if we\u0027re going to find the area covered in that time interval."},{"Start":"12:24.725 ","End":"12:30.920","Text":"The ratio between the area in our time interval is going to be a constant value,"},{"Start":"12:30.920 ","End":"12:33.055","Text":"which means that it\u0027s fixed."},{"Start":"12:33.055 ","End":"12:38.350","Text":"That also means that if our Delta t_1 is equal to our Delta t_2,"},{"Start":"12:38.350 ","End":"12:42.125","Text":"then the corresponding areas are going to be equal."},{"Start":"12:42.125 ","End":"12:45.650","Text":"It doesn\u0027t matter where we\u0027re looking on our orbit."},{"Start":"12:45.650 ","End":"12:47.825","Text":"If we\u0027re orbiting from this point to this point,"},{"Start":"12:47.825 ","End":"12:49.400","Text":"or this point to this point."},{"Start":"12:49.400 ","End":"12:51.440","Text":"If our time intervals are equal,"},{"Start":"12:51.440 ","End":"12:54.665","Text":"then our areas covered will also be equal."},{"Start":"12:54.665 ","End":"12:57.660","Text":"That\u0027s the end of this lesson."}],"ID":9468},{"Watched":false,"Name":"Kepler\u0026#39;s Third Law","Duration":"3m 38s","ChapterTopicVideoID":9199,"CourseChapterTopicPlaylistID":5362,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"Hello. In our previous video,"},{"Start":"00:02.340 ","End":"00:04.335","Text":"we spoke about Kepler\u0027s laws,"},{"Start":"00:04.335 ","End":"00:05.655","Text":"number 1 and 2."},{"Start":"00:05.655 ","End":"00:09.555","Text":"In this video, we\u0027re going to be speaking about Kepler\u0027s 3rd law."},{"Start":"00:09.555 ","End":"00:14.250","Text":"Now, Kepler\u0027s third law says that the square of a planet\u0027s orbital period is"},{"Start":"00:14.250 ","End":"00:19.590","Text":"proportional to the cube of the length of the orbit\u0027s semi-major axis."},{"Start":"00:19.590 ","End":"00:25.875","Text":"What does that mean? We know that from this point over here to"},{"Start":"00:25.875 ","End":"00:28.900","Text":"the furthest point over here and"},{"Start":"00:28.900 ","End":"00:33.635","Text":"the line joining these 2 points together is the major axis."},{"Start":"00:33.635 ","End":"00:36.800","Text":"The semi-major axis, semi is half."},{"Start":"00:36.800 ","End":"00:38.240","Text":"We know that that\u0027s going to be half of"},{"Start":"00:38.240 ","End":"00:41.605","Text":"the major axis and that\u0027s going to be this length a."},{"Start":"00:41.605 ","End":"00:45.580","Text":"This b is the semi-minor axis."},{"Start":"00:45.710 ","End":"00:50.030","Text":"That means that the square of the planet\u0027s orbital period is"},{"Start":"00:50.030 ","End":"00:55.475","Text":"proportional to the cube of the length of the orbit\u0027s semi-major axis,"},{"Start":"00:55.475 ","End":"00:58.265","Text":"as given over here by this equation."},{"Start":"00:58.265 ","End":"00:59.600","Text":"I\u0027ve squared it in red."},{"Start":"00:59.600 ","End":"01:02.405","Text":"It\u0027s important, you should write this equation down."},{"Start":"01:02.405 ","End":"01:04.820","Text":"T over here is our period,"},{"Start":"01:04.820 ","End":"01:10.690","Text":"a is simply the length of the semi-major axis,"},{"Start":"01:10.690 ","End":"01:13.165","Text":"our G is our gravitational constant,"},{"Start":"01:13.165 ","End":"01:17.805","Text":"and our M is the mass of the large body."},{"Start":"01:17.805 ","End":"01:19.850","Text":"If our sun is, for instance,"},{"Start":"01:19.850 ","End":"01:23.330","Text":"located over here and our Earth is located here,"},{"Start":"01:23.330 ","End":"01:25.400","Text":"obviously, the large body,"},{"Start":"01:25.400 ","End":"01:27.620","Text":"the one with a significantly larger mass,"},{"Start":"01:27.620 ","End":"01:29.395","Text":"is going to be the sun."},{"Start":"01:29.395 ","End":"01:33.395","Text":"We\u0027re referring to the mass of this over here."},{"Start":"01:33.395 ","End":"01:37.040","Text":"Now, a little pointer on the off chance that we have"},{"Start":"01:37.040 ","End":"01:41.465","Text":"some planet orbiting a large body in a perfect circle."},{"Start":"01:41.465 ","End":"01:45.245","Text":"In this equation, instead of writing an a over here,"},{"Start":"01:45.245 ","End":"01:49.440","Text":"we\u0027ll simply write the radius of the circle."},{"Start":"01:49.440 ","End":"01:53.460","Text":"Instead of a, we\u0027ll just substitute the radius of the circle."},{"Start":"01:53.480 ","End":"02:00.965","Text":"A circle is just a specific type of ellipse where our a and our b are equal."},{"Start":"02:00.965 ","End":"02:05.445","Text":"That means we just substitute in the a."},{"Start":"02:05.445 ","End":"02:09.770","Text":"Now, let\u0027s see what Kepler\u0027s 3rd law says about 2 bodies"},{"Start":"02:09.770 ","End":"02:12.335","Text":"which have around about the same mass."},{"Start":"02:12.335 ","End":"02:18.020","Text":"This equation we\u0027re assuming that the sun, for instance,"},{"Start":"02:18.020 ","End":"02:21.800","Text":"or the body that another planet,"},{"Start":"02:21.800 ","End":"02:23.990","Text":"a smaller planet is orbiting around."},{"Start":"02:23.990 ","End":"02:26.480","Text":"Assuming that the mass of this is"},{"Start":"02:26.480 ","End":"02:30.925","Text":"significantly larger than the mass of the orbiting planet."},{"Start":"02:30.925 ","End":"02:34.400","Text":"Now, let\u0027s speak about the case over here,"},{"Start":"02:34.400 ","End":"02:37.010","Text":"such as binary stars,"},{"Start":"02:37.010 ","End":"02:43.445","Text":"where the mass of this body is approximately equal to the mass of this body."},{"Start":"02:43.445 ","End":"02:47.425","Text":"How can we adapt this equation to that case?"},{"Start":"02:47.425 ","End":"02:53.540","Text":"What we do in this type of case is instead of taking the mass of just the largest body,"},{"Start":"02:53.540 ","End":"02:56.735","Text":"we take the sum of the masses of the 2 bodies."},{"Start":"02:56.735 ","End":"02:58.715","Text":"Instead of an M over here,"},{"Start":"02:58.715 ","End":"03:01.010","Text":"we\u0027ll have m_1 and m_2,"},{"Start":"03:01.010 ","End":"03:05.980","Text":"where this is our m_1 and this is our m_2."},{"Start":"03:06.080 ","End":"03:10.010","Text":"Then the equations are exactly the same."},{"Start":"03:10.010 ","End":"03:13.190","Text":"We still have that the orbital periods,"},{"Start":"03:13.190 ","End":"03:18.515","Text":"so the square of it is proportional to the cube length of the orbit semi-major axis."},{"Start":"03:18.515 ","End":"03:21.860","Text":"Except this time we\u0027re taking into account both of"},{"Start":"03:21.860 ","End":"03:26.520","Text":"the masses because they\u0027re each approximately equal to one another."},{"Start":"03:27.470 ","End":"03:31.580","Text":"These 2 laws are super important to remember"},{"Start":"03:31.580 ","End":"03:35.570","Text":"and they make it really easy to find our time period."},{"Start":"03:35.570 ","End":"03:38.340","Text":"That\u0027s the end of this lesson."}],"ID":9469},{"Watched":false,"Name":"Finding Time Period Kepler\u0026#39;s Second Law","Duration":"5m 26s","ChapterTopicVideoID":9200,"CourseChapterTopicPlaylistID":5362,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"Hello. In this question,"},{"Start":"00:02.280 ","End":"00:07.470","Text":"we\u0027re being told that a body orbits the sun with an elliptical trajectory such that"},{"Start":"00:07.470 ","End":"00:13.065","Text":"its maximum velocity corresponds to its minimum distance from the sun,"},{"Start":"00:13.065 ","End":"00:15.660","Text":"and both of those values are given."},{"Start":"00:15.660 ","End":"00:19.845","Text":"We\u0027re also being told that the area of the ellipse is also given."},{"Start":"00:19.845 ","End":"00:23.790","Text":"We\u0027re being asked to find the time period of the body."},{"Start":"00:23.790 ","End":"00:26.460","Text":"Let\u0027s see how to solve this."},{"Start":"00:26.460 ","End":"00:30.990","Text":"What I want to do is I want to look at the body and see what"},{"Start":"00:30.990 ","End":"00:36.705","Text":"happens when it moves a tiny bit down here."},{"Start":"00:36.705 ","End":"00:41.890","Text":"It\u0027s covered an infinitely small area."},{"Start":"00:41.890 ","End":"00:45.965","Text":"That means that exactly when it\u0027s at the point of r_min."},{"Start":"00:45.965 ","End":"00:49.580","Text":"It moves some kind of small distance,"},{"Start":"00:49.580 ","End":"00:52.094","Text":"which we\u0027ll call dx."},{"Start":"00:52.094 ","End":"00:58.085","Text":"In the lesson, I spoke about 2 ways that we can find this small area, ds."},{"Start":"00:58.085 ","End":"01:00.515","Text":"Right now we\u0027re going to use"},{"Start":"01:00.515 ","End":"01:06.605","Text":"the way of considering this as a triangle and working out the area."},{"Start":"01:06.605 ","End":"01:11.780","Text":"What we\u0027re going to do is we\u0027re going to draw a line like this and like this,"},{"Start":"01:11.780 ","End":"01:14.195","Text":"and we\u0027re going to make a triangle."},{"Start":"01:14.195 ","End":"01:20.170","Text":"Now we know that the area inside this triangle is called ds,"},{"Start":"01:20.170 ","End":"01:22.325","Text":"and because we know it\u0027s a triangle,"},{"Start":"01:22.325 ","End":"01:26.435","Text":"we know that the equation for our triangle is going to be the base,"},{"Start":"01:26.435 ","End":"01:28.580","Text":"which is our dx,"},{"Start":"01:28.580 ","End":"01:31.204","Text":"multiplied by the height,"},{"Start":"01:31.204 ","End":"01:32.850","Text":"which is our r_min,"},{"Start":"01:32.850 ","End":"01:36.080","Text":"and then divided by 2 because"},{"Start":"01:36.080 ","End":"01:40.030","Text":"the equation for the area of a triangle is half base times height."},{"Start":"01:40.030 ","End":"01:43.220","Text":"What we\u0027re going to do is we\u0027re going to rewrite this,"},{"Start":"01:43.220 ","End":"01:49.265","Text":"so we\u0027ll have half of our r_min and then our dx is some kind of distance."},{"Start":"01:49.265 ","End":"01:56.690","Text":"If we remember, we have the equation of speed is distance divided by time."},{"Start":"01:56.690 ","End":"02:03.361","Text":"That means that our distance is equal to speed or velocity multiplied by time."},{"Start":"02:03.361 ","End":"02:05.285","Text":"Our velocity at this point,"},{"Start":"02:05.285 ","End":"02:07.360","Text":"we know, we\u0027re being given that in the question,"},{"Start":"02:07.360 ","End":"02:12.680","Text":"so it\u0027s our v_max and then we\u0027re going to multiply it by dt,"},{"Start":"02:12.680 ","End":"02:13.970","Text":"the change in time."