[{"Name":"Explanation On Galilean Transformations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Basic Explanation and Formula","Duration":"6m 4s","ChapterTopicVideoID":8947,"CourseChapterTopicPlaylistID":5403,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8947.jpeg","UploadDate":"2017-03-21T08:44:41.8530000","DurationForVideoObject":"PT6M4S","Description":null,"MetaTitle":"Basic Explanation and Formula: Video + Workbook | Proprep","MetaDescription":"Relative Motion - 1. Explanation on Galilean Transformations. 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/relative-motion/1.-explanation-on-galilean-transformations/vid9240","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Hello. In this lecture,"},{"Start":"00:02.730 ","End":"00:04.650","Text":"we\u0027re going to talk about relative motion."},{"Start":"00:04.650 ","End":"00:07.395","Text":"Relative motion is when we measure the position,"},{"Start":"00:07.395 ","End":"00:12.615","Text":"velocity or acceleration of 1 object in relation or respect to another object."},{"Start":"00:12.615 ","End":"00:15.510","Text":"This is not the simplest subject we\u0027ll talk about,"},{"Start":"00:15.510 ","End":"00:17.190","Text":"in fact, it\u0027s rather confusing."},{"Start":"00:17.190 ","End":"00:20.856","Text":"Oftentimes, we\u0027re dealing with 3 or more different objects,"},{"Start":"00:20.856 ","End":"00:22.890","Text":"but a lot of units will return to"},{"Start":"00:22.890 ","End":"00:26.450","Text":"relative motion and this will be the base of many other things we do,"},{"Start":"00:26.450 ","End":"00:30.070","Text":"so it\u0027s rather important to pay a special attention here."},{"Start":"00:30.620 ","End":"00:33.660","Text":"The first thing we need are 2 objects."},{"Start":"00:33.660 ","End":"00:36.015","Text":"Our first object will be a car."},{"Start":"00:36.015 ","End":"00:38.415","Text":"Excuse my drawing ability."},{"Start":"00:38.415 ","End":"00:42.260","Text":"The second object will be a man."},{"Start":"00:42.260 ","End":"00:45.350","Text":"The man, he\u0027s going to move later on in the problem,"},{"Start":"00:45.350 ","End":"00:47.450","Text":"so let\u0027s put him on a skateboard."},{"Start":"00:47.450 ","End":"00:51.710","Text":"Now, the first thing we want to do is measure the position,"},{"Start":"00:51.710 ","End":"00:53.300","Text":"not velocity, not acceleration."},{"Start":"00:53.300 ","End":"00:57.380","Text":"We\u0027ll start with position of each of these objects in relation to the ground."},{"Start":"00:57.380 ","End":"00:59.765","Text":"The best way for me to think about this,"},{"Start":"00:59.765 ","End":"01:04.340","Text":"or my preferred method is to imagine that there\u0027s a lab and inside the lab is"},{"Start":"01:04.340 ","End":"01:09.770","Text":"some scientist who\u0027s taking measurements every moment of the position,"},{"Start":"01:09.770 ","End":"01:13.325","Text":"the velocity or the acceleration of the man or the car,"},{"Start":"01:13.325 ","End":"01:14.869","Text":"either of these 2 objects,"},{"Start":"01:14.869 ","End":"01:16.775","Text":"whatever that is at any given moment,"},{"Start":"01:16.775 ","End":"01:18.865","Text":"depending on what I want to measure."},{"Start":"01:18.865 ","End":"01:21.905","Text":"If I want to draw the vector,"},{"Start":"01:21.905 ","End":"01:25.370","Text":"that is the distance of the man from the lab,"},{"Start":"01:25.370 ","End":"01:26.840","Text":"I\u0027ll draw it in red here."},{"Start":"01:26.840 ","End":"01:29.815","Text":"It\u0027ll be something along these lines."},{"Start":"01:29.815 ","End":"01:32.220","Text":"For now we\u0027ll work in 1-dimension,"},{"Start":"01:32.220 ","End":"01:34.110","Text":"but later we\u0027ll add 2-dimensional and"},{"Start":"01:34.110 ","End":"01:36.965","Text":"3-dimensional figures for the sake of generalization."},{"Start":"01:36.965 ","End":"01:39.425","Text":"But if I want to do the same thing for the car,"},{"Start":"01:39.425 ","End":"01:40.955","Text":"measure the position of the car,"},{"Start":"01:40.955 ","End":"01:42.710","Text":"the distance of the car from the lab,"},{"Start":"01:42.710 ","End":"01:47.020","Text":"I can draw another vector also from the lab to the car like this."},{"Start":"01:47.020 ","End":"01:49.730","Text":"I didn\u0027t put it in writing before and actually,"},{"Start":"01:49.730 ","End":"01:53.450","Text":"it\u0027s a little educational because if I don\u0027t say where I\u0027m measuring from,"},{"Start":"01:53.450 ","End":"01:55.340","Text":"it should be assumed that are measuring from the lab,"},{"Start":"01:55.340 ","End":"01:56.600","Text":"but for the sake of simplicity,"},{"Start":"01:56.600 ","End":"01:59.040","Text":"I\u0027ll put it in here."},{"Start":"01:59.040 ","End":"02:02.300","Text":"Now, what I want to measure is the relative position of the car."},{"Start":"02:02.300 ","End":"02:03.365","Text":"Relative to what?"},{"Start":"02:03.365 ","End":"02:05.270","Text":"Relative to the man."},{"Start":"02:05.270 ","End":"02:08.540","Text":"Imagine that the man is my new measurer,"},{"Start":"02:08.540 ","End":"02:11.360","Text":"the one who\u0027s taking measurements and I want to measure"},{"Start":"02:11.360 ","End":"02:14.690","Text":"the distance from the man to the car."},{"Start":"02:14.690 ","End":"02:16.430","Text":"We can draw this out."},{"Start":"02:16.430 ","End":"02:18.695","Text":"The man is measuring the distance to the car."},{"Start":"02:18.695 ","End":"02:21.830","Text":"We\u0027re drawing the distance from the car to the man and"},{"Start":"02:21.830 ","End":"02:25.340","Text":"that we have in blue now. This is relative."},{"Start":"02:25.340 ","End":"02:28.730","Text":"Imagine that the distance from the lab to the man is"},{"Start":"02:28.730 ","End":"02:32.870","Text":"10 meters and the distance from the lab to the car is 30 meters,"},{"Start":"02:32.870 ","End":"02:37.340","Text":"then the distance from the man to the car is 20 meters."},{"Start":"02:37.340 ","End":"02:40.175","Text":"You can think of it in terms of the following."},{"Start":"02:40.175 ","End":"02:41.990","Text":"Imagine you add the distance,"},{"Start":"02:41.990 ","End":"02:44.300","Text":"the first vector that we drew,"},{"Start":"02:44.300 ","End":"02:46.780","Text":"the distance from the lab to the man"},{"Start":"02:46.780 ","End":"02:49.923","Text":"and add to that the distance from the man to the car,"},{"Start":"02:49.923 ","End":"02:53.000","Text":"that will give you the distance from the lab to the car,"},{"Start":"02:53.000 ","End":"02:55.280","Text":"and this is how the relative position works."},{"Start":"02:55.280 ","End":"02:59.125","Text":"We can even think of this in terms of a formula and write it out like so."},{"Start":"02:59.125 ","End":"03:03.410","Text":"We have the distance from the lab to the car equals"},{"Start":"03:03.410 ","End":"03:09.100","Text":"the distance from the lab to the man plus the distance from the man to the car."},{"Start":"03:09.100 ","End":"03:13.490","Text":"This is your formula. I don\u0027t always want to write out all of these words,"},{"Start":"03:13.490 ","End":"03:14.690","Text":"car relative to lab,"},{"Start":"03:14.690 ","End":"03:15.950","Text":"man relative to lab,"},{"Start":"03:15.950 ","End":"03:17.390","Text":"car relative to man,"},{"Start":"03:17.390 ","End":"03:21.440","Text":"so I\u0027ll simply put a tag next to the blue term,"},{"Start":"03:21.440 ","End":"03:24.343","Text":"which is a relative term to show that it is relative,"},{"Start":"03:24.343 ","End":"03:27.320","Text":"so it\u0027d be x tag would be my relative term."},{"Start":"03:27.320 ","End":"03:33.080","Text":"We\u0027re measuring the car relative to the man as opposed to relative to the lab."},{"Start":"03:33.080 ","End":"03:36.950","Text":"Another thing is that it\u0027s often more accepted to take"},{"Start":"03:36.950 ","End":"03:40.910","Text":"the relative term and put on the left side of the equation and isolate it from the rest."},{"Start":"03:40.910 ","End":"03:43.340","Text":"Oftentimes, that\u0027s what we\u0027re trying to measure so we can"},{"Start":"03:43.340 ","End":"03:46.226","Text":"write this in another way as follows,"},{"Start":"03:46.226 ","End":"03:47.750","Text":"car relative to man,"},{"Start":"03:47.750 ","End":"03:49.070","Text":"our blue distance,"},{"Start":"03:49.070 ","End":"03:53.275","Text":"equals car relative to lab minus man relative to lab."},{"Start":"03:53.275 ","End":"03:57.725","Text":"This is a more useful way perhaps of writing out the same formula."},{"Start":"03:57.725 ","End":"04:00.395","Text":"Each person has their own preferences."},{"Start":"04:00.395 ","End":"04:02.030","Text":"I\u0027ve noticed that some students prefer"},{"Start":"04:02.030 ","End":"04:05.015","Text":"the second way because it isolates the relative term,"},{"Start":"04:05.015 ","End":"04:08.600","Text":"relative meaning that we\u0027re using a different form of measurement."},{"Start":"04:08.600 ","End":"04:10.955","Text":"We\u0027re measuring from the man instead of measuring from the lab,"},{"Start":"04:10.955 ","End":"04:12.290","Text":"that makes it relative."},{"Start":"04:12.290 ","End":"04:14.450","Text":"When something\u0027s relative, it should always get"},{"Start":"04:14.450 ","End":"04:17.945","Text":"the tag symbols that we know that we\u0027re using a different observer."},{"Start":"04:17.945 ","End":"04:19.730","Text":"In terms of what you write down,"},{"Start":"04:19.730 ","End":"04:21.765","Text":"you should choose either the first or the second,"},{"Start":"04:21.765 ","End":"04:23.810","Text":"put it in your notebook, whatever you prefer,"},{"Start":"04:23.810 ","End":"04:25.374","Text":"whatever you find easiest,"},{"Start":"04:25.374 ","End":"04:29.285","Text":"whether it\u0027s adding things in 1 order or subtracting in the other,"},{"Start":"04:29.285 ","End":"04:33.055","Text":"choose what you\u0027d prefer and put that formula in your formula book."},{"Start":"04:33.055 ","End":"04:36.702","Text":"This explains what to do in terms of position,"},{"Start":"04:36.702 ","End":"04:39.830","Text":"but if we\u0027re dealing in terms of velocity that we want to measure,"},{"Start":"04:39.830 ","End":"04:41.605","Text":"we need to figure that out as well."},{"Start":"04:41.605 ","End":"04:44.915","Text":"Each object has a velocity that could be v_man,"},{"Start":"04:44.915 ","End":"04:46.070","Text":"for the car, we would have,"},{"Start":"04:46.070 ","End":"04:48.170","Text":"v_car and each of those, although,"},{"Start":"04:48.170 ","End":"04:49.610","Text":"I didn\u0027t say so explicitly,"},{"Start":"04:49.610 ","End":"04:52.625","Text":"is relative to the ground or relative to the lab."},{"Start":"04:52.625 ","End":"04:56.660","Text":"But what happens when we want to measure the relative velocity, that is,"},{"Start":"04:56.660 ","End":"04:58.970","Text":"how the man observes the car going at"},{"Start":"04:58.970 ","End":"05:02.125","Text":"what speed or what velocity the man observes the car?"},{"Start":"05:02.125 ","End":"05:04.640","Text":"Well, I need to find a formula for that and it turns"},{"Start":"05:04.640 ","End":"05:07.360","Text":"out it\u0027s the exact same formula we used before."},{"Start":"05:07.360 ","End":"05:10.400","Text":"We can say that, just as we had here,"},{"Start":"05:10.400 ","End":"05:13.280","Text":"the position of the car relative to the man equals the position of"},{"Start":"05:13.280 ","End":"05:17.249","Text":"the car relative to the lab minus the position of the man relative to the lab,"},{"Start":"05:17.249 ","End":"05:22.280","Text":"that the velocity of the car relative to the man equals the velocity of"},{"Start":"05:22.280 ","End":"05:27.800","Text":"the car relative to the lab minus the velocity of the man relative to the lab."},{"Start":"05:27.800 ","End":"05:31.100","Text":"The way that we come to this formula is if we\u0027re measuring"},{"Start":"05:31.100 ","End":"05:34.000","Text":"the position of each of these objects and they start to move,"},{"Start":"05:34.000 ","End":"05:37.760","Text":"we can take the derivative of each one and instead of position,"},{"Start":"05:37.760 ","End":"05:40.970","Text":"we\u0027re left with velocity of each of these objects so that we end up"},{"Start":"05:40.970 ","End":"05:44.180","Text":"with the car\u0027s velocity relative to the man,"},{"Start":"05:44.180 ","End":"05:46.265","Text":"the car\u0027s velocity relative to the lab,"},{"Start":"05:46.265 ","End":"05:48.860","Text":"and the man\u0027s velocity relative to the lab."},{"Start":"05:48.860 ","End":"05:50.630","Text":"In the case of acceleration,"},{"Start":"05:50.630 ","End":"05:51.680","Text":"we do the same thing."},{"Start":"05:51.680 ","End":"05:53.780","Text":"We can take the derivative of the velocity of"},{"Start":"05:53.780 ","End":"05:56.420","Text":"each and we\u0027re left with a very similar formula,"},{"Start":"05:56.420 ","End":"05:58.535","Text":"the acceleration of the car relative to the man"},{"Start":"05:58.535 ","End":"06:01.235","Text":"equals the acceleration of the car relative to the lab,"},{"Start":"06:01.235 ","End":"06:05.040","Text":"minus the acceleration of the man relative to the lab."}],"ID":9240},{"Watched":false,"Name":"Two Dimensions","Duration":"2m 48s","ChapterTopicVideoID":8948,"CourseChapterTopicPlaylistID":5403,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.205","Text":"Let\u0027s see what happens when we\u0027re dealing with multiple dimensions, 2 or 3-dimensions."},{"Start":"00:05.205 ","End":"00:06.330","Text":"For the sake of simplicity,"},{"Start":"00:06.330 ","End":"00:08.775","Text":"we\u0027ll do a 2-dimensional example where the car,"},{"Start":"00:08.775 ","End":"00:10.650","Text":"as opposed to being at y=0,"},{"Start":"00:10.650 ","End":"00:14.175","Text":"is somewhere higher up on the y-axis, say up here."},{"Start":"00:14.175 ","End":"00:18.555","Text":"It could also be the man that\u0027s on the y-axis or both, it doesn\u0027t really matter."},{"Start":"00:18.555 ","End":"00:23.865","Text":"But the point is, when you have a 2-dimensional relative motion problem,"},{"Start":"00:23.865 ","End":"00:25.860","Text":"you do each dimension on its own."},{"Start":"00:25.860 ","End":"00:29.025","Text":"In the same way that we did the position of x,"},{"Start":"00:29.025 ","End":"00:31.125","Text":"we can also do the position of y,"},{"Start":"00:31.125 ","End":"00:35.924","Text":"y tag which is the car relative to the man, equals y,"},{"Start":"00:35.924 ","End":"00:41.535","Text":"which would be the car relative to the lab minus the y of the man relative to the lab."},{"Start":"00:41.535 ","End":"00:44.420","Text":"As goes for x and y in 3-dimensions,"},{"Start":"00:44.420 ","End":"00:45.980","Text":"we could do the same thing for z,"},{"Start":"00:45.980 ","End":"00:47.930","Text":"and the same goes for velocity as well."},{"Start":"00:47.930 ","End":"00:50.540","Text":"We break our velocity down into 2-dimensions."},{"Start":"00:50.540 ","End":"00:52.159","Text":"For measuring the velocity,"},{"Start":"00:52.159 ","End":"00:55.945","Text":"we can measure our x velocity and our y velocity separately."},{"Start":"00:55.945 ","End":"01:00.200","Text":"V_x tag, which is the velocity of the car relative to the man,"},{"Start":"01:00.200 ","End":"01:04.160","Text":"equals the velocity of the car relative to the lab,"},{"Start":"01:04.160 ","End":"01:07.430","Text":"minus the velocity of the man relative to the lab."},{"Start":"01:07.430 ","End":"01:11.960","Text":"Of course, both in terms of the x component and you do the same for y."},{"Start":"01:11.960 ","End":"01:19.895","Text":"V_y equals the y velocity of the car minus the y velocity the man,"},{"Start":"01:19.895 ","End":"01:21.810","Text":"both relative to the lab."},{"Start":"01:21.810 ","End":"01:24.280","Text":"The same holds true for acceleration."},{"Start":"01:24.280 ","End":"01:28.370","Text":"We use the same principle of breaking things down based on dimension."},{"Start":"01:28.370 ","End":"01:30.620","Text":"We can also do this in terms of vectors."},{"Start":"01:30.620 ","End":"01:34.655","Text":"We can say the position vector or r. So r tag,"},{"Start":"01:34.655 ","End":"01:39.620","Text":"which would be the position vector of the car relative to the man, equals r,"},{"Start":"01:39.620 ","End":"01:42.260","Text":"the position vector of the car relative to"},{"Start":"01:42.260 ","End":"01:45.260","Text":"the lab minus the r of the man relative to the lab."},{"Start":"01:45.260 ","End":"01:48.050","Text":"Let\u0027s imagine this same scenario in a different way."},{"Start":"01:48.050 ","End":"01:52.310","Text":"We have our labs here and our man is"},{"Start":"01:52.310 ","End":"01:57.035","Text":"down here a little bit below the lab in terms of y obviously to the right in terms of x,"},{"Start":"01:57.035 ","End":"01:59.600","Text":"the car above the lab in terms of y,"},{"Start":"01:59.600 ","End":"02:01.060","Text":"and to the right in terms of x."},{"Start":"02:01.060 ","End":"02:04.925","Text":"This would be r of the car relative to the lab."},{"Start":"02:04.925 ","End":"02:09.545","Text":"That\u0027s r of the man relative to the lab and our relative vector,"},{"Start":"02:09.545 ","End":"02:11.810","Text":"our tag is there as you can see."},{"Start":"02:11.810 ","End":"02:15.590","Text":"You can see in terms of vectors that r tag does equal in fact,"},{"Start":"02:15.590 ","End":"02:17.795","Text":"the vector of the car relative to the lab"},{"Start":"02:17.795 ","End":"02:20.825","Text":"minus the vector of the man relative to the lab."},{"Start":"02:20.825 ","End":"02:24.530","Text":"You can approach this in whatever way is easiest for you,"},{"Start":"02:24.530 ","End":"02:27.950","Text":"whether that\u0027s in terms of vectors or in terms of x, y, and z."},{"Start":"02:27.950 ","End":"02:30.035","Text":"The same is true for velocity."},{"Start":"02:30.035 ","End":"02:33.650","Text":"The total velocity tag equals the total velocity of"},{"Start":"02:33.650 ","End":"02:39.410","Text":"the car relative to the lab minus the total velocity of the man relative to the lab."},{"Start":"02:39.410 ","End":"02:42.630","Text":"The same is true for acceleration as well of course."},{"Start":"02:42.630 ","End":"02:44.465","Text":"This is the end of the lecture."},{"Start":"02:44.465 ","End":"02:48.870","Text":"Now let\u0027s look at 1 example problem to put our theory into practice."}],"ID":9241}],"Thumbnail":null,"ID":5403},{"Name":"Exercises For Galilean Transforms","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"13m 48s","ChapterTopicVideoID":8949,"CourseChapterTopicPlaylistID":5404,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this example we have a bus traveling"},{"Start":"00:02.760 ","End":"00:06.450","Text":"50 meters per second to the right along the x-axis,"},{"Start":"00:06.450 ","End":"00:07.920","Text":"and the car traveling at"},{"Start":"00:07.920 ","End":"00:12.915","Text":"30 meters per second at an angle of 30 degrees relative to the x-axis."},{"Start":"00:12.915 ","End":"00:17.715","Text":"In Part A we\u0027re supposed to find the velocity of the car relative to the bus."},{"Start":"00:17.715 ","End":"00:20.370","Text":"Now, in a lot of these problems we\u0027re given"},{"Start":"00:20.370 ","End":"00:23.055","Text":"velocities and we have to understand intuitively that"},{"Start":"00:23.055 ","End":"00:28.710","Text":"these velocities are relative to the ground or the lab from our previous lecture."},{"Start":"00:28.710 ","End":"00:32.090","Text":"Now what we need to find is the relative velocity."},{"Start":"00:32.090 ","End":"00:34.550","Text":"We\u0027re going to call that V_2 tag."},{"Start":"00:34.550 ","End":"00:36.960","Text":"It\u0027s the relative velocity of the second object,"},{"Start":"00:36.960 ","End":"00:41.135","Text":"in our case the car, relative to the bus the first object."},{"Start":"00:41.135 ","End":"00:44.390","Text":"We can write that down and start filling in our equation."},{"Start":"00:44.390 ","End":"00:50.525","Text":"We know that the relative velocity of the car to the bus equals V_2,"},{"Start":"00:50.525 ","End":"00:55.135","Text":"which is the velocity of the car relative to the ground, minus V_1,"},{"Start":"00:55.135 ","End":"00:58.115","Text":"which is the velocity of the bus relative to the ground,"},{"Start":"00:58.115 ","End":"01:00.665","Text":"or again the lab from our former example."},{"Start":"01:00.665 ","End":"01:03.410","Text":"Now remember we\u0027re talking in terms of vectors here."},{"Start":"01:03.410 ","End":"01:06.560","Text":"If you really want to use these velocities,"},{"Start":"01:06.560 ","End":"01:09.140","Text":"we need to break them down into their different elements,"},{"Start":"01:09.140 ","End":"01:10.735","Text":"the x and the y element."},{"Start":"01:10.735 ","End":"01:12.465","Text":"Starting with the x element,"},{"Start":"01:12.465 ","End":"01:18.480","Text":"we have V_2 tag x equals V_2x, minus V_1x."},{"Start":"01:18.480 ","End":"01:22.940","Text":"That is the x velocity of the car relative to the bus,"},{"Start":"01:22.940 ","End":"01:25.985","Text":"equals the x velocity of the car relative to the ground"},{"Start":"01:25.985 ","End":"01:30.005","Text":"minus the x velocity of the bus relative to the ground."},{"Start":"01:30.005 ","End":"01:37.850","Text":"We can do the same thing for y. V_2 tag y equals V_2y minus V_1y."},{"Start":"01:37.850 ","End":"01:41.885","Text":"Again, this is velocity along the y axis."},{"Start":"01:41.885 ","End":"01:45.620","Text":"Now we can fill in some of our values and solve."},{"Start":"01:45.620 ","End":"01:48.470","Text":"The first thing we should do is start with the bus."},{"Start":"01:48.470 ","End":"01:54.770","Text":"The velocity of the bus relative to the ground along the x-axis is 50 meters per second,"},{"Start":"01:54.770 ","End":"01:56.745","Text":"and along the y-axis it\u0027s 0."},{"Start":"01:56.745 ","End":"01:59.265","Text":"Because it\u0027s going only along the x axis."},{"Start":"01:59.265 ","End":"02:01.525","Text":"As for the car V_2,"},{"Start":"02:01.525 ","End":"02:06.305","Text":"the x velocity is 30 times the cosine of 30 degrees,"},{"Start":"02:06.305 ","End":"02:10.690","Text":"which equals 30 times the square root of 3 over 2."},{"Start":"02:10.690 ","End":"02:18.815","Text":"V_2y is 30 times the sine of 30 degrees, which equals 15."},{"Start":"02:18.815 ","End":"02:22.190","Text":"Now I can take these values and plug them into"},{"Start":"02:22.190 ","End":"02:25.490","Text":"the original equation to solve for V_2 tag x,"},{"Start":"02:25.490 ","End":"02:29.090","Text":"that is the velocity of the car relative to the bus."},{"Start":"02:29.090 ","End":"02:35.175","Text":"V_2x equals 30 times square root of 3 over 2,"},{"Start":"02:35.175 ","End":"02:38.400","Text":"and V_1x is 50."},{"Start":"02:38.400 ","End":"02:40.475","Text":"As for the y element,"},{"Start":"02:40.475 ","End":"02:43.010","Text":"I have 15 minus 0."},{"Start":"02:43.010 ","End":"02:44.675","Text":"For the x element,"},{"Start":"02:44.675 ","End":"02:49.470","Text":"my solution is approximately negative 24.01,"},{"Start":"02:51.470 ","End":"02:54.975","Text":"and that\u0027s in meters per second,"},{"Start":"02:54.975 ","End":"02:59.750","Text":"and for my y element the solution is 15 meters per second."},{"Start":"02:59.750 ","End":"03:02.450","Text":"You have your x and your y components."},{"Start":"03:02.450 ","End":"03:05.635","Text":"You essentially solved the problem in Part A."},{"Start":"03:05.635 ","End":"03:11.600","Text":"In Part B we need to find the direction that the car is traveling relative to the bus."},{"Start":"03:11.600 ","End":"03:16.690","Text":"Now what we\u0027re really looking for is the direction of the vector V_2 tag."},{"Start":"03:16.690 ","End":"03:20.375","Text":"We know that relative to the lab at the ground the vector V2,"},{"Start":"03:20.375 ","End":"03:23.390","Text":"that is the velocity of the car relative to the ground,"},{"Start":"03:23.390 ","End":"03:25.490","Text":"is at an angle of 30 degrees."},{"Start":"03:25.490 ","End":"03:28.324","Text":"Interestingly enough, relative to the bus,"},{"Start":"03:28.324 ","End":"03:32.870","Text":"the vector V_2 tag does not come out to that same angle."},{"Start":"03:32.870 ","End":"03:35.045","Text":"I\u0027ll show you why here."},{"Start":"03:35.045 ","End":"03:39.853","Text":"The direction of a vector is defined by tangent of the angle Theta."},{"Start":"03:39.853 ","End":"03:41.720","Text":"In our case we\u0027ll put a tag here,"},{"Start":"03:41.720 ","End":"03:45.515","Text":"because we\u0027re talking about the relative angle between the car and the bus."},{"Start":"03:45.515 ","End":"03:48.650","Text":"Whenever we\u0027re talking about relativity we\u0027re going to use these tags."},{"Start":"03:48.650 ","End":"03:51.656","Text":"In that case, it\u0027s tangent Theta tag."},{"Start":"03:51.656 ","End":"03:56.230","Text":"A tangent equals V_y over V_x."},{"Start":"03:56.230 ","End":"03:59.525","Text":"Once again, we\u0027re talking in relative terms,"},{"Start":"03:59.525 ","End":"04:04.475","Text":"so actually be V_y tag over V_x tag."},{"Start":"04:04.475 ","End":"04:08.045","Text":"Notice because we\u0027re doing a relative calculation,"},{"Start":"04:08.045 ","End":"04:11.240","Text":"it doesn\u0027t mean we have to change how we do our calculations."},{"Start":"04:11.240 ","End":"04:14.480","Text":"Generally, this is the exact same calculation we do to find"},{"Start":"04:14.480 ","End":"04:19.180","Text":"the tangent of the velocity of the car relative to the lab at the ground."},{"Start":"04:19.180 ","End":"04:21.975","Text":"It\u0027s the same thing, here we just have tags everywhere."},{"Start":"04:21.975 ","End":"04:24.900","Text":"We can fill things in from our answer before,"},{"Start":"04:24.900 ","End":"04:27.403","Text":"and we know that the V_y,"},{"Start":"04:27.403 ","End":"04:30.780","Text":"velocity on the y-axis is 15 meters per second,"},{"Start":"04:30.780 ","End":"04:36.995","Text":"and on the x-axis we have negative 24.01 meters per second."},{"Start":"04:36.995 ","End":"04:39.080","Text":"If you put that in a calculator,"},{"Start":"04:39.080 ","End":"04:46.410","Text":"you\u0027ll find your answer is approximately negative 0.625."},{"Start":"04:47.020 ","End":"04:49.803","Text":"In order to take this value,"},{"Start":"04:49.803 ","End":"04:55.865","Text":"negative 0.625 and find the angle itself of Theta tag,"},{"Start":"04:55.865 ","End":"04:57.935","Text":"we need to do the inverse tangent,"},{"Start":"04:57.935 ","End":"05:01.460","Text":"or on your computer or calculator Shift Tan."},{"Start":"05:01.460 ","End":"05:06.560","Text":"We take the inverse tangent of negative 0.625,"},{"Start":"05:06.560 ","End":"05:10.535","Text":"and that will equal the angle Theta tag."