},{"Start":"02:13.970 ","End":"02:19.535","Text":"We\u0027re assuming that in this area our v_max is going to be constant."},{"Start":"02:19.535 ","End":"02:26.480","Text":"It\u0027s changing throughout the orbit but we\u0027re assuming that this dx is so small"},{"Start":"02:26.480 ","End":"02:27.950","Text":"that we can say that"},{"Start":"02:27.950 ","End":"02:34.280","Text":"the change in our velocity from our v_max to the different velocity,"},{"Start":"02:34.280 ","End":"02:39.155","Text":"is going to be so small that it\u0027s insignificant for our calculations."},{"Start":"02:39.155 ","End":"02:41.105","Text":"Then we\u0027re going to be left with this."},{"Start":"02:41.105 ","End":"02:43.730","Text":"What we\u0027re going to do is we\u0027re going to divide"},{"Start":"02:43.730 ","End":"02:48.095","Text":"both sides by our dt in order to get our relationship"},{"Start":"02:48.095 ","End":"02:52.280","Text":"between our change in area covered or"},{"Start":"02:52.280 ","End":"02:57.060","Text":"our area covered in our period of time and our time interval."},{"Start":"02:57.060 ","End":"03:03.920","Text":"We\u0027ll be left with ds divided by dt and that is going to be equal"},{"Start":"03:03.920 ","End":"03:11.860","Text":"to r_min multiplied by v_max divided by 2."},{"Start":"03:11.860 ","End":"03:15.710","Text":"In order to find what the time period of the body is,"},{"Start":"03:15.710 ","End":"03:18.779","Text":"so I\u0027ll remind you that that\u0027s t, we\u0027re trying to find t,"},{"Start":"03:18.779 ","End":"03:23.105","Text":"we\u0027re going to use Kepler\u0027s 2nd law."},{"Start":"03:23.105 ","End":"03:30.953","Text":"Kepler\u0027s 2nd law states that given a certain area covered in a certain time period,"},{"Start":"03:30.953 ","End":"03:34.820","Text":"so the ratio of the area divided by the time period,"},{"Start":"03:34.820 ","End":"03:39.720","Text":"is always going to be some constant value."},{"Start":"03:39.830 ","End":"03:42.460","Text":"In that case, our ds by dt,"},{"Start":"03:42.460 ","End":"03:43.820","Text":"which is equal to this,"},{"Start":"03:43.820 ","End":"03:49.025","Text":"is also going to be equal to the total area of the ellipse,"},{"Start":"03:49.025 ","End":"03:50.533","Text":"which we\u0027re given in the question,"},{"Start":"03:50.533 ","End":"03:52.820","Text":"the area of the ellipse is also given,"},{"Start":"03:52.820 ","End":"03:57.695","Text":"divided by the total time taken to cover that area,"},{"Start":"03:57.695 ","End":"04:00.785","Text":"which as we know, is 1 period."},{"Start":"04:00.785 ","End":"04:07.234","Text":"It takes the time of 1 period in order to complete this orbital."},{"Start":"04:07.234 ","End":"04:10.910","Text":"Now we have this relationship and all we have to do is we have to"},{"Start":"04:10.910 ","End":"04:14.911","Text":"isolate out our T, our time period."},{"Start":"04:14.911 ","End":"04:21.645","Text":"Then all we have to do is we have to say that our 2S,"},{"Start":"04:21.645 ","End":"04:23.915","Text":"2 times the area of the ellipse,"},{"Start":"04:23.915 ","End":"04:32.699","Text":"divided by r_min multiplied by v_max is equal to our time period."},{"Start":"04:32.699 ","End":"04:34.760","Text":"That\u0027s our final answer."},{"Start":"04:34.760 ","End":"04:39.770","Text":"Another way that we could have solved this is that when we got to this stage,"},{"Start":"04:39.770 ","End":"04:49.079","Text":"we would have had that our ds divided by our dt is equal to L divided by 2m."},{"Start":"04:49.079 ","End":"04:53.750","Text":"This is what we got in one of the previous lessons,"},{"Start":"04:53.750 ","End":"05:00.665","Text":"where L is given by our mass multiplied by our v_max,"},{"Start":"05:00.665 ","End":"05:03.290","Text":"multiplied by our r_min,"},{"Start":"05:03.290 ","End":"05:06.365","Text":"multiplied by sine of the angle between the 2,"},{"Start":"05:06.365 ","End":"05:09.140","Text":"which at this point of r_min and v_max,"},{"Start":"05:09.140 ","End":"05:11.240","Text":"we know that the angle between the 2 is 90,"},{"Start":"05:11.240 ","End":"05:13.230","Text":"so sin(90) is simply 1."},{"Start":"05:13.230 ","End":"05:17.750","Text":"Then once we substitute this in over here,"},{"Start":"05:17.750 ","End":"05:23.120","Text":"so we\u0027ll see that we\u0027ll get the exact same equation that we got over here."},{"Start":"05:23.120 ","End":"05:26.280","Text":"That\u0027s the end of this lesson."}],"ID":9470}],"Thumbnail":null,"ID":5362},{"Name":"The Two Body Problem and Reduced Mass","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Two Body Problem And Reduced Mass","Duration":"28m 2s","ChapterTopicVideoID":9201,"CourseChapterTopicPlaylistID":5363,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"Hello. What are we going to speak about in"},{"Start":"00:02.490 ","End":"00:05.910","Text":"this lesson is about how to solve questions when we have"},{"Start":"00:05.910 ","End":"00:12.540","Text":"two bodies where both of their masses are relatively similar one to another."},{"Start":"00:12.540 ","End":"00:14.820","Text":"Up until now, we\u0027ve been dealing with"},{"Start":"00:14.820 ","End":"00:18.447","Text":"one body having a significantly larger mass than the other body,"},{"Start":"00:18.447 ","End":"00:20.160","Text":"and we\u0027ve seen how to deal with that."},{"Start":"00:20.160 ","End":"00:25.244","Text":"Now, we\u0027re going to see how to deal with this case."},{"Start":"00:25.244 ","End":"00:27.360","Text":"Up until now, we\u0027ve been looking at, for instance,"},{"Start":"00:27.360 ","End":"00:31.808","Text":"the sun, where it\u0027s stationary and we have the Earth orbiting around it,"},{"Start":"00:31.808 ","End":"00:34.620","Text":"or if we have the Earth and we think of it as stationary,"},{"Start":"00:34.620 ","End":"00:41.090","Text":"and we have another body orbiting around the Earth such as a satellite."},{"Start":"00:41.090 ","End":"00:43.370","Text":"Right now, we\u0027re going to speak about having"},{"Start":"00:43.370 ","End":"00:48.230","Text":"two large bodies where they\u0027re both approximately the same size as one another,"},{"Start":"00:48.230 ","End":"00:53.631","Text":"which means that they\u0027re both moving simultaneously around each other."},{"Start":"00:53.631 ","End":"00:56.465","Text":"In order to solve this type of question,"},{"Start":"00:56.465 ","End":"01:01.940","Text":"instead of having two different systems and trying to calculate it,"},{"Start":"01:01.940 ","End":"01:03.290","Text":"we\u0027re going to try and change"},{"Start":"01:03.290 ","End":"01:07.610","Text":"those two different systems into an easier way to interpret,"},{"Start":"01:07.610 ","End":"01:12.140","Text":"which means that we\u0027re going to only use one variable to describe"},{"Start":"01:12.140 ","End":"01:18.290","Text":"the two systems and that variable is going to be the distance between the two bodies."},{"Start":"01:18.290 ","End":"01:27.270","Text":"I\u0027m going to express the entire problem in relation to this variable."},{"Start":"01:27.270 ","End":"01:29.895","Text":"We have these two bodies, M_1 and M_2."},{"Start":"01:29.895 ","End":"01:31.880","Text":"M_1 is slightly bigger than M_2,"},{"Start":"01:31.880 ","End":"01:34.255","Text":"but not significantly bigger."},{"Start":"01:34.255 ","End":"01:41.290","Text":"From our origin, we have a vector r_1 pointing to our mass M_1,"},{"Start":"01:41.290 ","End":"01:45.965","Text":"and we have our vector r_2 pointing to our mass M_2."},{"Start":"01:45.965 ","End":"01:52.655","Text":"From each mass, they\u0027re experiencing gravitational force from the other mass."},{"Start":"01:52.655 ","End":"01:54.700","Text":"So from M_1 to M_2,"},{"Start":"01:54.700 ","End":"01:58.070","Text":"we\u0027re experiencing F_1, 2,"},{"Start":"01:58.070 ","End":"02:02.810","Text":"and from M_2 to M_1 we\u0027re experiencing F_2, 1."},{"Start":"02:02.810 ","End":"02:06.830","Text":"Let\u0027s write out our energy."},{"Start":"02:06.830 ","End":"02:11.461","Text":"Our energy is going to be equal to the kinetic energy of mass 1 so"},{"Start":"02:11.461 ","End":"02:16.340","Text":"that\u0027s going to be equal to 1/2 M_1 multiplied"},{"Start":"02:16.340 ","End":"02:20.900","Text":"by its velocity squared plus the kinetic energy of mass 2 so"},{"Start":"02:20.900 ","End":"02:25.715","Text":"that\u0027s going to be 1/2 M_2 multiplied by its velocity squared."},{"Start":"02:25.715 ","End":"02:28.114","Text":"And then minus our Alpha,"},{"Start":"02:28.114 ","End":"02:32.330","Text":"where I\u0027m reminding you that our Alpha is equal to"},{"Start":"02:32.330 ","End":"02:41.390","Text":"G M_1 M_2 and then divided by the distance between the two masses."},{"Start":"02:41.390 ","End":"02:46.880","Text":"The distance between the two masses is going to be the size of"},{"Start":"02:46.880 ","End":"02:53.141","Text":"vector r_2 minus vector r_1."},{"Start":"02:53.141 ","End":"03:00.935","Text":"This vector over here is going to be called this r_2, 1 or r_relative."},{"Start":"03:00.935 ","End":"03:04.592","Text":"This over here."},{"Start":"03:04.592 ","End":"03:07.850","Text":"This expression over here is this vector and"},{"Start":"03:07.850 ","End":"03:11.780","Text":"our potential energy is dependent on the size of this vector."},{"Start":"03:11.780 ","End":"03:16.040","Text":"It\u0027s dependent on the distance between the two masses."},{"Start":"03:16.040 ","End":"03:20.450","Text":"Now what we want to do is variable substitution,"},{"Start":"03:20.450 ","End":"03:27.860","Text":"so we\u0027re going to say that our r_2, 1,"},{"Start":"03:27.860 ","End":"03:30.362","Text":"which is the distance,"},{"Start":"03:30.362 ","End":"03:33.890","Text":"so that\u0027s also going to be called r_relative and that\u0027s going"},{"Start":"03:33.890 ","End":"03:39.820","Text":"to be equal to our r_2 minus our r_1."},{"Start":"03:39.820 ","End":"03:45.035","Text":"Then the next variable that we\u0027re going to substitute in is going to be r_cm,"},{"Start":"03:45.035 ","End":"03:47.450","Text":"which we, of course, know the equation for that,"},{"Start":"03:47.450 ","End":"03:50.930","Text":"and that\u0027s going to be the mass of our first body multiplied"},{"Start":"03:50.930 ","End":"03:56.244","Text":"by its position from the origin."