},{"Start":"05:10.535 ","End":"05:12.575","Text":"When you put that in your calculator,"},{"Start":"05:12.575 ","End":"05:17.960","Text":"the result is approximately negative 32 degrees."},{"Start":"05:17.960 ","End":"05:20.620","Text":"Now this is a slightly weird answer,"},{"Start":"05:20.620 ","End":"05:24.860","Text":"and if you recall on your calculator when you put in Shift Tan function,"},{"Start":"05:24.860 ","End":"05:26.210","Text":"or the inverse tangent,"},{"Start":"05:26.210 ","End":"05:28.685","Text":"it\u0027ll give you 1 of 2 possible answers."},{"Start":"05:28.685 ","End":"05:31.400","Text":"Your calculator doesn\u0027t know which of the 2 you\u0027re looking for,"},{"Start":"05:31.400 ","End":"05:35.855","Text":"so it always gives you the answer between negative 90 and positive 90."},{"Start":"05:35.855 ","End":"05:38.215","Text":"We\u0027ll call this answer Theta_1,"},{"Start":"05:38.215 ","End":"05:40.875","Text":"and Theta_2 the alternative answer,"},{"Start":"05:40.875 ","End":"05:45.195","Text":"equals a 180 degrees plus your initial answer."},{"Start":"05:45.195 ","End":"05:49.215","Text":"In our case, a 180 plus Theta 1,"},{"Start":"05:49.215 ","End":"05:52.380","Text":"which is your first answer negative 32,"},{"Start":"05:52.380 ","End":"05:57.255","Text":"equals a 148 degrees."},{"Start":"05:57.255 ","End":"06:02.550","Text":"A 180 plus negative 32 equals a 148 degrees."},{"Start":"06:02.550 ","End":"06:05.435","Text":"Either of these answers could be correct."},{"Start":"06:05.435 ","End":"06:10.310","Text":"Before moving on and figuring out whether to use Theta_1 or Theta_2,"},{"Start":"06:10.310 ","End":"06:15.830","Text":"I want to give a little bit of mathematical background to explain what\u0027s going on here."},{"Start":"06:15.830 ","End":"06:20.375","Text":"If we have some coordinates here x and y, we take Theta_1."},{"Start":"06:20.375 ","End":"06:23.120","Text":"That\u0027s going to give us some vector in"},{"Start":"06:23.120 ","End":"06:27.125","Text":"the fourth quadrant at an angle of negative 32 degrees,"},{"Start":"06:27.125 ","End":"06:29.660","Text":"and we know in that quadrant that"},{"Start":"06:29.660 ","End":"06:33.260","Text":"the x value will be positive and the y value will be negative."},{"Start":"06:33.260 ","End":"06:38.060","Text":"Now if we take the exact opposite vector in the second quadrant,"},{"Start":"06:38.060 ","End":"06:40.310","Text":"it will have a positive y value,"},{"Start":"06:40.310 ","End":"06:42.140","Text":"and a negative x value."},{"Start":"06:42.140 ","End":"06:49.190","Text":"Whereas the first vector will have a value of 24 for x,"},{"Start":"06:49.190 ","End":"06:53.856","Text":"and negative 15 for y, approximately of course."},{"Start":"06:53.856 ","End":"06:57.380","Text":"Then the second one we\u0027ll have negative 24 for x,"},{"Start":"06:57.380 ","End":"06:59.255","Text":"and 15 for y."},{"Start":"06:59.255 ","End":"07:03.880","Text":"These are vectors of the same length and have exact opposite angles."},{"Start":"07:03.880 ","End":"07:06.145","Text":"Your computer or your calculator,"},{"Start":"07:06.145 ","End":"07:10.580","Text":"doesn\u0027t know which you\u0027re trying to do because the result of 15 over negative"},{"Start":"07:10.580 ","End":"07:15.895","Text":"24 is the same as the result of negative 15 over 24."},{"Start":"07:15.895 ","End":"07:19.760","Text":"To make this easy, the calculator always gives you a result in"},{"Start":"07:19.760 ","End":"07:23.840","Text":"the first or fourth quadrants between negative 90 and 90 degrees."},{"Start":"07:23.840 ","End":"07:28.535","Text":"You need to figure out for yourself which of these two solutions is correct for you."},{"Start":"07:28.535 ","End":"07:30.020","Text":"Now in our case,"},{"Start":"07:30.020 ","End":"07:35.435","Text":"we can see that our vector has length of 15 and negative 24."},{"Start":"07:35.435 ","End":"07:37.895","Text":"That mean that it\u0027s the vector in the second quadrant."},{"Start":"07:37.895 ","End":"07:41.820","Text":"The angle will choose is a 148 degrees."},{"Start":"07:42.350 ","End":"07:46.325","Text":"Our procedure here is not so complicated."},{"Start":"07:46.325 ","End":"07:49.130","Text":"First, we want to find our answer and do Shift Tan,"},{"Start":"07:49.130 ","End":"07:50.500","Text":"the inverse tangent,"},{"Start":"07:50.500 ","End":"07:55.640","Text":"and we notice based on the y and the x element which quadrant we should be looking at."},{"Start":"07:55.640 ","End":"07:57.895","Text":"From there we can find our correct answer."},{"Start":"07:57.895 ","End":"08:02.690","Text":"Now this angle, a 148 degrees sounds a little weird because it\u0027s"},{"Start":"08:02.690 ","End":"08:07.880","Text":"totally opposite from 30 degree angle that our car has relative to the ground."},{"Start":"08:07.880 ","End":"08:09.890","Text":"Our bus sees the car going in"},{"Start":"08:09.890 ","End":"08:14.680","Text":"a totally different direction than the car is going relative to the ground."},{"Start":"08:14.680 ","End":"08:17.565","Text":"This solution isn\u0027t very intuitive."},{"Start":"08:17.565 ","End":"08:19.655","Text":"I think it\u0027s worth a deeper explanation."},{"Start":"08:19.655 ","End":"08:24.320","Text":"The first thing that can be confusing is we expect the car to move forward,"},{"Start":"08:24.320 ","End":"08:27.469","Text":"to move, so to speak, in the direction of its headlights."},{"Start":"08:27.469 ","End":"08:30.785","Text":"But that\u0027s not really the case relative to the bus."},{"Start":"08:30.785 ","End":"08:32.060","Text":"Now relative to the ground,"},{"Start":"08:32.060 ","End":"08:33.260","Text":"relative to the origin,"},{"Start":"08:33.260 ","End":"08:35.405","Text":"relative to the lab, that is the case."},{"Start":"08:35.405 ","End":"08:37.850","Text":"But the bus will actually see the car with"},{"Start":"08:37.850 ","End":"08:40.155","Text":"its headlights pointing forward the same direction,"},{"Start":"08:40.155 ","End":"08:44.165","Text":"but the car will move in a totally different direction from the headlights."},{"Start":"08:44.165 ","End":"08:50.405","Text":"We can look at an example here using the origin and putting some coordinates down here,"},{"Start":"08:50.405 ","End":"08:52.615","Text":"and maybe that\u0027ll make it more clear."},{"Start":"08:52.615 ","End":"08:55.580","Text":"Let\u0027s assume that we have an origin."},{"Start":"08:55.580 ","End":"08:57.935","Text":"This is where the lab is situated."},{"Start":"08:57.935 ","End":"09:00.620","Text":"We can write it over here on the side, we\u0027ll come back to this."},{"Start":"09:00.620 ","End":"09:03.605","Text":"For now let\u0027s assume it\u0027s at the same point as the bus."},{"Start":"09:03.605 ","End":"09:07.685","Text":"This is our origin at the beginning of the problem at t equals 0."},{"Start":"09:07.685 ","End":"09:12.020","Text":"The car is at the exact point that the car is at now at t equals 0."},{"Start":"09:12.020 ","End":"09:14.060","Text":"However at some point t,"},{"Start":"09:14.060 ","End":"09:16.115","Text":"the car has moved forward."},{"Start":"09:16.115 ","End":"09:18.050","Text":"We see that relative to the origin,"},{"Start":"09:18.050 ","End":"09:19.430","Text":"relative to the lab,"},{"Start":"09:19.430 ","End":"09:22.135","Text":"the car is moving at an angle of 30 degrees."},{"Start":"09:22.135 ","End":"09:24.485","Text":"Let\u0027s assume we have a second origin,"},{"Start":"09:24.485 ","End":"09:25.895","Text":"a second set of axis,"},{"Start":"09:25.895 ","End":"09:27.650","Text":"and it\u0027s on the same point with the bus,"},{"Start":"09:27.650 ","End":"09:30.440","Text":"but it\u0027s relative to the bus. We\u0027ll do this in blue."},{"Start":"09:30.440 ","End":"09:32.180","Text":"I\u0027m making 2 points here,"},{"Start":"09:32.180 ","End":"09:34.010","Text":"they\u0027re really in the same spot,"},{"Start":"09:34.010 ","End":"09:35.030","Text":"but I want to make 2 points."},{"Start":"09:35.030 ","End":"09:37.895","Text":"You can see we\u0027re talking about 2 different observers."},{"Start":"09:37.895 ","End":"09:41.975","Text":"At t=0 they see the car in the exact same spot."},{"Start":"09:41.975 ","End":"09:45.695","Text":"But as time goes forward and we reach t,"},{"Start":"09:45.695 ","End":"09:50.700","Text":"the bus will actually see the car in a totally different location."},{"Start":"09:51.080 ","End":"09:56.640","Text":"It\u0027s not that the car has moved 2 places at once rather that the bus has moved."},{"Start":"09:56.640 ","End":"09:59.585","Text":"It observes the same point from a different perspective."},{"Start":"09:59.585 ","End":"10:03.995","Text":"If we go back to the origin and coordinates that we drew earlier,"},{"Start":"10:03.995 ","End":"10:09.215","Text":"we can imagine the car at t equals 0 and t random time after."},{"Start":"10:09.215 ","End":"10:11.060","Text":"Now in the blue coordinates,"},{"Start":"10:11.060 ","End":"10:12.425","Text":"this is relative to the bus."},{"Start":"10:12.425 ","End":"10:14.255","Text":"t=0 is the same."},{"Start":"10:14.255 ","End":"10:16.385","Text":"But once we reach the time t,"},{"Start":"10:16.385 ","End":"10:21.050","Text":"the relative or origin and coordinates of the bus has moved forward."},{"Start":"10:21.050 ","End":"10:25.175","Text":"Relative to the original point and relative to the original origin,"},{"Start":"10:25.175 ","End":"10:27.125","Text":"it sees the car at a different spot."},{"Start":"10:27.125 ","End":"10:29.630","Text":"This x value for time equals t for"},{"Start":"10:29.630 ","End":"10:33.695","Text":"the car is going to have a different x value than that for the lab."},{"Start":"10:33.695 ","End":"10:37.600","Text":"In fact, it\u0027ll be close to the original time in this example,"},{"Start":"10:37.600 ","End":"10:39.375","Text":"it\u0027ll look something like that."},{"Start":"10:39.375 ","End":"10:43.340","Text":"That is the trajectory that the bus sees for the car."},{"Start":"10:43.340 ","End":"10:46.430","Text":"Because the relative velocity is slower,"},{"Start":"10:46.430 ","End":"10:50.465","Text":"we see a much greater angle relative to the x-axis."},{"Start":"10:50.465 ","End":"10:51.930","Text":"Instead of 30 degrees here,"},{"Start":"10:51.930 ","End":"10:55.175","Text":"we get in this example 50 degrees, something like that,"},{"Start":"10:55.175 ","End":"10:57.740","Text":"and in the example from a problem we actually"},{"Start":"10:57.740 ","End":"11:01.085","Text":"see because the bus is going faster than the car,"},{"Start":"11:01.085 ","End":"11:03.620","Text":"that instead of just having a larger angle,"},{"Start":"11:03.620 ","End":"11:05.465","Text":"the angle is larger than 90."},{"Start":"11:05.465 ","End":"11:08.274","Text":"Meaning that for us instead of that point,"},{"Start":"11:08.274 ","End":"11:09.780","Text":"we\u0027d actually see this point."},{"Start":"11:09.780 ","End":"11:11.720","Text":"Where it looks like the car is going in"},{"Start":"11:11.720 ","End":"11:13.760","Text":"a totally different direction than its headlights."},{"Start":"11:13.760 ","End":"11:18.170","Text":"In our case with an angle of a 148 degrees,"},{"Start":"11:18.170 ","End":"11:25.880","Text":"what the bus observes is not the car going at a 30 degree angle relative to the x-axis,"},{"Start":"11:25.880 ","End":"11:28.524","Text":"rather it sees the car with its headlights"},{"Start":"11:28.524 ","End":"11:31.585","Text":"pointed at a 30 degree angle relative to the x-axis,"},{"Start":"11:31.585 ","End":"11:34.480","Text":"going into the totally opposite direction."},{"Start":"11:34.480 ","End":"11:39.115","Text":"I want to show you a short animation that may make things easier to understand."},{"Start":"11:39.115 ","End":"11:42.205","Text":"We\u0027ll clear the screen, and we have just the bus and the car"},{"Start":"11:42.205 ","End":"11:45.785","Text":"moving as at first we see from the lab."},{"Start":"11:45.785 ","End":"11:50.440","Text":"This is what you see. The bus going straight along the x-axis,"},{"Start":"11:50.440 ","End":"11:52.645","Text":"the car going at a 30 degree angle."},{"Start":"11:52.645 ","End":"11:56.590","Text":"Now if we\u0027re looking from the buses perspective, it\u0027s going to be different."},{"Start":"11:56.590 ","End":"12:01.300","Text":"We may naturally assume that we look at things from the labs perspective,"},{"Start":"12:01.300 ","End":"12:04.420","Text":"but we need to remember 2 things when we\u0027re looking from the buses perspective."},{"Start":"12:04.420 ","End":"12:06.310","Text":"First, the bus doesn\u0027t move."},{"Start":"12:06.310 ","End":"12:07.570","Text":"When you\u0027re inside the bus,"},{"Start":"12:07.570 ","End":"12:10.235","Text":"it looks like other things are moving relative to you,"},{"Start":"12:10.235 ","End":"12:11.525","Text":"but you\u0027re not moving."},{"Start":"12:11.525 ","End":"12:16.325","Text":"The second thing is the car will move at a 148 degrees relative to the bus."},{"Start":"12:16.325 ","End":"12:18.720","Text":"What you\u0027ll see is this."},{"Start":"12:19.090 ","End":"12:21.380","Text":"I hope this helped."},{"Start":"12:21.380 ","End":"12:22.580","Text":"If you\u0027re still confused,"},{"Start":"12:22.580 ","End":"12:26.180","Text":"the best thing you can do is rely on the mathematical explanation."},{"Start":"12:26.180 ","End":"12:30.135","Text":"Remember, we\u0027re looking for the tangent of the angle Theta tag."},{"Start":"12:30.135 ","End":"12:36.325","Text":"We can use our relative velocity and take the velocity of y over the velocity of x."},{"Start":"12:36.325 ","End":"12:39.260","Text":"As long as you deal with all your tagged properties,"},{"Start":"12:39.260 ","End":"12:43.190","Text":"the relative velocity, the relative everything, you\u0027ll be okay."},{"Start":"12:43.190 ","End":"12:46.145","Text":"Now another thing to remember is this was"},{"Start":"12:46.145 ","End":"12:50.105","Text":"the relative velocity of the car vis-a-vis the bus."},{"Start":"12:50.105 ","End":"12:52.120","Text":"If we were looking for the relative position,"},{"Start":"12:52.120 ","End":"12:54.125","Text":"we would do a slightly different operation."},{"Start":"12:54.125 ","End":"12:58.400","Text":"We\u0027re looking for this vector as opposed to the vector of the car,"},{"Start":"12:58.400 ","End":"13:00.515","Text":"and the relationship between the 2."},{"Start":"13:00.515 ","End":"13:06.290","Text":"Now instead of taking the tangent of the velocity y over the velocity x,"},{"Start":"13:06.290 ","End":"13:07.775","Text":"we could do the same operation."},{"Start":"13:07.775 ","End":"13:09.245","Text":"We\u0027re still finding a tangent,"},{"Start":"13:09.245 ","End":"13:12.290","Text":"but instead of the velocity y over the velocity x,"},{"Start":"13:12.290 ","End":"13:17.760","Text":"we take position y. Y tag over position x, x tag."},{"Start":"13:17.840 ","End":"13:21.268","Text":"Again, for position we\u0027re using y and x."},{"Start":"13:21.268 ","End":"13:23.570","Text":"For velocity V_y and V_x,"},{"Start":"13:23.570 ","End":"13:26.615","Text":"and of course for acceleration we could do A_y and A_x."},{"Start":"13:26.615 ","End":"13:31.400","Text":"As long as we\u0027re dealing with our relative attributes,"},{"Start":"13:31.400 ","End":"13:34.230","Text":"we\u0027re finding the relative angle."},{"Start":"13:34.230 ","End":"13:38.280","Text":"We need to find again the elements of the velocity for example,"},{"Start":"13:38.280 ","End":"13:39.660","Text":"but the relative velocity."},{"Start":"13:39.660 ","End":"13:42.110","Text":"The velocity of the car relative to the bus,"},{"Start":"13:42.110 ","End":"13:44.020","Text":"not relative to the ground."},{"Start":"13:44.020 ","End":"13:47.530","Text":"That\u0027s the end of the lecture, thank you for listening."}],"ID":9242},{"Watched":false,"Name":"Exercise 2","Duration":"14m 52s","ChapterTopicVideoID":10432,"CourseChapterTopicPlaylistID":5404,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.615","Text":"A driver is driving in his car at a velocity of 100 kilometers per hour and sees"},{"Start":"00:06.615 ","End":"00:10.170","Text":"raindrops running down the windowpane in the opposite direction to"},{"Start":"00:10.170 ","End":"00:15.120","Text":"the car\u0027s motion at an angle of 45 degrees."},{"Start":"00:15.120 ","End":"00:23.130","Text":"What does that mean? If we take our axis to be this and call this the y-axis,"},{"Start":"00:23.130 ","End":"00:30.843","Text":"so we can say that the raindrops are falling like so on the window,"},{"Start":"00:30.843 ","End":"00:37.450","Text":"and this angle over here is 45 degrees."},{"Start":"00:37.450 ","End":"00:42.410","Text":"Another driver is driving at 70 kilometers per hour and sees"},{"Start":"00:42.410 ","End":"00:47.905","Text":"the raindrops running down his windowpane at an angle of 30 degrees."},{"Start":"00:47.905 ","End":"00:52.040","Text":"That\u0027s in another car. We\u0027re being asked to find the velocity"},{"Start":"00:52.040 ","End":"00:56.430","Text":"of the raindrops relative to the ground."},{"Start":"00:56.560 ","End":"01:01.880","Text":"This is the axis over here perpendicular to"},{"Start":"01:01.880 ","End":"01:07.970","Text":"the car where we measure how the rain is striking the windowpane,"},{"Start":"01:07.970 ","End":"01:13.560","Text":"and the direction of motion of the car is in the x direction."},{"Start":"01:13.690 ","End":"01:19.520","Text":"What we\u0027re going to do is we\u0027re going to rub out this 45 degrees,"},{"Start":"01:19.520 ","End":"01:23.750","Text":"and we\u0027re going to say that this angle over here is Theta."},{"Start":"01:23.750 ","End":"01:28.775","Text":"At the beginning, we\u0027re going to say that Theta is equal to"},{"Start":"01:28.775 ","End":"01:32.360","Text":"45 degrees because this way I can solve"},{"Start":"01:32.360 ","End":"01:36.890","Text":"the question for the 2 cars just using this variable Theta,"},{"Start":"01:36.890 ","End":"01:38.165","Text":"and then right at the end,"},{"Start":"01:38.165 ","End":"01:39.995","Text":"I can substitute in the angle"},{"Start":"01:39.995 ","End":"01:45.140","Text":"45 degrees and 30 degrees and not have to go through the calculation again."},{"Start":"01:45.140 ","End":"01:49.160","Text":"Now, the next thing that we\u0027re going to do is we\u0027re"},{"Start":"01:49.160 ","End":"01:54.245","Text":"going to mark this Theta as Theta tag instead of just Theta."},{"Start":"01:54.245 ","End":"01:57.830","Text":"Why is that? We\u0027re being told that the rain is"},{"Start":"01:57.830 ","End":"02:01.505","Text":"striking the car at this angle, Theta tag,"},{"Start":"02:01.505 ","End":"02:04.751","Text":"relative to the driver,"},{"Start":"02:04.751 ","End":"02:08.180","Text":"so it\u0027s relative to some moving entity,"},{"Start":"02:08.180 ","End":"02:10.010","Text":"and it\u0027s not relative to the ground."},{"Start":"02:10.010 ","End":"02:11.630","Text":"So whenever we have an angle,"},{"Start":"02:11.630 ","End":"02:13.250","Text":"or velocity, or position,"},{"Start":"02:13.250 ","End":"02:17.345","Text":"or something relative to some moving body,"},{"Start":"02:17.345 ","End":"02:20.430","Text":"we always have to include a tag."},{"Start":"02:21.340 ","End":"02:27.560","Text":"Now what we\u0027re going to do is we\u0027re going to define this vector,"},{"Start":"02:27.560 ","End":"02:35.470","Text":"u, which is going to be the velocity of the raindrops relative to the driver."},{"Start":"02:35.470 ","End":"02:41.085","Text":"We\u0027ll have the velocity of the raindrops relative to the driver in the x direction"},{"Start":"02:41.085 ","End":"02:47.460","Text":"and the velocity of the raindrops relative to the driver in the y direction."},{"Start":"02:48.650 ","End":"02:53.510","Text":"Now we\u0027re going to have some angle, Theta,"},{"Start":"02:53.510 ","End":"02:59.585","Text":"which is the angle between the raindrops and relative to the ground,"},{"Start":"02:59.585 ","End":"03:01.130","Text":"so this we don\u0027t know,"},{"Start":"03:01.130 ","End":"03:05.675","Text":"and because it\u0027s relative to the ground which is stationary, so there\u0027s no tag."},{"Start":"03:05.675 ","End":"03:10.850","Text":"Then we\u0027re going to define the velocity of the raindrops relative to the ground,"},{"Start":"03:10.850 ","End":"03:12.875","Text":"so relative to something not moving."},{"Start":"03:12.875 ","End":"03:14.360","Text":"Again, we don\u0027t have a tag,"},{"Start":"03:14.360 ","End":"03:18.350","Text":"and that\u0027s going to be the velocity of the raindrops relative to the ground in"},{"Start":"03:18.350 ","End":"03:20.270","Text":"the x direction and the velocity of"},{"Start":"03:20.270 ","End":"03:23.945","Text":"the raindrops relative to the ground in the y direction."},{"Start":"03:23.945 ","End":"03:31.415","Text":"Finally, let\u0027s say that the velocity of the car is equal to V_0,"},{"Start":"03:31.415 ","End":"03:33.575","Text":"which has also a variable,"},{"Start":"03:33.575 ","End":"03:37.820","Text":"which once we\u0027re going to say it\u0027s equal to 100 kilometers per hour,"},{"Start":"03:37.820 ","End":"03:43.620","Text":"and next, we\u0027re going to say that it\u0027s equal to 70 kilometers per hour."},{"Start":"03:44.840 ","End":"03:53.130","Text":"Now, let\u0027s begin with the driver who is driving at 100 kilometers per hour."},{"Start":"03:53.990 ","End":"03:58.835","Text":"We can say that the velocity of the raindrops"},{"Start":"03:58.835 ","End":"04:03.605","Text":"relative to the driver is going to be equal to"},{"Start":"04:03.605 ","End":"04:13.830","Text":"the velocity of the raindrops relative to the ground minus the velocity of the car."},{"Start":"04:14.650 ","End":"04:19.573","Text":"Then the same thing with u_y tag,"},{"Start":"04:19.573 ","End":"04:23.240","Text":"so the velocity of the raindrops in the y direction relative to"},{"Start":"04:23.240 ","End":"04:27.680","Text":"the car or relative to the driver is going to equal the velocity"},{"Start":"04:27.680 ","End":"04:32.570","Text":"of the raindrops relative to the ground minus the velocity in"},{"Start":"04:32.570 ","End":"04:38.795","Text":"the y direction minus the velocity of the car of the driver in the y direction."},{"Start":"04:38.795 ","End":"04:43.010","Text":"We\u0027re being told that our car is just moving in this straight line in the x direction,"},{"Start":"04:43.010 ","End":"04:46.110","Text":"which means that there is no y component."},{"Start":"04:46.110 ","End":"04:49.524","Text":"This is an unknown,"},{"Start":"04:49.524 ","End":"04:52.210","Text":"this is an unknown, this is an unknown,"},{"Start":"04:52.210 ","End":"04:55.060","Text":"this is an unknown, but this we know."},{"Start":"04:55.060 ","End":"04:56.470","Text":"This here, specifically,"},{"Start":"04:56.470 ","End":"04:59.100","Text":"is going to be 100."},{"Start":"04:59.100 ","End":"05:01.525","Text":"So we need another equation."},{"Start":"05:01.525 ","End":"05:12.250","Text":"We can see over here that this vector over here is going to be our vector, u tag."},{"Start":"05:12.250 ","End":"05:16.540","Text":"What we want to do, we know what our angle Theta tag is equal to,"},{"Start":"05:16.540 ","End":"05:21.880","Text":"so we can write an equation involving our Theta tag,"},{"Start":"05:21.880 ","End":"05:23.395","Text":"and our u_x tag,"},{"Start":"05:23.395 ","End":"05:25.640","Text":"and our u_y tag."},{"Start":"05:25.970 ","End":"05:28.620","Text":"Let\u0027s just draw this out."},{"Start":"05:28.620 ","End":"05:31.950","Text":"Now, here is our axis,"},{"Start":"05:31.950 ","End":"05:38.585","Text":"and here we have our vector, Theta tag."},{"Start":"05:38.585 ","End":"05:42.530","Text":"Now usually what we do is we have some vector,"},{"Start":"05:42.530 ","End":"05:44.900","Text":"and we say that this is our angle over here,"},{"Start":"05:44.900 ","End":"05:53.940","Text":"Alpha or whatever, and then we\u0027ll get the tangent of Alpha over here."},{"Start":"05:53.940 ","End":"05:54.810","Text":"It"},{"Start":"05:54.810 ","End":"06:05.209","Text":"will"},{"Start":"06:05.209 ","End":"06:05.880","Text":"be equal to,"},{"Start":"06:05.880 ","End":"06:11.410","Text":"actually, our y component divided by our x component."},{"Start":"06:11.410 ","End":"06:19.030","Text":"However, because we are given our angle relative to our y-axis,"},{"Start":"06:19.030 ","End":"06:21.760","Text":"it\u0027s relative to this arrow over here,"},{"Start":"06:21.760 ","End":"06:28.900","Text":"so that means that our tan of Theta tag will just be the opposite of this."},{"Start":"06:28.900 ","End":"06:30.235","Text":"So it will be here,"},{"Start":"06:30.235 ","End":"06:34.855","Text":"our x component divided by our y component."},{"Start":"06:34.855 ","End":"06:40.465","Text":"Let\u0027s just rub this out so that we don\u0027t get confused."},{"Start":"06:40.465 ","End":"06:42.400","Text":"We can scroll down a little,"},{"Start":"06:42.400 ","End":"06:47.365","Text":"and then we can say that tan of this angle,"},{"Start":"06:47.365 ","End":"06:51.070","Text":"Theta tag, because it\u0027s relative to our y-axis,"},{"Start":"06:51.070 ","End":"06:54.725","Text":"is going to be equal to our x component,"},{"Start":"06:54.725 ","End":"07:00.185","Text":"which is u_x tag divided by our y component,"},{"Start":"07:00.185 ","End":"07:05.089","Text":"which is u_y tag."},{"Start":"07:05.089 ","End":"07:10.255","Text":"Now we can substitute in our values for u_x tag,"},{"Start":"07:10.255 ","End":"07:11.905","Text":"which is equal to v_x,"},{"Start":"07:11.905 ","End":"07:14.845","Text":"which we don\u0027t know, minus v_0,"},{"Start":"07:14.845 ","End":"07:16.855","Text":"which we do know,"},{"Start":"07:16.855 ","End":"07:19.375","Text":"divided by v_y,"},{"Start":"07:19.375 ","End":"07:21.175","Text":"which we also don\u0027t know."},{"Start":"07:21.175 ","End":"07:24.880","Text":"Now what we can see is tan of Theta tag,"},{"Start":"07:24.880 ","End":"07:28.113","Text":"I know what that is because I know what my Theta tag is equal to,"},{"Start":"07:28.113 ","End":"07:31.210","Text":"my v_0, I know what my v_0 is equal to,"},{"Start":"07:31.210 ","End":"07:35.875","Text":"and then I just have my v_x and v_y, which are unknowns."},{"Start":"07:35.875 ","End":"07:38.125","Text":"That means that I have 2 unknowns,"},{"Start":"07:38.125 ","End":"07:41.440","Text":"which means that I need 2 equations in order to solve them."},{"Start":"07:41.440 ","End":"07:44.470","Text":"What I can do is I can move up here."},{"Start":"07:44.470 ","End":"07:49.179","Text":"In the first time, I can say that my Theta tag is equal to 45 degrees."},{"Start":"07:49.179 ","End":"07:56.020","Text":"I can say that tan of 45 degrees is equal to my v_x,"},{"Start":"07:56.020 ","End":"07:57.595","Text":"which is an unknown,"},{"Start":"07:57.595 ","End":"07:59.275","Text":"minus my v_0,"},{"Start":"07:59.275 ","End":"08:01.840","Text":"which corresponds to the 45 degrees."