},{"Start":"03:56.244 ","End":"04:00.545","Text":"Plus the mass of our second body multiplied by"},{"Start":"04:00.545 ","End":"04:05.840","Text":"its position from the origin divided by the total mass of this system,"},{"Start":"04:05.840 ","End":"04:09.478","Text":"which is M_1 plus M_2."},{"Start":"04:09.478 ","End":"04:12.710","Text":"Now I\u0027ll substitute in my position,"},{"Start":"04:12.710 ","End":"04:16.548","Text":"but I also wanted to substitute in my velocities,"},{"Start":"04:16.548 ","End":"04:20.210","Text":"so I\u0027ll have my variable called V_2, 1,"},{"Start":"04:20.210 ","End":"04:24.650","Text":"which is also known as my V_relative."},{"Start":"04:24.650 ","End":"04:31.140","Text":"That\u0027s going to be my first derivative of my r_2, 1."},{"Start":"04:31.140 ","End":"04:34.865","Text":"The first derivative, so when I take the first derivative of this,"},{"Start":"04:34.865 ","End":"04:42.465","Text":"it\u0027s going to be r_2 dot minus r_1 dots."},{"Start":"04:42.465 ","End":"04:51.315","Text":"That\u0027s going to be simply equal to my V_2 minus my V_1,"},{"Start":"04:51.315 ","End":"04:58.600","Text":"because my r_2 dots is equal to V_2 and same with my r_1."},{"Start":"04:58.600 ","End":"05:01.685","Text":"Now the next thing that I wanted to do,"},{"Start":"05:01.685 ","End":"05:06.868","Text":"is I want to write out my equation for energy in terms of my r_2,"},{"Start":"05:06.868 ","End":"05:10.395","Text":"1, my r_cm, and my V_2, 1."},{"Start":"05:10.395 ","End":"05:17.730","Text":"Now what I\u0027m going to try and do is to isolate out r_1 and r_2,"},{"Start":"05:17.730 ","End":"05:24.620","Text":"so I\u0027m going to now rearrange this equation in order to find out what my r_2 is equal to."},{"Start":"05:24.620 ","End":"05:28.550","Text":"I\u0027m going to add to both sides by r_1,"},{"Start":"05:28.550 ","End":"05:31.790","Text":"so I\u0027ll get that my r_2,"},{"Start":"05:31.790 ","End":"05:40.450","Text":"1 vector plus my r_1 vector is going to be equal to my r_2 vector."},{"Start":"05:40.450 ","End":"05:45.575","Text":"Now what I\u0027m going to do,"},{"Start":"05:45.575 ","End":"05:49.426","Text":"is I\u0027m going to substitute my r_2 value,"},{"Start":"05:49.426 ","End":"05:54.725","Text":"so this over here into here in my r_cm."},{"Start":"05:54.725 ","End":"05:57.538","Text":"Let\u0027s substitute that in,"},{"Start":"05:57.538 ","End":"06:02.330","Text":"so we\u0027ll have that our r_cm is going to be equal"},{"Start":"06:02.330 ","End":"06:08.685","Text":"to our M_1 r_1 vector plus our M_2,"},{"Start":"06:08.685 ","End":"06:10.290","Text":"and instead of r_2 vector,"},{"Start":"06:10.290 ","End":"06:11.760","Text":"I\u0027ll substitute this in,"},{"Start":"06:11.760 ","End":"06:13.500","Text":"so my r_2,"},{"Start":"06:13.500 ","End":"06:18.065","Text":"1 vector plus my r_1 vector."},{"Start":"06:18.065 ","End":"06:24.200","Text":"Then this is going to be divided my M_1 plus my M_2."},{"Start":"06:24.200 ","End":"06:27.515","Text":"The total mass of the system."},{"Start":"06:27.515 ","End":"06:29.980","Text":"Now let\u0027s simplify this a little bit."},{"Start":"06:29.980 ","End":"06:37.890","Text":"Let\u0027s say that just this M is going to be equal to my M_1 plus my M_2,"},{"Start":"06:37.890 ","End":"06:41.208","Text":"so my M alone is my total mass over here."},{"Start":"06:41.208 ","End":"06:43.610","Text":"Then I\u0027m going to multiply both sides of"},{"Start":"06:43.610 ","End":"06:48.650","Text":"the equal sign by my M_1 plus M_2 so by this capital M,"},{"Start":"06:48.650 ","End":"06:56.595","Text":"so I\u0027m going to have that my m multiplied by my r_cm is equal to M_1 r_1,"},{"Start":"06:56.595 ","End":"07:05.095","Text":"plus M_2, and then r_2, 1 plus r_1."},{"Start":"07:05.095 ","End":"07:06.635","Text":"Now what we\u0027re going to do,"},{"Start":"07:06.635 ","End":"07:11.045","Text":"is we\u0027re going to isolate out our r_1."},{"Start":"07:11.045 ","End":"07:12.425","Text":"Let\u0027s take a look at what we get."},{"Start":"07:12.425 ","End":"07:16.715","Text":"We\u0027ll get that r_1 vector is going to be equal to"},{"Start":"07:16.715 ","End":"07:26.030","Text":"our r_cm minus our M_2 divided by our total mass,"},{"Start":"07:26.030 ","End":"07:33.320","Text":"so our M_1 plus M_2 multiplied by r_2, 1 vector."},{"Start":"07:33.320 ","End":"07:35.900","Text":"What I did here was simple algebra."},{"Start":"07:35.900 ","End":"07:40.753","Text":"Then now I have my r_1."},{"Start":"07:40.753 ","End":"07:42.710","Text":"Now in order to find my r_2,"},{"Start":"07:42.710 ","End":"07:49.508","Text":"I\u0027m going to substitute in this r_1 into this equation over here in order to get my r_2."},{"Start":"07:49.508 ","End":"07:51.110","Text":"Then with simple algebra,"},{"Start":"07:51.110 ","End":"07:56.405","Text":"you can do this on a sheet of paper if you want to see exactly how it\u0027s done."},{"Start":"07:56.405 ","End":"08:03.350","Text":"This will be equal to our r_cm plus our M_1 divided by our total mass,"},{"Start":"08:03.350 ","End":"08:10.870","Text":"so M_1 plus M_2 multiplied by our r_2, 1 vector."},{"Start":"08:10.870 ","End":"08:12.630","Text":"What we did over here,"},{"Start":"08:12.630 ","End":"08:16.390","Text":"is we found what our r_2,"},{"Start":"08:16.390 ","End":"08:23.530","Text":"1 and our r_cm are equal to if we have our r_2 and our r_1 vectors."},{"Start":"08:23.530 ","End":"08:26.240","Text":"If we have this vector and this vector,"},{"Start":"08:26.240 ","End":"08:31.420","Text":"then we can find this vector and our r_cm vector."},{"Start":"08:31.420 ","End":"08:35.834","Text":"If alternatively, we have our r_cm vector and our r_2,"},{"Start":"08:35.834 ","End":"08:37.864","Text":"1 vector given in the question,"},{"Start":"08:37.864 ","End":"08:43.070","Text":"then we can find our r_1 and our r_2: this vector and this vector."},{"Start":"08:43.070 ","End":"08:45.380","Text":"These 2 equations are very useful"},{"Start":"08:45.380 ","End":"08:49.410","Text":"depending on the information given to you in the question."},{"Start":"08:50.090 ","End":"08:57.690","Text":"Just how we found our position and similarly to how we found our velocity over here,"},{"Start":"08:57.690 ","End":"08:59.775","Text":"we\u0027re going to do the exact same thing."},{"Start":"08:59.775 ","End":"09:08.100","Text":"Our V_1 vector is simply going to be the derivative of this r_1 vector."},{"Start":"09:08.100 ","End":"09:13.500","Text":"Then we\u0027re going to have that it\u0027s equal to our V_cm minus"},{"Start":"09:13.500 ","End":"09:21.840","Text":"our M_2 divided by our total mass multiplied by our V_2, 1 vector."},{"Start":"09:21.840 ","End":"09:27.095","Text":"In the same way, our V_2 vector is going to be the derivative of this."},{"Start":"09:27.095 ","End":"09:30.785","Text":"It\u0027s going to be our V_cm plus"},{"Start":"09:30.785 ","End":"09:39.420","Text":"our M_1 divided by our total mass multiplied by our V_2, 1."},{"Start":"09:40.520 ","End":"09:48.370","Text":"Now what we\u0027re going to do is we\u0027re going to substitute in our V_1 over here,"},{"Start":"09:48.800 ","End":"09:57.165","Text":"didn\u0027t draw a line, and we\u0027re going to substitute in our V_2 over here."},{"Start":"09:57.165 ","End":"10:01.590","Text":"Now we\u0027re going to rewrite this equation,"},{"Start":"10:01.590 ","End":"10:06.540","Text":"substituting our V_1 and our V_2 that we\u0027ve already found."},{"Start":"10:06.540 ","End":"10:13.335","Text":"Now this is going to be a little bit long."},{"Start":"10:13.335 ","End":"10:18.555","Text":"Now I\u0027ve done solve the algebra of this already it\u0027s separately in order to save time."},{"Start":"10:18.555 ","End":"10:20.190","Text":"If you want to do it yourself,"},{"Start":"10:20.190 ","End":"10:25.095","Text":"then you\u0027re more than welcome to pause the video and see if you get the same expression."},{"Start":"10:25.095 ","End":"10:30.698","Text":"Our E is going to be equal to a half multiplied by our M_1,"},{"Start":"10:30.698 ","End":"10:39.814","Text":"and then multiplied by our V_cm squared minus 2M_2,"},{"Start":"10:39.814 ","End":"10:46.455","Text":"V_cm vector multiplied by our V_relative vector,"},{"Start":"10:46.455 ","End":"10:49.380","Text":"which I\u0027m reminding you is this over here,"},{"Start":"10:49.380 ","End":"10:58.080","Text":"and then this is going to be divided over here by M_1 plus M_2,"},{"Start":"10:58.080 ","End":"11:06.975","Text":"and then we\u0027re going to have plus our M_2 divided by M_1 plus M_2."},{"Start":"11:06.975 ","End":"11:13.290","Text":"This is going to be squared multiplied by our V_relative squared."},{"Start":"11:13.290 ","End":"11:16.005","Text":"Again, this is our V_relative over here."},{"Start":"11:16.005 ","End":"11:22.800","Text":"Then we\u0027ll have plus our half M_2"},{"Start":"11:22.800 ","End":"11:30.203","Text":"multiplied by our V_cm squared plus 2M_1,"},{"Start":"11:30.203 ","End":"11:40.125","Text":"V_cm multiplied by our V relative divided by our M_1 plus"},{"Start":"11:40.125 ","End":"11:47.279","Text":"M_2 plus our M_1 divided by our M_1"},{"Start":"11:47.279 ","End":"11:55.815","Text":"plus M_2 squared multiplied by our V relative squared."},{"Start":"11:55.815 ","End":"12:04.980","Text":"Then close the brackets and then minus our Alpha divided by our r_relative,"},{"Start":"12:04.980 ","End":"12:07.800","Text":"where our r_relative is,"},{"Start":"12:07.800 ","End":"12:09.690","Text":"of course, this over here,"},{"Start":"12:09.690 ","End":"12:11.955","Text":"our r_2 minus our r_1."},{"Start":"12:11.955 ","End":"12:14.100","Text":"This is exactly what we wanted to do."},{"Start":"12:14.100 ","End":"12:20.800","Text":"We wanted to get our entire expression in terms of this one variable."},{"Start":"12:21.620 ","End":"12:24.270","Text":"This is all very confusing,"},{"Start":"12:24.270 ","End":"12:28.230","Text":"so let\u0027s take a look at how we can simplify things a little bit."},{"Start":"12:28.230 ","End":"12:33.465","Text":"We\u0027ll see that this over here cancels out with this over here."},{"Start":"12:33.465 ","End":"12:36.480","Text":"Because here we have half M_1 and then here we have"},{"Start":"12:36.480 ","End":"12:39.690","Text":"our M_2 and everything else is exactly the same."},{"Start":"12:39.690 ","End":"12:41.325","Text":"Here we have half M_2,"},{"Start":"12:41.325 ","End":"12:43.185","Text":"and then here we have our M_1."