},{"Start":"08:01.840 ","End":"08:05.170","Text":"So here I said that I have an angle of 45 degrees when my driver is"},{"Start":"08:05.170 ","End":"08:09.006","Text":"at 100 kilometers per hour,"},{"Start":"08:09.006 ","End":"08:12.625","Text":"and then divided by v_y, which is my unknown."},{"Start":"08:12.625 ","End":"08:18.625","Text":"Then I can say that tan of my other car,"},{"Start":"08:18.625 ","End":"08:20.680","Text":"so the angle of the rain on the other car,"},{"Start":"08:20.680 ","End":"08:22.990","Text":"so that\u0027s tan of 30 degrees,"},{"Start":"08:22.990 ","End":"08:25.930","Text":"is equal to my v_x, which I don\u0027t know,"},{"Start":"08:25.930 ","End":"08:30.310","Text":"minus the velocity of the car corresponding to 30 degrees,"},{"Start":"08:30.310 ","End":"08:32.305","Text":"which here is 70,"},{"Start":"08:32.305 ","End":"08:34.885","Text":"and then divided by v_y,"},{"Start":"08:34.885 ","End":"08:37.640","Text":"which is my second unknown."},{"Start":"08:38.910 ","End":"08:41.260","Text":"Now let\u0027s solve this."},{"Start":"08:41.260 ","End":"08:44.290","Text":"I\u0027m going to move to the side over here."},{"Start":"08:44.290 ","End":"08:48.936","Text":"What I\u0027m going to do is I\u0027m going to try and get rid of my v_y right now,"},{"Start":"08:48.936 ","End":"08:51.535","Text":"and then I\u0027m just going to solve for my v_x."},{"Start":"08:51.535 ","End":"08:52.765","Text":"Then afterwards at the end,"},{"Start":"08:52.765 ","End":"08:56.440","Text":"I\u0027ll substitute in my v_x and find out what my v_y is equal to."},{"Start":"08:56.440 ","End":"09:00.770","Text":"So what I\u0027m going to do is I\u0027m going to divide the 2 equations."},{"Start":"09:01.140 ","End":"09:09.985","Text":"So I\u0027ll have tan of 45 degrees divided by tan of 30 degrees."},{"Start":"09:09.985 ","End":"09:13.082","Text":"It\u0027s going to be equal to v_x"},{"Start":"09:13.082 ","End":"09:20.220","Text":"minus 100 divided v_x minus 70,"},{"Start":"09:20.220 ","End":"09:22.370","Text":"and then our v_y\u0027s cancel out."},{"Start":"09:22.370 ","End":"09:24.525","Text":"So now what I\u0027m going to do is I\u0027m going to do"},{"Start":"09:24.525 ","End":"09:29.325","Text":"cross multiplication to get rid of these fractions."},{"Start":"09:29.325 ","End":"09:35.620","Text":"So I\u0027ll have v_x minus 70 multiplied by tan"},{"Start":"09:36.330 ","End":"09:40.859","Text":"of 45 degrees will be equal to v_x minus 100"},{"Start":"09:40.859 ","End":"09:50.935","Text":"multiplied by tan of 30 degrees."},{"Start":"09:50.935 ","End":"09:54.580","Text":"Then what I want to do is I want to isolate out my v_x,"},{"Start":"09:54.580 ","End":"09:56.530","Text":"so I\u0027m going to open up the brackets."},{"Start":"09:56.530 ","End":"10:06.183","Text":"So I\u0027ll get v_x multiplied by tan of 45 degrees minus"},{"Start":"10:06.183 ","End":"10:09.610","Text":"70 tan of 45 degrees is going to be equal to v_x multiplied by"},{"Start":"10:09.610 ","End":"10:18.655","Text":"tan of 30 degrees"},{"Start":"10:18.655 ","End":"10:25.630","Text":"minus 100 times tan of 30 degrees."},{"Start":"10:25.630 ","End":"10:32.200","Text":"Then I\u0027m going to try and get my v_x on the same side of the equal sign."},{"Start":"10:32.200 ","End":"10:36.160","Text":"So I\u0027m just going to move over this v_x over here and move"},{"Start":"10:36.160 ","End":"10:41.335","Text":"this 70 tan of 45 degrees to this side of the equation,"},{"Start":"10:41.335 ","End":"10:43.570","Text":"and then I\u0027m going to put my like terms together,"},{"Start":"10:43.570 ","End":"10:52.480","Text":"so I\u0027ll get v_x multiplied by tan of 45 degrees minus v_x,"},{"Start":"10:52.480 ","End":"10:54.205","Text":"which is a common denominator,"},{"Start":"10:54.205 ","End":"11:00.700","Text":"tan of 30 degrees is going to be equal to"},{"Start":"11:00.700 ","End":"11:09.490","Text":"70 tan of 45 degrees minus"},{"Start":"11:09.490 ","End":"11:15.530","Text":"100 tan of 30 degrees."},{"Start":"11:16.080 ","End":"11:24.805","Text":"My final stage to isolate out my v_x is to divide both sides by these brackets over here,"},{"Start":"11:24.805 ","End":"11:28.285","Text":"tan of 45 minus tan of 30 degrees."},{"Start":"11:28.285 ","End":"11:33.610","Text":"So I\u0027ll finally get that my v_x is equal to,"},{"Start":"11:33.610 ","End":"11:38.424","Text":"so I\u0027ll just plug in what 70 tan of 45 is,"},{"Start":"11:38.424 ","End":"11:41.130","Text":"so tan of 45 degrees is equal to 1,"},{"Start":"11:41.130 ","End":"11:49.540","Text":"so 70 times 1 is 70 minus 100 tan of 30 degrees is approximately 57.73."},{"Start":"11:50.040 ","End":"11:53.695","Text":"You can just plug that into your calculator, and you\u0027ll see that,"},{"Start":"11:53.695 ","End":"12:03.290","Text":"divided by tan of 45 degrees is 1 minus tan of 30 degrees is approximately 0.58."},{"Start":"12:03.620 ","End":"12:06.180","Text":"Then I\u0027ll get that my answer,"},{"Start":"12:06.180 ","End":"12:08.190","Text":"once we plug this into the calculator,"},{"Start":"12:08.190 ","End":"12:11.790","Text":"is equal to 29.21."},{"Start":"12:11.790 ","End":"12:14.025","Text":"We\u0027re working in kilometers per hour,"},{"Start":"12:14.025 ","End":"12:18.280","Text":"so this will be kilometers per hour."},{"Start":"12:18.630 ","End":"12:24.745","Text":"Now I have my v_x and what I want to do is I want to find my v_y."},{"Start":"12:24.745 ","End":"12:27.700","Text":"I\u0027m reminding you that v_y is my velocity of"},{"Start":"12:27.700 ","End":"12:32.005","Text":"the raindrops in the y direction relative to the ground."},{"Start":"12:32.005 ","End":"12:37.315","Text":"What I can do is I can just plug in my v_x into 1 of these equations."},{"Start":"12:37.315 ","End":"12:40.720","Text":"Because I know that tan of 45 degrees is equal to 1,"},{"Start":"12:40.720 ","End":"12:43.660","Text":"I\u0027ll prefer to use this equation because it\u0027s just easier."},{"Start":"12:43.660 ","End":"12:46.705","Text":"So I\u0027ll multiply both sides by v_y."},{"Start":"12:46.705 ","End":"12:52.165","Text":"So I\u0027ll have v_y multiplied by 1 will simply be equal to v_y,"},{"Start":"12:52.165 ","End":"12:55.225","Text":"which is equal to my v_x,"},{"Start":"12:55.225 ","End":"12:56.740","Text":"which I just worked out over here,"},{"Start":"12:56.740 ","End":"13:00.610","Text":"which is 29.21 kilometers an hour,"},{"Start":"13:00.610 ","End":"13:05.545","Text":"minus 100 kilometers per hour, which,"},{"Start":"13:05.545 ","End":"13:06.910","Text":"once we work this out,"},{"Start":"13:06.910 ","End":"13:16.010","Text":"is equal to negative 70.79 kilometers per hour."},{"Start":"13:17.820 ","End":"13:22.195","Text":"These are our components, v_x and v_y."},{"Start":"13:22.195 ","End":"13:26.305","Text":"Now if we scroll back to the question, we\u0027re being asked,"},{"Start":"13:26.305 ","End":"13:31.405","Text":"what is the velocity of the raindrops relative to the ground?"},{"Start":"13:31.405 ","End":"13:35.140","Text":"The velocity includes size and direction."},{"Start":"13:35.140 ","End":"13:36.940","Text":"If we want the size,"},{"Start":"13:36.940 ","End":"13:39.160","Text":"so let\u0027s write this out,"},{"Start":"13:39.160 ","End":"13:43.255","Text":"the size of our vector, v,"},{"Start":"13:43.255 ","End":"13:47.050","Text":"is simply going to be equal to the square root of"},{"Start":"13:47.050 ","End":"13:52.630","Text":"our v_x component squared plus our v_y component squared."},{"Start":"13:52.630 ","End":"13:59.815","Text":"This is the size. You can just plug in these numbers with regards to the direction."},{"Start":"13:59.815 ","End":"14:02.800","Text":"Then if I want the direction,"},{"Start":"14:02.800 ","End":"14:06.625","Text":"so I have to use tan of the angle."},{"Start":"14:06.625 ","End":"14:10.525","Text":"So if I\u0027m taking the angle relative to the x-axis,"},{"Start":"14:10.525 ","End":"14:16.163","Text":"it will be v_y divided by v_x,"},{"Start":"14:16.163 ","End":"14:18.890","Text":"and if I\u0027m taking the angle relative to the y-axis,"},{"Start":"14:18.890 ","End":"14:23.000","Text":"it will be v_x divided by v_y."},{"Start":"14:23.000 ","End":"14:27.035","Text":"I\u0027ll just write that. So tan of the angle,"},{"Start":"14:27.035 ","End":"14:29.408","Text":"let\u0027s say relative to the x-axis,"},{"Start":"14:29.408 ","End":"14:31.940","Text":"so that\u0027s relative to the x-axis,"},{"Start":"14:31.940 ","End":"14:35.030","Text":"I mean that if this is our vector,"},{"Start":"14:35.030 ","End":"14:39.123","Text":"this is the definition of our angle,"},{"Start":"14:39.123 ","End":"14:45.080","Text":"so this is simply going to be equal to our v_y component,"},{"Start":"14:45.080 ","End":"14:50.300","Text":"which is this, divided by our v_x component, which is this."},{"Start":"14:50.300 ","End":"14:53.370","Text":"That\u0027s the end of this lesson."}],"ID":10789},{"Watched":false,"Name":"Exercise 3","Duration":"13m 29s","ChapterTopicVideoID":10433,"CourseChapterTopicPlaylistID":5404,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"Hello, In this question,"},{"Start":"00:01.890 ","End":"00:07.320","Text":"we\u0027re being told that a river is flowing northwards with a velocity of v_r,"},{"Start":"00:07.320 ","End":"00:12.780","Text":"and we\u0027re being told that Patricia is located on the western bank of the river,"},{"Start":"00:12.780 ","End":"00:15.750","Text":"and she wants to reach the exact same point that she is"},{"Start":"00:15.750 ","End":"00:19.920","Text":"at but on the eastern side of the river."},{"Start":"00:19.920 ","End":"00:25.140","Text":"She has a boat, and the velocity of her boat is v_br."},{"Start":"00:25.140 ","End":"00:29.490","Text":"So I\u0027m going to add a tag over here because our v\u0027_br is"},{"Start":"00:29.490 ","End":"00:34.120","Text":"the velocity of her boat relative to the river."},{"Start":"00:34.120 ","End":"00:36.200","Text":"It\u0027s a relative velocity,"},{"Start":"00:36.200 ","End":"00:37.955","Text":"so I\u0027m adding a tag here."},{"Start":"00:37.955 ","End":"00:43.625","Text":"We\u0027re being told that the width of the river is d. Question Number 1 is,"},{"Start":"00:43.625 ","End":"00:46.970","Text":"in which direction will Patricia need to sail her boat,"},{"Start":"00:46.970 ","End":"00:50.518","Text":"such that she\u0027ll be going in this straight line,"},{"Start":"00:50.518 ","End":"00:55.400","Text":"that she\u0027ll only have an x component of velocity when we\u0027re dealing"},{"Start":"00:55.400 ","End":"01:01.350","Text":"with velocity relative to the ground and not relative to the river, of course."},{"Start":"01:01.350 ","End":"01:05.630","Text":"As we can see, if she sets sail in this straight line,"},{"Start":"01:05.630 ","End":"01:08.225","Text":"because the river is moving northwards,"},{"Start":"01:08.225 ","End":"01:11.360","Text":"her boat will travel in some diagonal,"},{"Start":"01:11.360 ","End":"01:17.995","Text":"and she\u0027ll reach some point upstream and not to this point that she wants over here."},{"Start":"01:17.995 ","End":"01:21.350","Text":"So first of all, we\u0027re going to write that our v\u0027_br is"},{"Start":"01:21.350 ","End":"01:25.486","Text":"the velocity of Patricia\u0027s boat relative to the river."},{"Start":"01:25.486 ","End":"01:27.980","Text":"It\u0027s simply going to be equal to the velocity of"},{"Start":"01:27.980 ","End":"01:32.800","Text":"her boat minus the velocity of the river."},{"Start":"01:32.800 ","End":"01:37.790","Text":"Now what we want to do is we want to find an x component for our v\u0027_br,"},{"Start":"01:37.790 ","End":"01:42.085","Text":"so that\u0027s v\u0027_br_x,"},{"Start":"01:42.085 ","End":"01:46.145","Text":"which is going to have to position her boat at some angle,"},{"Start":"01:46.145 ","End":"01:51.470","Text":"Theta, relative to this arrow over here,"},{"Start":"01:51.470 ","End":"01:59.285","Text":"such that, as she sets sail together with her v\u0027_br and with the velocity of the river,"},{"Start":"01:59.285 ","End":"02:04.465","Text":"she\u0027ll end up traveling just in a straight line like so."},{"Start":"02:04.465 ","End":"02:08.145","Text":"So let\u0027s say that she positions her boat at some angle, Theta,"},{"Start":"02:08.145 ","End":"02:13.050","Text":"so we\u0027ll say that our x component of v\u0027_br will be equal to"},{"Start":"02:13.050 ","End":"02:22.190","Text":"v\u0027_br multiplied by cosine of this angle,"},{"Start":"02:22.190 ","End":"02:25.730","Text":"Theta, and of course, this angle Theta has a tag because it\u0027s"},{"Start":"02:25.730 ","End":"02:30.085","Text":"the angle relative to the river which is moving."},{"Start":"02:30.085 ","End":"02:34.200","Text":"Then similarly, with the y component of v\u0027_br,"},{"Start":"02:34.200 ","End":"02:44.150","Text":"so that will simply be equal to v\u0027_br multiplied by sine of our angle,"},{"Start":"02:44.150 ","End":"02:47.430","Text":"Theta tag, which is relative to the river."},{"Start":"02:48.110 ","End":"02:50.405","Text":"So our angle, Theta tag,"},{"Start":"02:50.405 ","End":"02:51.485","Text":"is, as we said,"},{"Start":"02:51.485 ","End":"02:54.740","Text":"going to be the angle between the direction that Patricia"},{"Start":"02:54.740 ","End":"02:59.435","Text":"positions her boat and this line over here."},{"Start":"02:59.435 ","End":"03:01.715","Text":"But of course, in reality,"},{"Start":"03:01.715 ","End":"03:04.400","Text":"when we\u0027re taking the angle relative to the ground,"},{"Start":"03:04.400 ","End":"03:07.100","Text":"so if we are just looking at the river,"},{"Start":"03:07.100 ","End":"03:08.480","Text":"and we can\u0027t see that it\u0027s moving,"},{"Start":"03:08.480 ","End":"03:10.450","Text":"and we\u0027re just seeing the boat move,"},{"Start":"03:10.450 ","End":"03:15.965","Text":"so what we want in this question is for Patricia\u0027s boat to just move like so,"},{"Start":"03:15.965 ","End":"03:22.903","Text":"just across the river with no y component of the boat\u0027s velocity relative to the ground,"},{"Start":"03:22.903 ","End":"03:24.373","Text":"not relative to the river."},{"Start":"03:24.373 ","End":"03:27.830","Text":"The boat\u0027s velocity relative to the ground in the y direction,"},{"Start":"03:27.830 ","End":"03:29.920","Text":"we want to be equal to 0."},{"Start":"03:29.920 ","End":"03:31.995","Text":"So we know that our angle,"},{"Start":"03:31.995 ","End":"03:34.235","Text":"not Theta tag,"},{"Start":"03:34.235 ","End":"03:35.495","Text":"but our angle, Theta,"},{"Start":"03:35.495 ","End":"03:39.305","Text":"which is the angle relative to the boat and the ground,"},{"Start":"03:39.305 ","End":"03:42.595","Text":"we want that to be equal to 0"},{"Start":"03:42.595 ","End":"03:48.005","Text":"because we want the boat to move across here relative to the ground."},{"Start":"03:48.005 ","End":"03:52.625","Text":"So that means that the angle Theta relative to the ground,"},{"Start":"03:52.625 ","End":"03:56.190","Text":"we want it to be equal to 0."},{"Start":"03:57.620 ","End":"04:02.973","Text":"I\u0027m given my v\u0027_br in the question,"},{"Start":"04:02.973 ","End":"04:06.715","Text":"so that means that I have both the x and the y components."},{"Start":"04:06.715 ","End":"04:10.580","Text":"Now what I have to find is my Theta and my Theta tag."},{"Start":"04:10.580 ","End":"04:18.695","Text":"So I have to find some condition that my Thetas or my angles have to abide by"},{"Start":"04:18.695 ","End":"04:22.990","Text":"and that way my boat will just sail across the river so"},{"Start":"04:22.990 ","End":"04:28.320","Text":"that means sail in the x direction relative to the ground."},{"Start":"04:28.320 ","End":"04:32.210","Text":"That means that the condition I need to meet is what we just said"},{"Start":"04:32.210 ","End":"04:36.140","Text":"that the velocity of my boat not relative to"},{"Start":"04:36.140 ","End":"04:38.750","Text":"the river but the velocity of my boat relative to"},{"Start":"04:38.750 ","End":"04:45.365","Text":"the ground in the y direction has to be equal to 0."},{"Start":"04:45.365 ","End":"04:48.635","Text":"I don\u0027t want my boat to move up or downstream."},{"Start":"04:48.635 ","End":"04:54.860","Text":"I want it to just stay on the exact same line but just move across in the x direction."},{"Start":"04:54.860 ","End":"05:00.960","Text":"So I want no y component for the velocity of my boat."},{"Start":"05:02.740 ","End":"05:09.545","Text":"The velocity of my boat in the y direction relative to the ground has to be equal to 0."},{"Start":"05:09.545 ","End":"05:13.160","Text":"How can I rewrite this using the terms that I"},{"Start":"05:13.160 ","End":"05:16.970","Text":"was given in my question and what we just wrote out over here?"},{"Start":"05:16.970 ","End":"05:19.070","Text":"Here we have my equation."},{"Start":"05:19.070 ","End":"05:23.330","Text":"What I can do is I can just take the y components and substitute them in."},{"Start":"05:23.330 ","End":"05:29.450","Text":"So I can say that my velocity of the boat relative to the river in"},{"Start":"05:29.450 ","End":"05:33.245","Text":"the y direction is equal to the velocity of the boat"},{"Start":"05:33.245 ","End":"05:37.925","Text":"relative to the ground in the y direction minus the velocity of the river,"},{"Start":"05:37.925 ","End":"05:41.670","Text":"which we can see is already just as a component in the y direction,"},{"Start":"05:41.670 ","End":"05:44.270","Text":"so I don\u0027t have to write the subscript y."},{"Start":"05:44.270 ","End":"05:49.700","Text":"Then all I have to do is rearrange my equation over here."},{"Start":"05:49.700 ","End":"05:52.160","Text":"So I\u0027ll have v_b_y,"},{"Start":"05:52.160 ","End":"05:54.335","Text":"which is this term over here,"},{"Start":"05:54.335 ","End":"05:58.430","Text":"is going to be equal to the velocity of my boat relative to"},{"Start":"05:58.430 ","End":"06:03.131","Text":"the river in the y direction,"},{"Start":"06:03.131 ","End":"06:06.035","Text":"this I move over to this side of the equation,"},{"Start":"06:06.035 ","End":"06:10.115","Text":"plus the velocity of the river which only has a y component."},{"Start":"06:10.115 ","End":"06:12.740","Text":"Now I\u0027ll just rub out the subscript so we can use"},{"Start":"06:12.740 ","End":"06:15.960","Text":"this equation later for the x direction."},{"Start":"06:17.320 ","End":"06:21.515","Text":"Now we have our v_br in the y direction"},{"Start":"06:21.515 ","End":"06:25.520","Text":"which we can substitute with this equation over here."},{"Start":"06:25.520 ","End":"06:29.875","Text":"We get that 0 has to be equal to"},{"Start":"06:29.875 ","End":"06:35.975","Text":"v\u0027_br multiplied by sine"},{"Start":"06:35.975 ","End":"06:41.535","Text":"of Theta tag plus our velocity of the river."},{"Start":"06:41.535 ","End":"06:47.430","Text":"Now what we can do is we can isolate out our sine of Theta tag,"},{"Start":"06:47.430 ","End":"06:52.685","Text":"so what we\u0027ll get is that sine of Theta tag has to be equal to"},{"Start":"06:52.685 ","End":"07:00.690","Text":"negative v_r divided by v\u0027_br."},{"Start":"07:01.040 ","End":"07:04.560","Text":"Now if we arcsin both sides,"},{"Start":"07:04.560 ","End":"07:06.240","Text":"we\u0027ll get what our angle, Theta tag,"},{"Start":"07:06.240 ","End":"07:07.620","Text":"has to be equal to,"},{"Start":"07:07.620 ","End":"07:13.306","Text":"and I\u0027m reminding you that angle Theta tag is this angle over here,"},{"Start":"07:13.306 ","End":"07:16.340","Text":"so the angle that the boat has to be pointing in in"},{"Start":"07:16.340 ","End":"07:20.410","Text":"order for it to just move in this straight line."},{"Start":"07:20.410 ","End":"07:24.755","Text":"As we can see, we get a negative angle."},{"Start":"07:24.755 ","End":"07:26.540","Text":"Why do we have a negative angle?"},{"Start":"07:26.540 ","End":"07:28.715","Text":"It actually makes a lot of sense."},{"Start":"07:28.715 ","End":"07:33.420","Text":"Because the river is flowing in this positive y direction,"},{"Start":"07:33.420 ","End":"07:39.860","Text":"so we can see that if we have our boat pointing in this positive angle,"},{"Start":"07:39.860 ","End":"07:42.620","Text":"our boat is just going to move upstream."},{"Start":"07:42.620 ","End":"07:46.805","Text":"However, if our boat, let\u0027s rub this out,"},{"Start":"07:46.805 ","End":"07:52.700","Text":"is instead pointing in this direction, so we can see that,"},{"Start":"07:52.700 ","End":"07:55.820","Text":"slowly, it will be able to move across,"},{"Start":"07:55.820 ","End":"07:58.685","Text":"and the river will push it in this direction."},{"Start":"07:58.685 ","End":"08:00.050","Text":"But because it\u0027s pointing in"},{"Start":"08:00.050 ","End":"08:03.695","Text":"the opposite direction with a velocity in the negative y direction,"},{"Start":"08:03.695 ","End":"08:08.790","Text":"the y velocities of the river and of the boat will cancel out,"},{"Start":"08:08.790 ","End":"08:11.255","Text":"and eventually, the boat will just have"},{"Start":"08:11.255 ","End":"08:15.600","Text":"an x component of velocity relative to the ground."},{"Start":"08:16.880 ","End":"08:21.080","Text":"So this is the answer to Question Number 1."},{"Start":"08:21.080 ","End":"08:24.830","Text":"Now let\u0027s take a look at Question Number 2."},{"Start":"08:24.830 ","End":"08:29.080","Text":"What is the velocity of the boat relative to the ground?"},{"Start":"08:29.080 ","End":"08:34.000","Text":"So what we\u0027re trying to find right now is v_b."},{"Start":"08:34.430 ","End":"08:38.360","Text":"We already know that the y component of"},{"Start":"08:38.360 ","End":"08:41.555","Text":"the velocity of our boat relative to the ground is equal to 0."},{"Start":"08:41.555 ","End":"08:43.685","Text":"From Question Number 1, we already saw that,"},{"Start":"08:43.685 ","End":"08:48.755","Text":"so we can just say that v_by is equal to 0, and now,"},{"Start":"08:48.755 ","End":"08:53.750","Text":"all we have to do is we have to find what our x component of"},{"Start":"08:53.750 ","End":"08:59.385","Text":"the velocity of the boat relative to the ground is equal to 0."},{"Start":"08:59.385 ","End":"09:02.630","Text":"Now we\u0027re trying to find the velocity of our boat"},{"Start":"09:02.630 ","End":"09:06.185","Text":"relative to the ground but in the x direction is equal to."},{"Start":"09:06.185 ","End":"09:10.550","Text":"So what we\u0027re going to do is we\u0027re going to use this equation over here."},{"Start":"09:10.550 ","End":"09:14.780","Text":"Now if we substitute in the x subscript into here,"},{"Start":"09:14.780 ","End":"09:20.990","Text":"so we\u0027ll have that v_b_x is equal to the velocity of the boat relative to"},{"Start":"09:20.990 ","End":"09:27.950","Text":"the river in the x direction plus the velocity of the river in the x direction."},{"Start":"09:27.950 ","End":"09:30.275","Text":"I\u0027ve just rearranged this equation."},{"Start":"09:30.275 ","End":"09:34.390","Text":"Now, obviously, the velocity of the river has no x component,"},{"Start":"09:34.390 ","End":"09:35.780","Text":"it only has a y component,"},{"Start":"09:35.780 ","End":"09:37.940","Text":"so this is equal to 0."},{"Start":"09:38.390 ","End":"09:42.440","Text":"Now, I have the equation over here for"},{"Start":"09:42.440 ","End":"09:45.828","Text":"the velocity of the boat relative to the river in the x direction,"},{"Start":"09:45.828 ","End":"09:50.345","Text":"which we said was equal to the velocity of the boat relative to the river,"},{"Start":"09:50.345 ","End":"09:54.615","Text":"multiplied by cosine of my angle, Theta."},{"Start":"09:54.615 ","End":"09:59.600","Text":"So this is really great because I know that sine of my angle,"},{"Start":"09:59.600 ","End":"10:02.570","Text":"Theta, is equal to this value over here."},{"Start":"10:02.570 ","End":"10:06.380","Text":"So all I have to do is I have to use the trig identity saying"},{"Start":"10:06.380 ","End":"10:10.320","Text":"that cosine of my angle Theta or any angle,"},{"Start":"10:10.320 ","End":"10:13.895","Text":"Theta tag, is going to be equal to the square root of"},{"Start":"10:13.895 ","End":"10:19.605","Text":"1 minus sine squared of my Theta tag."},{"Start":"10:19.605 ","End":"10:25.775","Text":"This is a very useful trig identity which I suggest you write down somewhere or memorize."},{"Start":"10:25.775 ","End":"10:27.200","Text":"In this case, over here,"},{"Start":"10:27.200 ","End":"10:32.530","Text":"this will simply be the square root of 1 minus sine squared Theta."},{"Start":"10:32.530 ","End":"10:35.540","Text":"This will be a negative multiplied by a negative,"},{"Start":"10:35.540 ","End":"10:36.860","Text":"or a negative squared is a positive,"},{"Start":"10:36.860 ","End":"10:45.390","Text":"so I can just write v_r divided by v\u0027_br squared."},{"Start":"10:46.400 ","End":"10:50.530","Text":"So this is cosine of my angle, Theta."},{"Start":"10:52.070 ","End":"10:54.335","Text":"So then, therefore,"},{"Start":"10:54.335 ","End":"10:56.600","Text":"I can say that this,"},{"Start":"10:56.600 ","End":"10:59.044","Text":"taking this equation into account,"},{"Start":"10:59.044 ","End":"11:07.860","Text":"is simply going to be equal to my v_br multiplied by cosine Theta tag."},{"Start":"11:07.860 ","End":"11:15.120","Text":"Once I move my v\u0027_br into this square root sign,"},{"Start":"11:15.120 ","End":"11:16.875","Text":"this is just some algebra,"},{"Start":"11:16.875 ","End":"11:22.205","Text":"my v_b_x is simply going to be equal to the square root of"},{"Start":"11:22.205 ","End":"11:31.975","Text":"v_br squared minus the velocity of my river squared."},{"Start":"11:31.975 ","End":"11:37.520","Text":"So all I\u0027ve done is I\u0027ve substituted in my cosine Theta into this equation and"},{"Start":"11:37.520 ","End":"11:42.620","Text":"moved my v\u0027_br inside the square root sign."},{"Start":"11:42.620 ","End":"11:46.920","Text":"This is our answer to Question Number 2."},{"Start":"11:47.810 ","End":"11:51.780","Text":"That\u0027s our answer to Question 2."},{"Start":"11:51.780 ","End":"11:54.170","Text":"Now Question 3 is,"},{"Start":"11:54.170 ","End":"11:58.495","Text":"how long will it take Patricia to cross the river?"},{"Start":"11:58.495 ","End":"12:02.780","Text":"What we\u0027re searching for is an expression of time."},{"Start":"12:02.780 ","End":"12:07.269","Text":"How long will it take Patricia to get from here all the way to here,"},{"Start":"12:07.269 ","End":"12:09.560","Text":"so this is Question Number 3."},{"Start":"12:09.560 ","End":"12:14.975","Text":"I\u0027m just going to use my equation that I know and that means that my distance,"},{"Start":"12:14.975 ","End":"12:20.960","Text":"x, is equal to my velocity multiplied by the time taken."},{"Start":"12:20.960 ","End":"12:27.390","Text":"I know that my distance traveled is going to be d. She travels right across the river,"},{"Start":"12:27.390 ","End":"12:29.750","Text":"so that means that the distance,"},{"Start":"12:29.750 ","End":"12:32.550","Text":"d, is equal to my velocity."