},{"Start":"12:43.185 ","End":"12:45.840","Text":"If we open up the brackets, we\u0027ll see that these 2 are the same,"},{"Start":"12:45.840 ","End":"12:49.410","Text":"but here there\u0027s a minus and here there\u0027s a plus. They cancel out."},{"Start":"12:49.410 ","End":"12:54.915","Text":"Then we\u0027ll see that this expression over here is the same as this expression,"},{"Start":"12:54.915 ","End":"12:57.915","Text":"except here there\u0027s an M_1 and here there\u0027s an M_2."},{"Start":"12:57.915 ","End":"13:00.510","Text":"We can rewrite this."},{"Start":"13:00.510 ","End":"13:03.450","Text":"We\u0027ll have that our E is equal to,"},{"Start":"13:03.450 ","End":"13:08.295","Text":"we can have a half and then we\u0027ll have M_1"},{"Start":"13:08.295 ","End":"13:15.610","Text":"plus M_2 multiplied by our V_cm squared."},{"Start":"13:15.860 ","End":"13:18.645","Text":"Then we can look over here,"},{"Start":"13:18.645 ","End":"13:20.715","Text":"and here we have half M_1,"},{"Start":"13:20.715 ","End":"13:22.860","Text":"M_2 over the total mass,"},{"Start":"13:22.860 ","End":"13:25.725","Text":"and here we\u0027ll have half M_1,"},{"Start":"13:25.725 ","End":"13:28.208","Text":"M_2 divided by the total mass and, of course,"},{"Start":"13:28.208 ","End":"13:31.440","Text":"squared multiplied by our V_relative."},{"Start":"13:31.440 ","End":"13:35.910","Text":"What we can do is we can simply just add those on, so plus,"},{"Start":"13:35.910 ","End":"13:41.510","Text":"and then we\u0027ll have our half multiplied by our M_2 multiplied by"},{"Start":"13:41.510 ","End":"13:49.240","Text":"our M_1 divided by our M_1 plus our M_2."},{"Start":"13:49.240 ","End":"13:53.955","Text":"Then that\u0027s multiplied by our V_relative."},{"Start":"13:53.955 ","End":"14:01.900","Text":"Then finally, this minus our Alpha divided by our r_relative."},{"Start":"14:02.540 ","End":"14:07.200","Text":"Now all I added over here were these lines,"},{"Start":"14:07.200 ","End":"14:13.290","Text":"and they represent that I\u0027m dividing my Alpha by the distance between the 2 bodies."},{"Start":"14:13.290 ","End":"14:15.615","Text":"I only want the size of this vector."},{"Start":"14:15.615 ","End":"14:20.650","Text":"The direction of the vector is not my concern at the moment."},{"Start":"14:21.830 ","End":"14:26.700","Text":"What we have over here are M_1 plus our M_2."},{"Start":"14:26.700 ","End":"14:29.010","Text":"We already spoke about that. That\u0027s our total mass."},{"Start":"14:29.010 ","End":"14:34.020","Text":"We\u0027re going to just call this our capital M. Now,"},{"Start":"14:34.020 ","End":"14:40.380","Text":"what we\u0027re going to have over here is our reduced mass."},{"Start":"14:40.380 ","End":"14:42.090","Text":"What is our reduce mass?"},{"Start":"14:42.090 ","End":"14:45.810","Text":"It\u0027s this entire expression over here."},{"Start":"14:45.810 ","End":"14:50.625","Text":"This is denoted by the Greek letter Mu."},{"Start":"14:50.625 ","End":"14:53.985","Text":"It\u0027s not this M. Mu,"},{"Start":"14:53.985 ","End":"15:01.450","Text":"the reduced mass is equal to M_2 multiplied by M_1 divided by M_1 plus M_2."},{"Start":"15:02.360 ","End":"15:09.600","Text":"Now once we\u0027ve given our new names to our certain sections in this equation,"},{"Start":"15:09.600 ","End":"15:13.830","Text":"so we can see that our energy equation is starting to really take"},{"Start":"15:13.830 ","End":"15:20.580","Text":"form and look like what we want it to look like in order to solve this question."},{"Start":"15:20.580 ","End":"15:24.120","Text":"Let\u0027s start writing it in the form."},{"Start":"15:24.120 ","End":"15:33.780","Text":"We\u0027re going to have that our E is equal to half our mass multiplied by our V_cm squared."},{"Start":"15:33.780 ","End":"15:40.890","Text":"We\u0027re going to see that this over here is going to be equal to a constant."},{"Start":"15:40.890 ","End":"15:44.130","Text":"Why is this equal to a constant?"},{"Start":"15:44.130 ","End":"15:52.410","Text":"That is because the sum of the external forces is equal to 0."},{"Start":"15:52.410 ","End":"15:54.810","Text":"Let\u0027s just explain this."},{"Start":"15:54.810 ","End":"15:58.170","Text":"There are no external forces because all of"},{"Start":"15:58.170 ","End":"16:02.400","Text":"the forces are acting between the two large bodies."},{"Start":"16:02.400 ","End":"16:05.430","Text":"That means that there\u0027s no external forces whatsoever,"},{"Start":"16:05.430 ","End":"16:11.415","Text":"which means that the sum of these external forces which don\u0027t exist is equal to 0."},{"Start":"16:11.415 ","End":"16:15.045","Text":"If the sum of all of the external forces is equal to 0,"},{"Start":"16:15.045 ","End":"16:19.635","Text":"that means that our velocity is constant."},{"Start":"16:19.635 ","End":"16:22.320","Text":"We have no acceleration."},{"Start":"16:22.320 ","End":"16:24.210","Text":"That means that, of course,"},{"Start":"16:24.210 ","End":"16:26.730","Text":"our half is constant and our mass is going to be constant."},{"Start":"16:26.730 ","End":"16:28.080","Text":"If our velocity is constant,"},{"Start":"16:28.080 ","End":"16:33.420","Text":"then that means that this entire expression over here is going to be a constant as well."},{"Start":"16:33.420 ","End":"16:35.955","Text":"Because this is constant,"},{"Start":"16:35.955 ","End":"16:37.995","Text":"it interests me slightly less."},{"Start":"16:37.995 ","End":"16:43.635","Text":"We know that my value for my energy over here is going to be some constant value."},{"Start":"16:43.635 ","End":"16:46.035","Text":"Because this is also a constant value,"},{"Start":"16:46.035 ","End":"16:49.140","Text":"I can also put them together on the same side."},{"Start":"16:49.140 ","End":"16:51.675","Text":"That means that on this side of the equal sign,"},{"Start":"16:51.675 ","End":"16:54.720","Text":"I\u0027m still going to have a constant value and it will just be"},{"Start":"16:54.720 ","End":"16:58.770","Text":"slightly different to this value E over here."},{"Start":"16:58.770 ","End":"17:02.490","Text":"Now what we can do is we can add in the other terms,"},{"Start":"17:02.490 ","End":"17:06.390","Text":"so we\u0027ll have plus a half and then our reduced mass,"},{"Start":"17:06.390 ","End":"17:12.135","Text":"which is Mu multiplied by our V_relative squared,"},{"Start":"17:12.135 ","End":"17:18.210","Text":"and then minus our Alpha divided by our r_relative,"},{"Start":"17:18.210 ","End":"17:20.970","Text":"and again, the size."},{"Start":"17:20.970 ","End":"17:25.380","Text":"Now we can take a look at the equation that we had before when we were"},{"Start":"17:25.380 ","End":"17:31.500","Text":"dealing with some smaller mass orbiting a significantly larger mass."},{"Start":"17:31.500 ","End":"17:35.635","Text":"Then we had that our equation was equal to"},{"Start":"17:35.635 ","End":"17:40.805","Text":"half multiplied by the mass of the larger body,"},{"Start":"17:40.805 ","End":"17:46.159","Text":"multiplied by V_relative squared minus"},{"Start":"17:46.159 ","End":"17:55.370","Text":"our GMm divided by our r. Now instead of our M,"},{"Start":"17:55.370 ","End":"17:58.780","Text":"it\u0027s been switched to a reduced mass Mu,"},{"Start":"17:58.780 ","End":"18:01.710","Text":"and instead of our r,"},{"Start":"18:01.710 ","End":"18:05.060","Text":"the distance between the large body"},{"Start":"18:05.060 ","End":"18:07.940","Text":"and the smaller body when the large body is stationary,"},{"Start":"18:07.940 ","End":"18:10.550","Text":"so now we have our r_relative,"},{"Start":"18:10.550 ","End":"18:16.260","Text":"which is when both of the bodies are not stationary."},{"Start":"18:16.730 ","End":"18:24.110","Text":"Now we can see that mathematically our equation is exactly identical to"},{"Start":"18:24.110 ","End":"18:31.280","Text":"what we had before when we had one large object and a smaller object rotating about it."},{"Start":"18:31.280 ","End":"18:33.655","Text":"Mathematically it\u0027s exactly the same."},{"Start":"18:33.655 ","End":"18:39.605","Text":"Just like what we did in our lesson dealing with our u effective,"},{"Start":"18:39.605 ","End":"18:45.720","Text":"we saw that we can rewrite this type of equation to be in one variable."},{"Start":"18:45.720 ","End":"18:48.705","Text":"Dealing with our r_relative."},{"Start":"18:48.705 ","End":"18:51.590","Text":"From our equation for our u effective,"},{"Start":"18:51.590 ","End":"18:57.645","Text":"we can see that this is equal to half Mr dot squared"},{"Start":"18:57.645 ","End":"19:05.378","Text":"plus L squared divided by 2mr squared,"},{"Start":"19:05.378 ","End":"19:12.920","Text":"and negative Alpha divided by r. If you want to remember how we got to this,"},{"Start":"19:12.920 ","End":"19:18.570","Text":"please go back to our lesson about our u effective."},{"Start":"19:18.830 ","End":"19:22.005","Text":"This is in fact our u, effective."},{"Start":"19:22.005 ","End":"19:23.340","Text":"Then, of course,"},{"Start":"19:23.340 ","End":"19:25.315","Text":"again, instead of our M,"},{"Start":"19:25.315 ","End":"19:28.360","Text":"we\u0027re going to have our Mu and instead of r,"},{"Start":"19:28.360 ","End":"19:32.195","Text":"we\u0027re going to have our r_relative."},{"Start":"19:32.195 ","End":"19:36.885","Text":"Now let\u0027s go back to our equation."},{"Start":"19:36.885 ","End":"19:42.485","Text":"That means that we can therefore write out our energy equation as"},{"Start":"19:42.485 ","End":"19:49.055","Text":"half MV_cm squared plus,"},{"Start":"19:49.055 ","End":"19:50.720","Text":"going from this equation over here,"},{"Start":"19:50.720 ","End":"19:54.650","Text":"so it\u0027s going to be half of Mu,"},{"Start":"19:54.650 ","End":"20:00.465","Text":"r_relative dot plus L"},{"Start":"20:00.465 ","End":"20:06.450","Text":"squared cm divided by 2Mu,"},{"Start":"20:06.450 ","End":"20:14.710","Text":"r_relative squared minus Alpha divided by our r_relative."},{"Start":"20:15.210 ","End":"20:20.320","Text":"From now on we can see that we\u0027re using the exact same formula."},{"Start":"20:20.320 ","End":"20:26.335","Text":"The only difference being that instead of using our mass of the smaller object,"},{"Start":"20:26.335 ","End":"20:29.485","Text":"we\u0027re going to be using our reduced mass Mu."