},{"Start":"12:32.550 ","End":"12:36.530","Text":"Obviously, I\u0027m going to only take my velocity in the x direction because,"},{"Start":"12:36.530 ","End":"12:38.735","Text":"relative to the ground,"},{"Start":"12:38.735 ","End":"12:45.335","Text":"I\u0027m only moving with an x component of velocity and with a 0 y component for velocity."},{"Start":"12:45.335 ","End":"12:48.980","Text":"I\u0027m going to use this equation which is simply,"},{"Start":"12:48.980 ","End":"12:50.660","Text":"let\u0027s write that out afterwards,"},{"Start":"12:50.660 ","End":"12:58.985","Text":"so it\u0027s just going to v_b_x multiplied by t. So t is the value that I\u0027m trying to find."},{"Start":"12:58.985 ","End":"13:01.670","Text":"So the time that it will take Patricia to cross"},{"Start":"13:01.670 ","End":"13:04.790","Text":"the river is going to be equal to the distance,"},{"Start":"13:04.790 ","End":"13:06.665","Text":"d, which is given to us in the question,"},{"Start":"13:06.665 ","End":"13:11.105","Text":"divided by my v_bx which is simply the square root of"},{"Start":"13:11.105 ","End":"13:14.570","Text":"v\u0027_br squared"},{"Start":"13:14.570 ","End":"13:22.770","Text":"minus v_r^2."},{"Start":"13:22.770 ","End":"13:25.700","Text":"This is the answer to Question Number 3,"},{"Start":"13:25.700 ","End":"13:29.100","Text":"and that is the end of this lesson."}],"ID":10790},{"Watched":false,"Name":"Exercise 4","Duration":"21m 48s","ChapterTopicVideoID":10434,"CourseChapterTopicPlaylistID":5404,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"Hello. In this question,"},{"Start":"00:01.845 ","End":"00:06.735","Text":"we\u0027re being given 2 bullets which are shot at a time of t = 0."},{"Start":"00:06.735 ","End":"00:09.510","Text":"Their initial positions and velocities are given"},{"Start":"00:09.510 ","End":"00:13.860","Text":"by the position of bullet number 1 at t = 0,"},{"Start":"00:13.860 ","End":"00:17.780","Text":"is 0 and the position of bullet 2 is at i,"},{"Start":"00:17.780 ","End":"00:20.200","Text":"so 1 in the i direction,"},{"Start":"00:20.200 ","End":"00:24.480","Text":"and their initial velocity is for the first bullet is 2 in the i direction,"},{"Start":"00:24.480 ","End":"00:26.295","Text":"and 5 in the j direction,"},{"Start":"00:26.295 ","End":"00:28.905","Text":"and for the 2nd bullet,"},{"Start":"00:28.905 ","End":"00:34.755","Text":"the velocity is negative 1 in the i direction and 4 in the j direction."},{"Start":"00:34.755 ","End":"00:38.040","Text":"Both bullets experience a pulling force which causes them to"},{"Start":"00:38.040 ","End":"00:42.210","Text":"accelerate at a is equal to negative 3,"},{"Start":"00:42.210 ","End":"00:45.525","Text":"the i direction plus 1 in the j direction."},{"Start":"00:45.525 ","End":"00:49.550","Text":"All of the units that we\u0027re dealing with are in MKS."},{"Start":"00:49.550 ","End":"00:53.480","Text":"Now we\u0027re being asked in question number 1 to find"},{"Start":"00:53.480 ","End":"01:00.090","Text":"the positions of the bullets as a function of time."},{"Start":"01:00.800 ","End":"01:08.540","Text":"As we know, in order to find our possession r_1 as a function of time,"},{"Start":"01:08.540 ","End":"01:12.320","Text":"what we need to do is we have to integrate along"},{"Start":"01:12.320 ","End":"01:17.355","Text":"the velocity of the first bullet as a function of time dt."},{"Start":"01:17.355 ","End":"01:23.150","Text":"Now our problem is we don\u0027t have function v_1 as a function of time,"},{"Start":"01:23.150 ","End":"01:27.795","Text":"we have velocity v_1 at t is equal to 0,"},{"Start":"01:27.795 ","End":"01:29.710","Text":"so only at that 1 time,"},{"Start":"01:29.710 ","End":"01:33.310","Text":"but we want to have the function for all time."},{"Start":"01:33.310 ","End":"01:38.560","Text":"What do we have to do is we can see that we have this equation here for acceleration."},{"Start":"01:38.560 ","End":"01:41.050","Text":"You can also see that this has constant acceleration."},{"Start":"01:41.050 ","End":"01:47.155","Text":"All we\u0027re going to do in order to find what our v_1 as a function of time is equal to,"},{"Start":"01:47.155 ","End":"01:50.815","Text":"is we\u0027re simply going to do the same thing but on the acceleration."},{"Start":"01:50.815 ","End":"01:57.775","Text":"We\u0027re going to integrate along our acceleration as a function of time,"},{"Start":"01:57.775 ","End":"02:02.380","Text":"which is going to be the integral of negative 3 in"},{"Start":"02:02.380 ","End":"02:08.070","Text":"the i direction plus 1 in the j direction, dt."},{"Start":"02:08.070 ","End":"02:12.815","Text":"Then what this is going to be equal to because this is constant acceleration,"},{"Start":"02:12.815 ","End":"02:19.220","Text":"negative 3t plus our constant for integration,"},{"Start":"02:19.220 ","End":"02:21.380","Text":"because we have an indefinite integrals,"},{"Start":"02:21.380 ","End":"02:26.165","Text":"so plus some constant c_1 in the i direction,"},{"Start":"02:26.165 ","End":"02:31.340","Text":"and then we\u0027ll have plus t in"},{"Start":"02:31.340 ","End":"02:37.340","Text":"the j direction plus an integrating constant because it\u0027s still an indefinite integral,"},{"Start":"02:37.340 ","End":"02:42.090","Text":"so c_2 in the j direction."},{"Start":"02:43.070 ","End":"02:47.883","Text":"Now what we want to do is we want to find out what are c_1 and c_2 are equal to,"},{"Start":"02:47.883 ","End":"02:52.490","Text":"so we\u0027re going to do that by using our initial conditions."},{"Start":"02:52.490 ","End":"02:58.627","Text":"For v_1, we see that our initial velocity is going to be 2i plus 5j,"},{"Start":"02:58.627 ","End":"03:01.005","Text":"so let\u0027s substitute that in."},{"Start":"03:01.005 ","End":"03:03.090","Text":"We\u0027ll have v_1,"},{"Start":"03:03.090 ","End":"03:05.640","Text":"at t is equal to 0."},{"Start":"03:05.640 ","End":"03:10.970","Text":"We\u0027re going to substitute in here 0 wherever we see t. We\u0027ll get that that is equal to"},{"Start":"03:10.970 ","End":"03:17.775","Text":"c_1 in the i direction plus c_2 in the j direction,"},{"Start":"03:17.775 ","End":"03:21.365","Text":"and this is equal to what we\u0027re given in the question,"},{"Start":"03:21.365 ","End":"03:27.440","Text":"which is equal to 2i plus 5j."},{"Start":"03:27.440 ","End":"03:30.340","Text":"Therefore, we can say that our c_1,"},{"Start":"03:30.340 ","End":"03:32.730","Text":"our coefficient for the i direction,"},{"Start":"03:32.730 ","End":"03:35.245","Text":"is 2, and c_2,"},{"Start":"03:35.245 ","End":"03:40.335","Text":"our coefficient for the j direction is just going to be 5."},{"Start":"03:40.335 ","End":"03:44.330","Text":"Now we\u0027re going to plug all of that in and do"},{"Start":"03:44.330 ","End":"03:49.865","Text":"this integration on r. Let\u0027s go and do that."},{"Start":"03:49.865 ","End":"03:55.265","Text":"We can say that the position as a function of time of"},{"Start":"03:55.265 ","End":"04:01.985","Text":"our first bullet is going to be equal to the integral on our v_1 dt,"},{"Start":"04:01.985 ","End":"04:05.100","Text":"where our v_1 we got is this."},{"Start":"04:05.100 ","End":"04:12.785","Text":"We\u0027ll have negative 3t plus our c_1 which we found was equal to 2,"},{"Start":"04:12.785 ","End":"04:23.190","Text":"and this is in the i direction plus t plus our c_2,"},{"Start":"04:23.190 ","End":"04:25.170","Text":"which we found was equal to 5."},{"Start":"04:25.170 ","End":"04:29.325","Text":"So plus 5 in our j direction,"},{"Start":"04:29.325 ","End":"04:31.695","Text":"and then all of this dt,"},{"Start":"04:31.695 ","End":"04:34.710","Text":"we\u0027re integrating with respect to time."},{"Start":"04:34.710 ","End":"04:38.140","Text":"This is going to be equal to,"},{"Start":"04:38.140 ","End":"04:39.610","Text":"so let\u0027s just integrate."},{"Start":"04:39.610 ","End":"04:42.595","Text":"We\u0027ll have that this is equal to negative 3 over"},{"Start":"04:42.595 ","End":"04:49.320","Text":"2t^2 plus 2t plus some integrating constant,"},{"Start":"04:49.320 ","End":"04:53.250","Text":"let\u0027s call it c_3 in the i direction,"},{"Start":"04:53.250 ","End":"05:03.320","Text":"plus t^2 over 2 plus 5t plus another integration constant,"},{"Start":"05:03.320 ","End":"05:05.720","Text":"because again, this is an indefinite integral,"},{"Start":"05:05.720 ","End":"05:11.190","Text":"c_4 in the j direction."},{"Start":"05:11.660 ","End":"05:15.515","Text":"Now again, we have these constant c_3 and c_4,"},{"Start":"05:15.515 ","End":"05:17.225","Text":"so we want to find out what they are."},{"Start":"05:17.225 ","End":"05:20.490","Text":"How do we do that? We go to our initial conditions."},{"Start":"05:20.490 ","End":"05:25.730","Text":"We\u0027re being told that our position of bullet number 1 at t is equal to 0, is equal to 0."},{"Start":"05:25.730 ","End":"05:29.210","Text":"That means if we substitute that in here,"},{"Start":"05:29.210 ","End":"05:32.090","Text":"so r_1 at t is equal to 0,"},{"Start":"05:32.090 ","End":"05:35.735","Text":"so let\u0027s substitute in t is equal to 0 into this equation."},{"Start":"05:35.735 ","End":"05:44.055","Text":"We\u0027ll get c_3 in our i direction plus c_4 in our j direction,"},{"Start":"05:44.055 ","End":"05:47.375","Text":"and this, as we know from our initial conditions over here,"},{"Start":"05:47.375 ","End":"05:49.430","Text":"has to be equal to 0."},{"Start":"05:49.430 ","End":"05:54.285","Text":"That means therefore that c_3 is equal to c_4,"},{"Start":"05:54.285 ","End":"05:57.070","Text":"which is equal to 0."},{"Start":"05:58.040 ","End":"06:03.205","Text":"Now I\u0027ve just written out what r_1 as a function of t is equal to,"},{"Start":"06:03.205 ","End":"06:07.420","Text":"where I just substituted in for c_3 and c_4 0,"},{"Start":"06:07.420 ","End":"06:10.425","Text":"and then we get this answer over here."},{"Start":"06:10.425 ","End":"06:14.620","Text":"Now, we have to find what r_2 as a function of time is equal to,"},{"Start":"06:14.620 ","End":"06:18.720","Text":"so I\u0027m going to do the exact same process for r_2."},{"Start":"06:19.310 ","End":"06:26.585","Text":"Again, what I\u0027m going to do in order to find my v_2 as a function of time,"},{"Start":"06:26.585 ","End":"06:31.645","Text":"is I\u0027m going to integrate along my acceleration dt,"},{"Start":"06:31.645 ","End":"06:39.565","Text":"which is going to work out to this exact same thing over here."},{"Start":"06:39.565 ","End":"06:45.215","Text":"I get that my v_2 as a function of time is going to be equal to"},{"Start":"06:45.215 ","End":"06:50.760","Text":"negative 3t plus some c_1 in the i direction,"},{"Start":"06:50.760 ","End":"06:56.820","Text":"plus t plus c_2 in the j direction."},{"Start":"06:56.820 ","End":"06:59.690","Text":"Now, in order to find what my constants are,"},{"Start":"06:59.690 ","End":"07:04.025","Text":"my constants c_1 and c_2 are going to be different for my v_2."},{"Start":"07:04.025 ","End":"07:10.505","Text":"Because my initial conditions for v_2 are different to my initial conditions for v_1."},{"Start":"07:10.505 ","End":"07:18.420","Text":"I know that my v_2 at time t is equal to 0 is going to equal to,"},{"Start":"07:18.420 ","End":"07:21.440","Text":"so let\u0027s substitute t is equal to 0 into this equation."},{"Start":"07:21.440 ","End":"07:27.980","Text":"I\u0027ll have c_1 in the i direction plus c_2 in the j direction,"},{"Start":"07:27.980 ","End":"07:33.050","Text":"which is equal to my initial v_2 that\u0027s given to me in the question,"},{"Start":"07:33.050 ","End":"07:39.060","Text":"so I have negative 1 in the i direction plus 4 in the j direction."},{"Start":"07:39.060 ","End":"07:42.390","Text":"Therefore, my c_1 is the coefficient of the i direction,"},{"Start":"07:42.390 ","End":"07:43.590","Text":"so that\u0027s negative 1,"},{"Start":"07:43.590 ","End":"07:46.925","Text":"and my c_2 is the coefficient of the j direction,"},{"Start":"07:46.925 ","End":"07:49.885","Text":"which is positive 4."},{"Start":"07:49.885 ","End":"08:00.180","Text":"Now I can say that my v_2 as a function of time is equal to negative 3t plus my c_1,"},{"Start":"08:00.180 ","End":"08:01.605","Text":"which is negative 1."},{"Start":"08:01.605 ","End":"08:06.690","Text":"We can rub that out, so negative 1 in the i direction,"},{"Start":"08:06.690 ","End":"08:10.830","Text":"plus t plus my c_2,"},{"Start":"08:10.830 ","End":"08:16.810","Text":"which is 4 in the j direction."},{"Start":"08:17.330 ","End":"08:25.665","Text":"Now what I want to do is I want to find my position vector i_2 as a function of time."},{"Start":"08:25.665 ","End":"08:33.220","Text":"That is going to be the integral along my v_2 vector as a function of time dt,"},{"Start":"08:33.220 ","End":"08:37.115","Text":"which once I integrate all of this,"},{"Start":"08:37.115 ","End":"08:40.760","Text":"so I\u0027ll get that this is equal to negative 3 divided by"},{"Start":"08:40.760 ","End":"08:49.200","Text":"2t^2 minus t plus some integrating constant c_3,"},{"Start":"08:49.200 ","End":"08:51.666","Text":"and this is in the i direction."},{"Start":"08:51.666 ","End":"08:54.465","Text":"Plus in my j direction,"},{"Start":"08:54.465 ","End":"08:59.240","Text":"I have t^2 divided by 2 plus"},{"Start":"08:59.240 ","End":"09:08.430","Text":"4t plus an integration constant c_4 in the j direction."