},{"Start":"20:29.485 ","End":"20:33.250","Text":"Instead of dealing with our r over here,"},{"Start":"20:33.250 ","End":"20:36.535","Text":"we\u0027re going to be dealing with our r_relative."},{"Start":"20:36.535 ","End":"20:41.755","Text":"Now what we can do is we can write out our equations."},{"Start":"20:41.755 ","End":"20:52.280","Text":"We can see that our r_relative is still going to be a function of Theta."},{"Start":"20:52.410 ","End":"20:54.685","Text":"Just like we saw before,"},{"Start":"20:54.685 ","End":"20:59.050","Text":"our equation for our trajectory is going to be equal to our r_0"},{"Start":"20:59.050 ","End":"21:05.080","Text":"divided by 1 plus Epsilon multiplied by cosine of Theta."},{"Start":"21:05.080 ","End":"21:09.715","Text":"Where r_0 is equal to L squared divided by,"},{"Start":"21:09.715 ","End":"21:15.100","Text":"and then instead of m, we\u0027ll have our reduced mass multiplied by our Alpha,"},{"Start":"21:15.100 ","End":"21:18.895","Text":"and our Epsilon is going to be the same thing also,"},{"Start":"21:18.895 ","End":"21:26.215","Text":"which is the square root of 1 plus 2EL squared divided by,"},{"Start":"21:26.215 ","End":"21:33.440","Text":"and then instead of mass, it\u0027s again the reduced mass Mu multiplied by Alpha squared."},{"Start":"21:33.900 ","End":"21:35.920","Text":"Just as a reminder,"},{"Start":"21:35.920 ","End":"21:37.960","Text":"what our Alpha is equal to,"},{"Start":"21:37.960 ","End":"21:40.105","Text":"so our Alpha hasn\u0027t changed."},{"Start":"21:40.105 ","End":"21:45.460","Text":"It\u0027s still equal to GM_1 multiplied by M_2."},{"Start":"21:45.460 ","End":"21:50.710","Text":"The last thing to note is that our L over here,"},{"Start":"21:50.710 ","End":"21:52.870","Text":"that\u0027s our angular momentum,"},{"Start":"21:52.870 ","End":"21:57.100","Text":"it\u0027s equal to the angular momentum of the center of mass."},{"Start":"21:57.100 ","End":"22:03.700","Text":"Because we\u0027re dealing with a system with two bodies in and not two bodies which"},{"Start":"22:03.700 ","End":"22:06.850","Text":"are approximately the same size and not one large body and"},{"Start":"22:06.850 ","End":"22:11.330","Text":"one small body so with the angular momentum of the center of mass."},{"Start":"22:11.940 ","End":"22:14.440","Text":"Now just like in"},{"Start":"22:14.440 ","End":"22:20.425","Text":"any other normal question of this sort that I\u0027ve been dealing with up until now,"},{"Start":"22:20.425 ","End":"22:23.920","Text":"we have to use this equation for the trajectory."},{"Start":"22:23.920 ","End":"22:28.945","Text":"Then you can get what your trajectory will look like at certain points."},{"Start":"22:28.945 ","End":"22:36.770","Text":"Just like in the normal case of Earth orbiting the sun, etc."},{"Start":"22:36.960 ","End":"22:41.830","Text":"Now as for our equation for our r_cm,"},{"Start":"22:41.830 ","End":"22:45.805","Text":"let\u0027s see what that is as a function of time."},{"Start":"22:45.805 ","End":"22:52.570","Text":"It\u0027s going to be equal to our V_cm multiplied by t and then"},{"Start":"22:52.570 ","End":"23:00.265","Text":"plus our r_cm value when our t is equal to 0."},{"Start":"23:00.265 ","End":"23:02.521","Text":"That is going to be the value,"},{"Start":"23:02.521 ","End":"23:05.020","Text":"and we know that our V_cm is constant."},{"Start":"23:05.020 ","End":"23:07.300","Text":"It\u0027s not changing with the time."},{"Start":"23:07.300 ","End":"23:09.130","Text":"Of course, our V_cm,"},{"Start":"23:09.130 ","End":"23:13.015","Text":"we can work out from our equation for our V_cm,"},{"Start":"23:13.015 ","End":"23:15.175","Text":"which was up here."},{"Start":"23:15.175 ","End":"23:17.755","Text":"Let\u0027s take a look."},{"Start":"23:17.755 ","End":"23:21.730","Text":"Right over here, we just take the first derivative"},{"Start":"23:21.730 ","End":"23:25.240","Text":"of our r_cm and we\u0027ll get that our V_cm is simply"},{"Start":"23:25.240 ","End":"23:33.025","Text":"equal to M_1 V_1 plus M_2 V_2 divided by the total mass."},{"Start":"23:33.025 ","End":"23:35.724","Text":"Now I use my r_relative,"},{"Start":"23:35.724 ","End":"23:40.915","Text":"just like I would use my regular r as a function of Theta equation."},{"Start":"23:40.915 ","End":"23:43.435","Text":"I just have my r_cm which I found like this."},{"Start":"23:43.435 ","End":"23:46.555","Text":"Now that I\u0027ve found my r_relative and my r_cm,"},{"Start":"23:46.555 ","End":"23:50.515","Text":"if these were this time given to me in my question,"},{"Start":"23:50.515 ","End":"23:57.640","Text":"then all I have to do is substitute in their values into these two equations over here,"},{"Start":"23:57.640 ","End":"24:01.630","Text":"and I\u0027ll find out what my r_1 and my r_2 are."},{"Start":"24:01.630 ","End":"24:07.450","Text":"Now what we\u0027re going to do just before the end of the lesson is we\u0027re going to take"},{"Start":"24:07.450 ","End":"24:13.645","Text":"a look at what the movement or the motion of the system is going to look like."},{"Start":"24:13.645 ","End":"24:18.010","Text":"If we have our axes over here,"},{"Start":"24:18.010 ","End":"24:21.130","Text":"then we have our two bodies."},{"Start":"24:21.130 ","End":"24:24.940","Text":"Here is one body and here is the other body."},{"Start":"24:24.940 ","End":"24:28.600","Text":"We can see that they\u0027re approximately the same size."},{"Start":"24:28.600 ","End":"24:36.100","Text":"Now the vector going from here until here is our vector r_relative,"},{"Start":"24:36.100 ","End":"24:40.495","Text":"sorry not r center of mass our r_relative."},{"Start":"24:40.495 ","End":"24:44.500","Text":"Because we can see that this mass is slightly larger,"},{"Start":"24:44.500 ","End":"24:50.950","Text":"that means that our center of mass is going to be slightly closer to it."},{"Start":"24:50.950 ","End":"24:56.560","Text":"Now, we saw that the sum of the external forces"},{"Start":"24:56.560 ","End":"25:01.945","Text":"was equal to 0 because all of the forces are between these two masses."},{"Start":"25:01.945 ","End":"25:04.780","Text":"There\u0027s no external forces acting on the system."},{"Start":"25:04.780 ","End":"25:07.600","Text":"That means that our center of mass is going to"},{"Start":"25:07.600 ","End":"25:11.050","Text":"move in a straight line at a constant velocity."},{"Start":"25:11.050 ","End":"25:15.160","Text":"That means that our center of mass is just going to move in"},{"Start":"25:15.160 ","End":"25:20.270","Text":"this direction at a constant velocity."},{"Start":"25:20.460 ","End":"25:24.100","Text":"That\u0027s where our center of mass as for the bodies."},{"Start":"25:24.100 ","End":"25:25.270","Text":"What they\u0027re going to do is,"},{"Start":"25:25.270 ","End":"25:29.365","Text":"they\u0027re going to be rotating about the center of mass."},{"Start":"25:29.365 ","End":"25:32.875","Text":"The center of mass becomes the axes of rotation."},{"Start":"25:32.875 ","End":"25:38.895","Text":"The way that they\u0027re going to be rotating is in an elliptical form. What does that mean?"},{"Start":"25:38.895 ","End":"25:41.795","Text":"As the center of mass moves,"},{"Start":"25:41.795 ","End":"25:45.490","Text":"the bodies are going to be rotating about the center of mass"},{"Start":"25:45.490 ","End":"25:49.495","Text":"and our r_relative is going to be changing size."},{"Start":"25:49.495 ","End":"25:52.822","Text":"It\u0027s going to get smaller and bigger and smaller and bigger,"},{"Start":"25:52.822 ","End":"25:56.905","Text":"just like the radius looks when we\u0027re going through an ellipse."},{"Start":"25:56.905 ","End":"25:59.410","Text":"If here we have an ellipse,"},{"Start":"25:59.410 ","End":"26:02.019","Text":"so here our radius is at a minimum."},{"Start":"26:02.019 ","End":"26:04.315","Text":"Here, the radius is slightly bigger."},{"Start":"26:04.315 ","End":"26:05.800","Text":"Here it\u0027s bigger."},{"Start":"26:05.800 ","End":"26:08.620","Text":"Still here it reaches some maximum."},{"Start":"26:08.620 ","End":"26:14.785","Text":"Then again, our radius slowly decreases until it reaches the same point."},{"Start":"26:14.785 ","End":"26:17.380","Text":"That\u0027s what\u0027s going to happen with our r_relative."},{"Start":"26:17.380 ","End":"26:20.560","Text":"It\u0027s going to be increasing and decreasing as"},{"Start":"26:20.560 ","End":"26:24.505","Text":"the masses rotate about our axes of rotation,"},{"Start":"26:24.505 ","End":"26:29.540","Text":"which is moving in a straight line at a constant velocity."},{"Start":"26:30.000 ","End":"26:35.320","Text":"Lastly, due to this increasing and decreasing in the size of"},{"Start":"26:35.320 ","End":"26:38.440","Text":"our r_relative and because our center of mass is moving"},{"Start":"26:38.440 ","End":"26:41.770","Text":"and because the bodies are rotating about our center of mass,"},{"Start":"26:41.770 ","End":"26:49.045","Text":"so what\u0027s going to happen is that as well as our radius doing some elliptical motion,"},{"Start":"26:49.045 ","End":"26:52.390","Text":"we\u0027re also going to see on masses moving or"},{"Start":"26:52.390 ","End":"26:57.265","Text":"orbiting in an ellipse around the center of mass."},{"Start":"26:57.265 ","End":"27:01.285","Text":"If we take a look at this mass over here,"},{"Start":"27:01.285 ","End":"27:03.460","Text":"so right now it\u0027s located over here,"},{"Start":"27:03.460 ","End":"27:04.960","Text":"then it will be over here."},{"Start":"27:04.960 ","End":"27:07.090","Text":"Then as it rotates,"},{"Start":"27:07.090 ","End":"27:08.485","Text":"it will reach a minimum."},{"Start":"27:08.485 ","End":"27:13.015","Text":"Then it will increase outwards until it reaches this again."},{"Start":"27:13.015 ","End":"27:16.030","Text":"As we can see, this is an ellipse."},{"Start":"27:16.030 ","End":"27:19.990","Text":"The same thing is true for this mass as well."},{"Start":"27:19.990 ","End":"27:25.885","Text":"This motion is described by 2 objects which are approximately the same size."},{"Start":"27:25.885 ","End":"27:28.555","Text":"It\u0027s common amongst certain styles,"},{"Start":"27:28.555 ","End":"27:33.340","Text":"binary stars, and this is how they move pretty much for eternity."},{"Start":"27:33.340 ","End":"27:35.185","Text":"Now, this also happens,"},{"Start":"27:35.185 ","End":"27:37.600","Text":"believe it or not, between Earth and the sun."},{"Start":"27:37.600 ","End":"27:42.070","Text":"However, because the sun is so much larger than the Earth,"},{"Start":"27:42.070 ","End":"27:45.700","Text":"at the center of mass is almost at the center of the sun."},{"Start":"27:45.700 ","End":"27:49.870","Text":"It\u0027s located maybe here for instance."},{"Start":"27:49.870 ","End":"27:57.010","Text":"That means that we can\u0027t see the elliptic motion of our sun about the center of mass,"},{"Start":"27:57.010 ","End":"27:59.720","Text":"but it is in fact happening."},{"Start":"28:00.450 ","End":"28:03.740","Text":"That\u0027s the end of our lesson."}],"ID":9471}],"Thumbnail":null,"ID":5363},{"Name":"Further Questions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Satellite Orbiting","Duration":"22m 22s","ChapterTopicVideoID":9202,"CourseChapterTopicPlaylistID":5364,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9202.jpeg","UploadDate":"2017-03-24T12:46:22.4270000","DurationForVideoObject":"PT22M22S","Description":null,"MetaTitle":"Satellite Orbiting: Video + Workbook | Proprep","MetaDescription":"Gravity and Central Force - Further Questions. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/physics-1-mechanics-waves-and-thermodynamics/gravity-and-central-force/further-questions/vid9472","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.015","Text":"Hello, in this lesson we\u0027re being told that a satellite is shot vertically from Earth."},{"Start":"00:06.015 ","End":"00:11.910","Text":"The satellite then reaches a maximum height of 2R_E."},{"Start":"00:11.910 ","End":"00:14.262","Text":"R_E is the radius of the earth."},{"Start":"00:14.262 ","End":"00:18.195","Text":"When the satellite is shot from the surface of the earth,"},{"Start":"00:18.195 ","End":"00:22.410","Text":"it rises R_E and together with"},{"Start":"00:22.410 ","End":"00:26.744","Text":"the distance from the surface of the Earth to the center of the Earth in total,"},{"Start":"00:26.744 ","End":"00:29.520","Text":"this distance is 2R_E."},{"Start":"00:29.520 ","End":"00:33.465","Text":"At the moment when it reaches its maximum height,"},{"Start":"00:33.465 ","End":"00:38.205","Text":"the satellite is given a velocity u,"},{"Start":"00:38.205 ","End":"00:44.305","Text":"which is at an angle of 60 degrees to the vertical axis from Earth."},{"Start":"00:44.305 ","End":"00:50.030","Text":"We\u0027re being told in the question to ignore Earth\u0027s motion and its rotation."},{"Start":"00:50.030 ","End":"00:53.570","Text":"Question number 1 is to find a condition for"},{"Start":"00:53.570 ","End":"01:00.000","Text":"the velocity u such that the satellite will orbit in a closed circuit."},{"Start":"01:00.020 ","End":"01:04.175","Text":"A closed circuit orbit, just to remind you,"},{"Start":"01:04.175 ","End":"01:10.320","Text":"means that our satellite is orbiting in a perfect circle or an ellipse."},{"Start":"01:11.410 ","End":"01:18.350","Text":"The condition for the orbit so that the orbit will be in a closed circuit,"},{"Start":"01:18.350 ","End":"01:23.050","Text":"is that our energy will be smaller than 0."},{"Start":"01:23.050 ","End":"01:26.765","Text":"We can show this in two ways."},{"Start":"01:26.765 ","End":"01:33.135","Text":"The first way is that we said that our value for epsilon has to be less than 1."},{"Start":"01:33.135 ","End":"01:38.419","Text":"Just to remind you, our value for epsilon is the square root of 1 plus"},{"Start":"01:38.419 ","End":"01:46.630","Text":"2EL^2 divided by m Alpha squared."},{"Start":"01:46.630 ","End":"01:50.150","Text":"This whole expression has to be smaller than 1."},{"Start":"01:50.150 ","End":"01:55.690","Text":"The only way it\u0027s smaller than 1 is if our energy over here is a negative value,"},{"Start":"01:55.690 ","End":"02:02.420","Text":"and that\u0027s because all of the other variables in here are positive."},{"Start":"02:02.420 ","End":"02:09.005","Text":"Now, another way of thinking about this is if we remember, our u effective."},{"Start":"02:09.005 ","End":"02:13.639","Text":"So if we remember the graph for our u effective,"},{"Start":"02:13.639 ","End":"02:18.545","Text":"it looked something like this."},{"Start":"02:18.545 ","End":"02:25.130","Text":"When this is our u effective with a radius increasing"},{"Start":"02:25.130 ","End":"02:31.640","Text":"along the x-axis and we saw that when our energy was a negative value so,"},{"Start":"02:31.640 ","End":"02:33.380","Text":"it\u0027s some kind of constant function."},{"Start":"02:33.380 ","End":"02:39.785","Text":"Then that meant that our radius has to be between 2 values over here."},{"Start":"02:39.785 ","End":"02:43.430","Text":"Then we can see that we\u0027re going to have a closed circuit where"},{"Start":"02:43.430 ","End":"02:47.075","Text":"our radius is either one specific value,"},{"Start":"02:47.075 ","End":"02:53.280","Text":"if it were at the minimum point or a range of values if we\u0027re somewhere over here."},{"Start":"02:53.280 ","End":"02:56.060","Text":"The one specific point is when we\u0027re orbiting in"},{"Start":"02:56.060 ","End":"03:01.385","Text":"a perfect circle and the range of values represents an ellipse."},{"Start":"03:01.385 ","End":"03:07.415","Text":"But both of these options are when our energy is below 0."},{"Start":"03:07.415 ","End":"03:09.260","Text":"When our energy is above the 0,"},{"Start":"03:09.260 ","End":"03:12.725","Text":"then we can see that we have some starting radius,"},{"Start":"03:12.725 ","End":"03:16.220","Text":"but then our body can move off into infinity."},{"Start":"03:16.220 ","End":"03:20.070","Text":"That means that our orbit is not a closed circuit."},{"Start":"03:20.120 ","End":"03:24.345","Text":"Those are the two ways to think about this,"},{"Start":"03:24.345 ","End":"03:27.530","Text":"but the important point is that for a closed circuits,"},{"Start":"03:27.530 ","End":"03:31.740","Text":"we have to have our energy being smaller than 0."},{"Start":"03:31.850 ","End":"03:35.150","Text":"What we\u0027re going to do now is we\u0027re going to write out"},{"Start":"03:35.150 ","End":"03:39.665","Text":"our energy equation for a satellite at exactly this point."},{"Start":"03:39.665 ","End":"03:44.210","Text":"It\u0027s reached its maximum height and now it\u0027s switched"},{"Start":"03:44.210 ","End":"03:48.980","Text":"on one of its motors and it\u0027s began on moving at this velocity u,"},{"Start":"03:48.980 ","End":"03:53.300","Text":"so our equation for energy is going to be equal to 1/2 of"},{"Start":"03:53.300 ","End":"03:57.995","Text":"the mass of the satellite multiplied by its velocity squared,"},{"Start":"03:57.995 ","End":"04:00.850","Text":"and then plus its potential energy."},{"Start":"04:00.850 ","End":"04:04.955","Text":"It\u0027s potential energy is pointing in the opposite direction,"},{"Start":"04:04.955 ","End":"04:08.555","Text":"so it\u0027s going to be negative GM,"},{"Start":"04:08.555 ","End":"04:10.445","Text":"which is mass of the Earth,"},{"Start":"04:10.445 ","End":"04:17.225","Text":"multiplied by mass of the satellite and then divided by the distance between the two,"},{"Start":"04:17.225 ","End":"04:21.505","Text":"which is 2 times the radius of the Earth."},{"Start":"04:21.505 ","End":"04:27.575","Text":"Then I have to say that all of this has to be smaller than 0."},{"Start":"04:27.575 ","End":"04:34.925","Text":"Now what we can do is we can cross off this m and this m and this 1/2, and this 1/2,"},{"Start":"04:34.925 ","End":"04:40.250","Text":"and then we will see that our condition is that u^2 is"},{"Start":"04:40.250 ","End":"04:47.865","Text":"smaller than GM divided by R_E."},{"Start":"04:47.865 ","End":"04:50.530","Text":"Now we just have to take the square root."},{"Start":"04:50.530 ","End":"04:55.940","Text":"We\u0027ll get that the absolute value of our velocity so,"},{"Start":"04:55.940 ","End":"04:57.380","Text":"instead of using it as a vector,"},{"Start":"04:57.380 ","End":"05:00.230","Text":"we\u0027re not too concerned about its direction right now,"},{"Start":"05:00.230 ","End":"05:01.810","Text":"but rather its size."},{"Start":"05:01.810 ","End":"05:04.955","Text":"The size of the velocity therefore,"},{"Start":"05:04.955 ","End":"05:10.835","Text":"has to be smaller than the square root of G multiplied by"},{"Start":"05:10.835 ","End":"05:17.975","Text":"the mass of the Earth divided by the radius of the Earth."},{"Start":"05:17.975 ","End":"05:21.750","Text":"That\u0027s our answer to question Number 1."},{"Start":"05:22.040 ","End":"05:26.960","Text":"Now let\u0027s move on to question Number 2."},{"Start":"05:26.960 ","End":"05:30.935","Text":"Question Number 2 is asking us to find another condition for"},{"Start":"05:30.935 ","End":"05:36.655","Text":"our velocity u such that the satellite will not hit Earth."},{"Start":"05:36.655 ","End":"05:41.435","Text":"What we want is that our satellites is going to be orbiting Earth,"},{"Start":"05:41.435 ","End":"05:44.149","Text":"either in a circle or an ellipse,"},{"Start":"05:44.149 ","End":"05:47.325","Text":"and it will never hit the Earth."},{"Start":"05:47.325 ","End":"05:50.120","Text":"What does that mean? That inner circle,"},{"Start":"05:50.120 ","End":"05:53.480","Text":"that means that our radius has to be bigger than"},{"Start":"05:53.480 ","End":"05:56.870","Text":"our R_E and if we\u0027re dealing with an ellipse,"},{"Start":"05:56.870 ","End":"05:59.270","Text":"that means that the shortest distance between"},{"Start":"05:59.270 ","End":"06:02.615","Text":"the satellite and the Earth has to be bigger than our R_E,"},{"Start":"06:02.615 ","End":"06:05.290","Text":"which in that case is our minimum."},{"Start":"06:05.290 ","End":"06:12.785","Text":"Let\u0027s write that down so our r minimum must be bigger than the radius of the Earth."},{"Start":"06:12.785 ","End":"06:14.225","Text":"This is an ellipse,"},{"Start":"06:14.225 ","End":"06:16.970","Text":"and if we\u0027re dealing with a circle instead of r minimum,"},{"Start":"06:16.970 ","End":"06:20.550","Text":"it will just be r_min bigger than R_E."},{"Start":"06:20.690 ","End":"06:27.665","Text":"This means that even at the closest point that a satellite can be to Earth,"},{"Start":"06:27.665 ","End":"06:31.200","Text":"it\u0027s still not touching Earth."},{"Start":"06:31.580 ","End":"06:35.690","Text":"Now what I wanted to do is I wanted to find out what my r minimum"},{"Start":"06:35.690 ","End":"06:39.425","Text":"is and then I\u0027ll set this condition."},{"Start":"06:39.425 ","End":"06:42.605","Text":"Then my satellite will not hit Earth."},{"Start":"06:42.605 ","End":"06:45.215","Text":"How am I going to find my r minimum?"},{"Start":"06:45.215 ","End":"06:50.965","Text":"There\u0027s a few ways of doing this and 1 of them is saying that my energy,"},{"Start":"06:50.965 ","End":"06:55.865","Text":"or if my energy is simply equal to my u effective,"},{"Start":"06:55.865 ","End":"06:58.115","Text":"which is a function of r. Now,"},{"Start":"06:58.115 ","End":"07:01.790","Text":"another way to solve this is by using the idea of conservation"},{"Start":"07:01.790 ","End":"07:05.480","Text":"of energy and conservation of angular momentum and then to"},{"Start":"07:05.480 ","End":"07:08.210","Text":"find the velocity is at the maximum point and at"},{"Start":"07:08.210 ","End":"07:13.185","Text":"the minimum point and then from there to find our value for r_min however,"},{"Start":"07:13.185 ","End":"07:16.140","Text":"this way is significantly shorter."},{"Start":"07:16.140 ","End":"07:18.460","Text":"How come at my point r_min,"},{"Start":"07:18.460 ","End":"07:22.550","Text":"does my energy equal to my u effective?"},{"Start":"07:22.550 ","End":"07:27.230","Text":"If we go back to our diagram over here that we drew for our u"},{"Start":"07:27.230 ","End":"07:33.215","Text":"effective our function as a function of r. We can see that when,"},{"Start":"07:33.215 ","End":"07:37.925","Text":"we\u0027re over here, when I satellites orbiting in an ellipse so,"},{"Start":"07:37.925 ","End":"07:40.895","Text":"we saw that our energy has to be less than 0."},{"Start":"07:40.895 ","End":"07:44.000","Text":"Then we can see that our graph for energy,"},{"Start":"07:44.000 ","End":"07:46.685","Text":"this constant graph hits"},{"Start":"07:46.685 ","End":"07:51.700","Text":"our u effective graph at this point over here and at this point over here."},{"Start":"07:51.700 ","End":"07:56.255","Text":"These points, if we draw this up to our x-axis,"},{"Start":"07:56.255 ","End":"08:04.450","Text":"this point corresponds to our r_min and this point corresponds to our r_max."},{"Start":"08:04.450 ","End":"08:11.630","Text":"What this actually means is that when my energy is equal to my u effective so,"},{"Start":"08:11.630 ","End":"08:13.460","Text":"there are two points where that happens."},{"Start":"08:13.460 ","End":"08:17.695","Text":"Where my graph for energy cuts my graph for u effective so,"},{"Start":"08:17.695 ","End":"08:21.675","Text":"that is where my r_min and my r_max are."},{"Start":"08:21.675 ","End":"08:25.515","Text":"What I will get is I\u0027ll get two solutions to"},{"Start":"08:25.515 ","End":"08:30.844","Text":"this equation and then I just have to find which one is a smaller solution,"},{"Start":"08:30.844 ","End":"08:33.630","Text":"and that will be my r_min."},{"Start":"08:34.040 ","End":"08:36.465","Text":"Let\u0027s work this out."},{"Start":"08:36.465 ","End":"08:40.490","Text":"My energy, the general equation for energy of my body,"},{"Start":"08:40.490 ","End":"08:42.875","Text":"I already worked it out. It\u0027s over here."},{"Start":"08:42.875 ","End":"08:51.875","Text":"That\u0027s going to be equal to my 1/2 mu^2 minus G mass of the Earth,"},{"Start":"08:51.875 ","End":"08:53.270","Text":"mass of the satellite,"},{"Start":"08:53.270 ","End":"08:56.085","Text":"divided by 2 R_E,"},{"Start":"08:56.085 ","End":"09:02.295","Text":"the distance between the two and that has to be equal to my u effective."},{"Start":"09:02.295 ","End":"09:04.635","Text":"Let\u0027s scroll a little bit to the side."},{"Start":"09:04.635 ","End":"09:09.860","Text":"My u effective we saw from our lesson is going to be equal to,"},{"Start":"09:09.860 ","End":"09:13.207","Text":"and it\u0027s always like this when we\u0027re dealing with gravity"},{"Start":"09:13.207 ","End":"09:17.975","Text":"L^2 divided by 2 mass of the satellite,"},{"Start":"09:17.975 ","End":"09:23.695","Text":"r^2 minus GMm divided by"},{"Start":"09:23.695 ","End":"09:29.900","Text":"r. Now I have this and what I\u0027m trying to find is my r over here."},{"Start":"09:29.900 ","End":"09:32.270","Text":"This is what I\u0027m trying to isolate out,"},{"Start":"09:32.270 ","End":"09:35.600","Text":"and also this equation is true whenever we\u0027re"},{"Start":"09:35.600 ","End":"09:39.380","Text":"dealing with gravitational force if you can\u0027t remember how we got to this equation,"},{"Start":"09:39.380 ","End":"09:43.390","Text":"please go back to the lesson about u effective."},{"Start":"09:43.390 ","End":"09:48.845","Text":"Before we can isolate out r and then set it to be bigger than R_E."},{"Start":"09:48.845 ","End":"09:53.710","Text":"First we have to find out what our angular momentum is."},{"Start":"09:53.710 ","End":"09:59.225","Text":"Let\u0027s work out what our angular momentum L is equal to."},{"Start":"09:59.225 ","End":"10:05.400","Text":"As we know, it\u0027s going to be equal to the mass multiplied by the velocity,"},{"Start":"10:05.400 ","End":"10:08.480","Text":"so we\u0027re going to be using the information that we have over here."},{"Start":"10:08.480 ","End":"10:11.810","Text":"The mass multiplied by the velocity, which is u,"},{"Start":"10:11.810 ","End":"10:17.060","Text":"and then it\u0027s going to be multiplied by the distance to the Earth,"},{"Start":"10:17.060 ","End":"10:20.420","Text":"so that\u0027s R_E, and then multiplied by"},{"Start":"10:20.420 ","End":"10:24.935","Text":"sine of the angle between our velocity and our radius."},{"Start":"10:24.935 ","End":"10:31.060","Text":"We\u0027ve already been told that the angle is this angle over here, this 60 degrees."},{"Start":"10:31.060 ","End":"10:40.175","Text":"Sine of 60 and sine of 60 is going to be equal to root 3 over 2."},{"Start":"10:40.175 ","End":"10:43.555","Text":"Then, oh sorry, here we have a 2."},{"Start":"10:43.555 ","End":"10:52.860","Text":"What we\u0027re going to get is root 3 multiplied by muR_E."},{"Start":"10:52.860 ","End":"10:56.180","Text":"The 2 over here canceled out with our sine 60,"},{"Start":"10:56.180 ","End":"10:59.795","Text":"which is equal to root 3 over 2."},{"Start":"10:59.795 ","End":"11:04.620","Text":"2R _E comes from this distance over here, which is 2R_E."},{"Start":"11:04.860 ","End":"11:08.245","Text":"Just a reminder, this distance,"},{"Start":"11:08.245 ","End":"11:12.460","Text":"it\u0027s always going to be the distance from the axis of rotation."},{"Start":"11:12.460 ","End":"11:17.450","Text":"Here\u0027s specifically the axis of rotation is at the center of the Earth."},{"Start":"11:17.490 ","End":"11:21.070","Text":"Now what we\u0027re going to do is we\u0027re going to"},{"Start":"11:21.070 ","End":"11:25.120","Text":"substitute in our angular momentum into our equation,"},{"Start":"11:25.120 ","End":"11:27.010","Text":"so let\u0027s write this out again,"},{"Start":"11:27.010 ","End":"11:37.580","Text":"so we\u0027re going to have a half mass times velocity squared minus GMm divided by 2R_E,"},{"Start":"11:37.910 ","End":"11:42.120","Text":"and that\u0027s going to be equal to our angular momentum squared,"},{"Start":"11:42.120 ","End":"11:51.720","Text":"so our root 3 squared is just going to be 3 and then multiplied by mUR_E squared"},{"Start":"11:51.720 ","End":"12:02.420","Text":"divided by 2mr^2 negative GMm divided by r."},{"Start":"12:02.420 ","End":"12:10.690","Text":"Now what we\u0027re going to do is we are going to multiply both sides by"},{"Start":"12:10.690 ","End":"12:19.135","Text":"2r squared and divide both sides by m. Once we\u0027ve done that,"},{"Start":"12:19.135 ","End":"12:22.270","Text":"we\u0027re going to be left with an expression like so."},{"Start":"12:22.270 ","End":"12:31.210","Text":"I have r squared multiplied by u squared minus GM divided by R_E,"},{"Start":"12:31.210 ","End":"12:32.710","Text":"and that is going to be equal"},{"Start":"12:32.710 ","End":"12:42.230","Text":"to 3u^2R_E^2 minus 2GMr."},{"Start":"12:44.340 ","End":"12:49.240","Text":"We see that our m\u0027s are going to cancel out,"},{"Start":"12:49.240 ","End":"12:53.210","Text":"and we get this expression over here."},{"Start":"12:53.880 ","End":"12:59.740","Text":"Now we can see that we have some quadratic equation,"},{"Start":"12:59.740 ","End":"13:03.880","Text":"here we have a coefficient multiplied by r^2,"},{"Start":"13:03.880 ","End":"13:06.685","Text":"here we have a coefficient multiplied by r,"},{"Start":"13:06.685 ","End":"13:10.135","Text":"and here we have our free coefficients."},{"Start":"13:10.135 ","End":"13:15.415","Text":"Let\u0027s rearrange this so that it looks slightly more like a quadratic equation,"},{"Start":"13:15.415 ","End":"13:20.575","Text":"so we\u0027ll have our u^2 minus GM divided by R_E."},{"Start":"13:20.575 ","End":"13:25.180","Text":"As we can see, this is a constant coefficient multiplied by r^2,"},{"Start":"13:25.180 ","End":"13:31.210","Text":"and then we\u0027ll have plus 2GM multiplied by r minus"},{"Start":"13:31.210 ","End":"13:39.355","Text":"3u^2R_E squared and all of that is going to be equal to 0."},{"Start":"13:39.355 ","End":"13:46.480","Text":"Now we can label everything so we\u0027ll see that this over here is a,"},{"Start":"13:46.480 ","End":"13:49.630","Text":"this over here is b,"},{"Start":"13:49.630 ","End":"13:55.810","Text":"and all of this is equal to c. Now all"},{"Start":"13:55.810 ","End":"14:02.095","Text":"we have to do is we have to plug this in to our quadratic equation solver,"},{"Start":"14:02.095 ","End":"14:05.350","Text":"and I reminding you every time we use"},{"Start":"14:05.350 ","End":"14:08.965","Text":"the quadratic equation we\u0027re going to get 2 solutions,"},{"Start":"14:08.965 ","End":"14:10.540","Text":"r_1 and r_2,"},{"Start":"14:10.540 ","End":"14:15.655","Text":"which of course correspond to a minimum and a maximum."},{"Start":"14:15.655 ","End":"14:17.680","Text":"Our r_1 and r_2,"},{"Start":"14:17.680 ","End":"14:22.068","Text":"what do we have to do is we have to plug in negative b plus or minus."},{"Start":"14:22.068 ","End":"14:24.325","Text":"So this is where we get our 2 solutions,"},{"Start":"14:24.325 ","End":"14:30.760","Text":"the square root of b squared minus 4 times a times c,"},{"Start":"14:30.760 ","End":"14:35.570","Text":"and all of this is going to be divided by 2a."},{"Start":"14:36.480 ","End":"14:39.100","Text":"We\u0027re going to put negative b in here,"},{"Start":"14:39.100 ","End":"14:45.535","Text":"plus or minus the square root of our b squared minus 4 times this coefficient times this,"},{"Start":"14:45.535 ","End":"14:48.685","Text":"and including this negative sign over here."},{"Start":"14:48.685 ","End":"14:53.200","Text":"Then we want to get our r minimum,"},{"Start":"14:53.200 ","End":"14:56.800","Text":"so in order to find our r_min,"},{"Start":"14:56.800 ","End":"14:59.995","Text":"so let\u0027s see how we do this."},{"Start":"14:59.995 ","End":"15:02.830","Text":"Our r_min we have to decide if we\u0027re going to take"},{"Start":"15:02.830 ","End":"15:06.670","Text":"the solution with a positive or a negative."},{"Start":"15:06.670 ","End":"15:09.865","Text":"So this might be slightly counter-intuitive,"},{"Start":"15:09.865 ","End":"15:16.740","Text":"but we\u0027re going to take the solution with the positive in order to find our r_min."},{"Start":"15:16.740 ","End":"15:18.295","Text":"Why is that?"},{"Start":"15:18.295 ","End":"15:20.485","Text":"Because this is a negative number,"},{"Start":"15:20.485 ","End":"15:24.400","Text":"when we add this square root sign,"},{"Start":"15:24.400 ","End":"15:29.080","Text":"the size of this is going to be smaller."},{"Start":"15:29.080 ","End":"15:33.970","Text":"If we have a negative number and we take the negative root there,"},{"Start":"15:33.970 ","End":"15:37.480","Text":"we\u0027re going to have a negative number and make it more negative,"},{"Start":"15:37.480 ","End":"15:38.710","Text":"and then when we take the size,"},{"Start":"15:38.710 ","End":"15:40.165","Text":"it\u0027s going to be bigger,"},{"Start":"15:40.165 ","End":"15:43.540","Text":"so let\u0027s just put this on a line,"},{"Start":"15:43.540 ","End":"15:46.345","Text":"we\u0027re here, r is increasing."},{"Start":"15:46.345 ","End":"15:49.405","Text":"Then here we have our 0,"},{"Start":"15:49.405 ","End":"15:54.040","Text":"so if r starts off at negative b divided by 2a and"},{"Start":"15:54.040 ","End":"15:58.825","Text":"then we take off this square root divided by 2a,"},{"Start":"15:58.825 ","End":"16:02.290","Text":"so our new r is going to be here,"},{"Start":"16:02.290 ","End":"16:07.910","Text":"and then the size is going to be just the size of this distance, which is big."},{"Start":"16:07.910 ","End":"16:11.730","Text":"If however, we start off here at our negative b divided by 2a,"},{"Start":"16:11.730 ","End":"16:17.190","Text":"this is the value and then we add on the square root divided by 2a,"},{"Start":"16:17.190 ","End":"16:20.755","Text":"then we\u0027ll be adding on something positive,"},{"Start":"16:20.755 ","End":"16:22.405","Text":"which can be maybe here,"},{"Start":"16:22.405 ","End":"16:28.255","Text":"let\u0027s say, and then we can see that the size of this is significantly smaller."},{"Start":"16:28.255 ","End":"16:31.570","Text":"I know it\u0027s counter-intuitive when we want the r_min to"},{"Start":"16:31.570 ","End":"16:35.575","Text":"take the positive square root expression,"},{"Start":"16:35.575 ","End":"16:37.555","Text":"but that is what we\u0027re meant to do."},{"Start":"16:37.555 ","End":"16:46.330","Text":"Our value for r_min is going to be negative b plus the square root of b^2 minus 4ac,"},{"Start":"16:46.330 ","End":"16:51.940","Text":"now I\u0027m not going to substitute in these values now to save a little bit of time,"},{"Start":"16:51.940 ","End":"16:55.420","Text":"and then from what we saw up top over here,"},{"Start":"16:55.420 ","End":"17:01.270","Text":"that our condition is that our r_minimum is bigger than the radius of the Earth."},{"Start":"17:01.270 ","End":"17:07.310","Text":"Then I\u0027m going to set all of this to be bigger than R_E,"},{"Start":"17:07.470 ","End":"17:13.120","Text":"so we found our condition for r_min but what we\u0027re trying to find is"},{"Start":"17:13.120 ","End":"17:19.600","Text":"another condition actually for u our velocity such that the satellite will not hit Earth."},{"Start":"17:19.600 ","End":"17:26.515","Text":"Now we\u0027re trying to find a condition for our u corresponding to this condition for r_min."},{"Start":"17:26.515 ","End":"17:31.570","Text":"Let\u0027s go back down and let\u0027s carry on solving,"},{"Start":"17:31.570 ","End":"17:40.285","Text":"so now we can rewrite this equation by multiplying both sides by 2a,"},{"Start":"17:40.285 ","End":"17:48.385","Text":"so then we\u0027ll be left with negative b plus the square root of b^2 minus 4ac,"},{"Start":"17:48.385 ","End":"17:55.810","Text":"and this is going to be bigger than 2a multiplied by R_E."},{"Start":"17:55.810 ","End":"18:01.795","Text":"Now what we\u0027re going to do is we\u0027re going to add our b to both sides,"},{"Start":"18:01.795 ","End":"18:03.490","Text":"to get rid of this,"},{"Start":"18:03.490 ","End":"18:05.500","Text":"so we\u0027ll have here plus b,"},{"Start":"18:05.500 ","End":"18:09.700","Text":"and now what we want to do is we want to get rid of this square root sign."},{"Start":"18:09.700 ","End":"18:12.730","Text":"What we\u0027re going to do is we\u0027re going to square"},{"Start":"18:12.730 ","End":"18:17.185","Text":"both sides and we\u0027re going to be left with this,"},{"Start":"18:17.185 ","End":"18:22.675","Text":"so we\u0027ll have b squared minus 4ac is bigger than"},{"Start":"18:22.675 ","End":"18:28.330","Text":"4a^2R_E squared"},{"Start":"18:28.330 ","End":"18:36.655","Text":"plus 4aR_E b plus b squared."},{"Start":"18:36.655 ","End":"18:39.700","Text":"What I did is I squared both sides,"},{"Start":"18:39.700 ","End":"18:45.350","Text":"and then I was left with this equation over here."},{"Start":"18:45.870 ","End":"18:49.990","Text":"Here we can see that I have a b^2 and here I have a b^2,"},{"Start":"18:49.990 ","End":"18:53.605","Text":"so I can minus b^2 from both sides,"},{"Start":"18:53.605 ","End":"18:57.130","Text":"and now I can see that in every single term I have a 4a,"},{"Start":"18:57.130 ","End":"18:59.560","Text":"so I can cancel out the 4a here,"},{"Start":"18:59.560 ","End":"19:01.750","Text":"the 4a over here,"},{"Start":"19:01.750 ","End":"19:05.080","Text":"and the 4a over here."},{"Start":"19:05.080 ","End":"19:10.315","Text":"Now I\u0027m left with negative c is bigger than"},{"Start":"19:10.315 ","End":"19:19.885","Text":"aR_E squared plus R_E multiplied by b,"},{"Start":"19:19.885 ","End":"19:25.255","Text":"so now it\u0027s the time to substitute in my values for a,"},{"Start":"19:25.255 ","End":"19:29.305","Text":"b, and c. My negative c,"},{"Start":"19:29.305 ","End":"19:33.505","Text":"let\u0027s substitute in my c so I can see that my c is also a negative number,"},{"Start":"19:33.505 ","End":"19:35.560","Text":"so negative and negative is positive,"},{"Start":"19:35.560 ","End":"19:41.635","Text":"so I\u0027ll be left with simply 3u^2 multiplied by R_E squared,"},{"Start":"19:41.635 ","End":"19:42.850","Text":"so that\u0027s my c,"},{"Start":"19:42.850 ","End":"19:47.760","Text":"and it has to be bigger than my a multiplied by my R_E squared."},{"Start":"19:47.760 ","End":"19:49.247","Text":"This is my a over here."},{"Start":"19:49.247 ","End":"19:54.340","Text":"I\u0027ll have u squared minus GM divided by"},{"Start":"19:54.340 ","End":"20:00.475","Text":"R_E multiplied by R_E squared,"},{"Start":"20:00.475 ","End":"20:04.585","Text":"and then plus my arm multiplied by b."},{"Start":"20:04.585 ","End":"20:10.705","Text":"My b is 2GM and then multiplied by my R_E."},{"Start":"20:10.705 ","End":"20:14.815","Text":"My question was to find my condition for u,"},{"Start":"20:14.815 ","End":"20:18.250","Text":"so that means that I want to isolate out my u.,"},{"Start":"20:18.250 ","End":"20:24.325","Text":"and then I will know that corresponding to this velocity,"},{"Start":"20:24.325 ","End":"20:26.755","Text":"and the condition that I\u0027m setting for it,"},{"Start":"20:26.755 ","End":"20:29.425","Text":"it will never hit Earth."},{"Start":"20:29.425 ","End":"20:35.515","Text":"Let\u0027s start opening up all of our brackets and moving stuff around,"},{"Start":"20:35.515 ","End":"20:38.515","Text":"and then we will get to this,"},{"Start":"20:38.515 ","End":"20:42.939","Text":"that our 2u^2 multiplied by R_E"},{"Start":"20:42.939 ","End":"20:49.030","Text":"squared is bigger than GM multiplied by R_E."},{"Start":"20:49.030 ","End":"20:54.130","Text":"Then once again, we will isolate out our u and again,"},{"Start":"20:54.130 ","End":"20:56.093","Text":"we\u0027re going to take the square root."},{"Start":"20:56.093 ","End":"20:59.830","Text":"So because we don\u0027t really care in which direction it\u0027s going,"},{"Start":"20:59.830 ","End":"21:02.557","Text":"but rather we want to know the size of the velocity,"},{"Start":"21:02.557 ","End":"21:04.555","Text":"so we\u0027re going to the size,"},{"Start":"21:04.555 ","End":"21:06.580","Text":"and that means that when we take the square root,"},{"Start":"21:06.580 ","End":"21:09.325","Text":"we don\u0027t have to deal with a plus or minus."},{"Start":"21:09.325 ","End":"21:19.820","Text":"That means that the size of our u has to be bigger than GM divided by 2R_E."},{"Start":"21:21.360 ","End":"21:24.415","Text":"This is our second condition,"},{"Start":"21:24.415 ","End":"21:26.830","Text":"and if we adhere to this condition,"},{"Start":"21:26.830 ","End":"21:29.650","Text":"then our satellite shouldn\u0027t hit Earth,"},{"Start":"21:29.650 ","End":"21:33.820","Text":"so now in order to put our 2 answers together,"},{"Start":"21:33.820 ","End":"21:40.600","Text":"so we know that in order for our satellite to orbit the Earth in a closed orbit,"},{"Start":"21:40.600 ","End":"21:44.680","Text":"so that means in either an ellipse or a perfect circle,"},{"Start":"21:44.680 ","End":"21:50.275","Text":"and in order for our satellites to never hit Earth, so our velocity,"},{"Start":"21:50.275 ","End":"21:52.390","Text":"or rather the size of our velocity,"},{"Start":"21:52.390 ","End":"22:01.160","Text":"has to be bigger than the square root of GM divided by 2R_E,"},{"Start":"22:02.160 ","End":"22:12.010","Text":"and it has to be smaller than the square root of GM divided by R_E,"},{"Start":"22:12.010 ","End":"22:16.355","Text":"so the size of u as long as it\u0027s in this range,"},{"Start":"22:16.355 ","End":"22:18.660","Text":"we\u0027re good to go."},{"Start":"22:19.600 ","End":"22:23.009","Text":"That\u0027s the end of our question."}],"ID":9472}],"Thumbnail":null,"ID":5364}]

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