},{"Start":"09:08.430 ","End":"09:11.430","Text":"Now what I need to do is just like before,"},{"Start":"09:11.430 ","End":"09:14.820","Text":"I have to find out what my c_3 and c_4 are equal to,"},{"Start":"09:14.820 ","End":"09:18.945","Text":"so I\u0027m going to use my initial conditions given to me in the question."},{"Start":"09:18.945 ","End":"09:23.770","Text":"I know that my r_2 at time t is equal to 0,"},{"Start":"09:23.770 ","End":"09:25.400","Text":"so let\u0027s substitute in here,"},{"Start":"09:25.400 ","End":"09:26.825","Text":"t is equal to 0."},{"Start":"09:26.825 ","End":"09:28.630","Text":"In the i direction,"},{"Start":"09:28.630 ","End":"09:32.345","Text":"I\u0027ll have c_3 in the i direction,"},{"Start":"09:32.345 ","End":"09:34.135","Text":"and then in the j direction,"},{"Start":"09:34.135 ","End":"09:37.765","Text":"I\u0027ll have c_4 in the j direction,"},{"Start":"09:37.765 ","End":"09:44.775","Text":"which is equal to 1 in the i direction from my initial conditions."},{"Start":"09:44.775 ","End":"09:47.010","Text":"Let\u0027s scroll down a little bit,"},{"Start":"09:47.010 ","End":"09:48.990","Text":"and then I can say that my c_3,"},{"Start":"09:48.990 ","End":"09:53.460","Text":"so my coefficient for the i direction is simply going to be equal to 1,"},{"Start":"09:53.460 ","End":"09:56.255","Text":"and my c_4, which is my coefficient in the j direction,"},{"Start":"09:56.255 ","End":"09:59.185","Text":"I\u0027ve no j component from the initial conditions,"},{"Start":"09:59.185 ","End":"10:03.810","Text":"so my c_4 is going to be equal to 0."},{"Start":"10:03.980 ","End":"10:08.265","Text":"I\u0027ve made this a little bit smaller so that I have space."},{"Start":"10:08.265 ","End":"10:16.170","Text":"Then I can say finally that my r_2 vector as a function of time is simply going to be"},{"Start":"10:16.170 ","End":"10:20.220","Text":"equal to this equation over here but"},{"Start":"10:20.220 ","End":"10:24.765","Text":"when I substitute in my constant C_3 and C_4 that I found."},{"Start":"10:24.765 ","End":"10:34.230","Text":"It\u0027s going to be equal to negative 3 over 2t^2 minus t plus C_3 which was equal to 1,"},{"Start":"10:34.230 ","End":"10:38.475","Text":"so plus 1 in the I direction plus,"},{"Start":"10:38.475 ","End":"10:45.990","Text":"so I have t^2 divided by 2 plus 4t plus C_4,"},{"Start":"10:45.990 ","End":"10:50.590","Text":"which was equal to 0 in the j direction."},{"Start":"10:51.080 ","End":"10:54.450","Text":"Here we have our answer to question number 1,"},{"Start":"10:54.450 ","End":"11:00.705","Text":"which is finding vector r_1 as a function of time and r_2 as a function of time."},{"Start":"11:00.705 ","End":"11:03.850","Text":"Let\u0027s go on to question number 2."},{"Start":"11:03.890 ","End":"11:08.415","Text":"Question number 2 is asking us,"},{"Start":"11:08.415 ","End":"11:13.560","Text":"what is the distance between the 2 bullets as a function of time?"},{"Start":"11:13.560 ","End":"11:16.935","Text":"If we draw over here some axis,"},{"Start":"11:16.935 ","End":"11:20.130","Text":"and then we have our vector r_1 as a function of time,"},{"Start":"11:20.130 ","End":"11:21.825","Text":"which can be pointing here."},{"Start":"11:21.825 ","End":"11:26.640","Text":"This is r_1 as a function of time and then we have over here"},{"Start":"11:26.640 ","End":"11:32.205","Text":"our vector r_2 which is also as a function of time."},{"Start":"11:32.205 ","End":"11:41.835","Text":"The distance between these 2 vectors is this size over here, this in blue."},{"Start":"11:41.835 ","End":"11:46.620","Text":"In order to find this over here,"},{"Start":"11:46.620 ","End":"11:50.910","Text":"we can say that this is a vector r_1,2."},{"Start":"11:50.910 ","End":"11:53.520","Text":"What I\u0027m going to do is I\u0027m going to erase that."},{"Start":"11:53.520 ","End":"12:00.180","Text":"Now vector r_1,2 is"},{"Start":"12:00.180 ","End":"12:06.315","Text":"the vector going from r_2 to r_1."},{"Start":"12:06.315 ","End":"12:10.905","Text":"This is vector r_1,2."},{"Start":"12:10.905 ","End":"12:19.900","Text":"This is equal to vector r_1 minus vector r_2."},{"Start":"12:20.630 ","End":"12:24.525","Text":"Whenever you see r_1,2,"},{"Start":"12:24.525 ","End":"12:28.095","Text":"that means that you have to write r_1 minus r_2."},{"Start":"12:28.095 ","End":"12:34.335","Text":"Let\u0027s just see that this sorts out as we know when we\u0027re adding vectors."},{"Start":"12:34.335 ","End":"12:36.660","Text":"If we move this over,"},{"Start":"12:36.660 ","End":"12:46.080","Text":"so we can say that r_1,2 plus r_2 has to be equal to r_1."},{"Start":"12:46.080 ","End":"12:48.570","Text":"If we just rearrange this equation."},{"Start":"12:48.570 ","End":"12:55.665","Text":"That\u0027s the same r_1,2 plus r_2 is the same as r_2 plus r_1,2."},{"Start":"12:55.665 ","End":"12:58.845","Text":"It doesn\u0027t matter which way we\u0027re adding,"},{"Start":"12:58.845 ","End":"13:01.065","Text":"this plus this or this plus this."},{"Start":"13:01.065 ","End":"13:04.500","Text":"As we can see, according to laws of adding vectors,"},{"Start":"13:04.500 ","End":"13:07.920","Text":"if we go from the origin and we add on vector r_2,"},{"Start":"13:07.920 ","End":"13:09.330","Text":"we\u0027ll reach this point."},{"Start":"13:09.330 ","End":"13:13.740","Text":"Then if we add on vector r_1,2 which is this in blue,"},{"Start":"13:13.740 ","End":"13:21.132","Text":"we\u0027ll get r_1 which is exactly what we get when we rearrange this equation."},{"Start":"13:21.132 ","End":"13:23.880","Text":"That\u0027s just to check that we understand what\u0027s going"},{"Start":"13:23.880 ","End":"13:28.065","Text":"on and that we\u0027ve written our equations correctly."},{"Start":"13:28.065 ","End":"13:36.420","Text":"We\u0027re asking to find the distance between the 2 bullets as a function of time."},{"Start":"13:36.420 ","End":"13:45.450","Text":"The distance simply means what is the size of this vector r_1,2?"},{"Start":"13:45.450 ","End":"13:47.565","Text":"That\u0027s what we\u0027re trying to find."},{"Start":"13:47.565 ","End":"13:54.000","Text":"The size of this vector will give us the distance between r_2 and r_1."},{"Start":"13:54.000 ","End":"13:55.680","Text":"Let\u0027s work this out."},{"Start":"13:55.680 ","End":"14:03.075","Text":"We\u0027re just going to take the I components of r_1 and subtract the I components of r_2."},{"Start":"14:03.075 ","End":"14:10.305","Text":"We\u0027ll get negative 3 over 2t^2 minus 3 over 2t^2."},{"Start":"14:10.305 ","End":"14:16.470","Text":"Those will cancel out and then we\u0027ll have 2t minus t,"},{"Start":"14:16.470 ","End":"14:22.080","Text":"so 2t plus t will give us 3t."},{"Start":"14:22.080 ","End":"14:26.670","Text":"Then we have 0 minus positive 1."},{"Start":"14:26.670 ","End":"14:29.085","Text":"That will be negative 1,"},{"Start":"14:29.085 ","End":"14:33.795","Text":"and that is in the I direction, sorry."},{"Start":"14:33.795 ","End":"14:36.720","Text":"Then in the j direction,"},{"Start":"14:36.720 ","End":"14:39.855","Text":"we subtract the 2j times,"},{"Start":"14:39.855 ","End":"14:44.025","Text":"so t^2 over 2 negative t^2 over 2."},{"Start":"14:44.025 ","End":"14:45.510","Text":"That will cancel out."},{"Start":"14:45.510 ","End":"14:53.610","Text":"Then we have 5t minus 4t which will give us simply t in the j direction."},{"Start":"14:53.610 ","End":"14:57.645","Text":"Now in order to find the size of this vector,"},{"Start":"14:57.645 ","End":"14:59.670","Text":"we\u0027re just going to use our usual method,"},{"Start":"14:59.670 ","End":"15:01.530","Text":"which is via Pythagoras."},{"Start":"15:01.530 ","End":"15:04.590","Text":"I\u0027m going to take this component,"},{"Start":"15:04.590 ","End":"15:06.300","Text":"my I component and I\u0027ll square it."},{"Start":"15:06.300 ","End":"15:14.115","Text":"I have 3t minus 1 is by I component squared plus my j component squared, which is t^2."},{"Start":"15:14.115 ","End":"15:19.140","Text":"Then all I have to do is I can open up the brackets and add"},{"Start":"15:19.140 ","End":"15:23.880","Text":"on the like terms and I\u0027ll get that the size of this vector,"},{"Start":"15:23.880 ","End":"15:26.805","Text":"which means the size of this vector in blue,"},{"Start":"15:26.805 ","End":"15:31.620","Text":"which means the distance between my 2 bullets is simply going to be"},{"Start":"15:31.620 ","End":"15:37.930","Text":"equal to 10t^2 minus 6t"},{"Start":"15:38.120 ","End":"15:45.450","Text":"plus 1."},{"Start":"15:45.450 ","End":"15:49.240","Text":"This is our answer to question number 2."},{"Start":"15:49.520 ","End":"15:54.375","Text":"Now let\u0027s answer question number 3."},{"Start":"15:54.375 ","End":"16:02.910","Text":"Question number 3 is to find the angle between v_1 and v_2 at time t=3."},{"Start":"16:02.910 ","End":"16:08.115","Text":"First of all, before we find the angle between v_1 and v_2 at t=3,"},{"Start":"16:08.115 ","End":"16:12.165","Text":"let\u0027s find out what v_1 at t=3 is equal to,"},{"Start":"16:12.165 ","End":"16:15.615","Text":"and what v_2 at t=3 is equal to."},{"Start":"16:15.615 ","End":"16:19.560","Text":"First of all, I already rubbed it out,"},{"Start":"16:19.560 ","End":"16:24.720","Text":"but you can rewind this video in order to see for yourself but"},{"Start":"16:24.720 ","End":"16:30.420","Text":"I\u0027m just going to copy what we worked out for v_1 as a function of time."},{"Start":"16:30.420 ","End":"16:37.890","Text":"That was equal to negative 3t plus 2 in the I direction,"},{"Start":"16:37.890 ","End":"16:43.545","Text":"plus t plus 5 in the j direction."},{"Start":"16:43.545 ","End":"16:46.050","Text":"This we worked out for question number 1."},{"Start":"16:46.050 ","End":"16:49.455","Text":"Our equation for v_2 as a function of time,"},{"Start":"16:49.455 ","End":"16:55.830","Text":"we got to be negative 3t negative 1 in the I direction,"},{"Start":"16:55.830 ","End":"17:02.414","Text":"plus t plus 4 in the j direction."},{"Start":"17:02.414 ","End":"17:12.090","Text":"Now let\u0027s see what our v_1 at t=3 is equal to. Let\u0027s substitute in."},{"Start":"17:12.090 ","End":"17:16.020","Text":"We have negative 3 multiplied by 3 is negative 9,"},{"Start":"17:16.020 ","End":"17:21.375","Text":"negative 9 plus 2 is negative 7 in the I direction,"},{"Start":"17:21.375 ","End":"17:23.670","Text":"and then plus in the j direction,"},{"Start":"17:23.670 ","End":"17:29.715","Text":"3 plus 5 is equal to 8 in the j direction."},{"Start":"17:29.715 ","End":"17:38.775","Text":"Then our v_2 at time t=3 is going to be negative 3 multiplied by 3 is negative 9,"},{"Start":"17:38.775 ","End":"17:44.309","Text":"negative 9 minus 1 is equal to negative 10 in the I direction."},{"Start":"17:44.309 ","End":"17:49.725","Text":"Plus 3 plus 4 is 7 in the j direction."},{"Start":"17:49.725 ","End":"17:58.020","Text":"Great. Now we have the vectors v_1 and v_2 at time t=3."},{"Start":"17:58.020 ","End":"18:04.275","Text":"Now what we want to do is we want to find the angle between these 2 vectors."},{"Start":"18:04.275 ","End":"18:08.145","Text":"What we\u0027re going to do is we\u0027re going to use our equation."},{"Start":"18:08.145 ","End":"18:10.350","Text":"Cosine over angle Alpha,"},{"Start":"18:10.350 ","End":"18:12.345","Text":"which is angle between the 2 vectors,"},{"Start":"18:12.345 ","End":"18:19.200","Text":"is equal to v_1.v_2 divided by"},{"Start":"18:19.200 ","End":"18:27.580","Text":"the size of our v_1 vector multiplied by the size of our v_2 vector."},{"Start":"18:27.860 ","End":"18:31.875","Text":"Now, of course, when we\u0027re using this equation,"},{"Start":"18:31.875 ","End":"18:34.680","Text":"because specifically in this question they asked"},{"Start":"18:34.680 ","End":"18:39.720","Text":"the angle between these 2 vectors at a specific time t=3,"},{"Start":"18:39.720 ","End":"18:45.285","Text":"we\u0027re calculating our v_1 vector and v_2 vector t=3."},{"Start":"18:45.285 ","End":"18:48.375","Text":"I\u0027m just going to write that over here, t=3."},{"Start":"18:48.375 ","End":"18:54.125","Text":"If we weren\u0027t told at a specific time and it was just at any time t,"},{"Start":"18:54.125 ","End":"18:57.380","Text":"we would do this exact same equation."},{"Start":"18:57.380 ","End":"19:04.200","Text":"However, we would be using these vector equations where t is just some"},{"Start":"19:04.200 ","End":"19:07.836","Text":"unknown and we\u0027ll do the exact same process that we\u0027re about to do now"},{"Start":"19:07.836 ","End":"19:11.640","Text":"but here specifically because we\u0027re being asked at t=3,"},{"Start":"19:11.640 ","End":"19:18.405","Text":"so we\u0027re going to be using this equation with these 2 vector quantities."},{"Start":"19:18.405 ","End":"19:24.015","Text":"The first thing that we\u0027re going to do is let\u0027s find the size of our v_1 vector."},{"Start":"19:24.015 ","End":"19:26.520","Text":"As we know that\u0027s using Pythagoras,"},{"Start":"19:26.520 ","End":"19:32.340","Text":"we\u0027re going to take the square root of this component,"},{"Start":"19:32.340 ","End":"19:33.480","Text":"our I component squared,"},{"Start":"19:33.480 ","End":"19:38.490","Text":"so negative 7 squared is simply 49 plus our j components squared,"},{"Start":"19:38.490 ","End":"19:41.530","Text":"which 8^2 is 64."},{"Start":"19:41.570 ","End":"19:49.020","Text":"Then that\u0027s simply going to be equal to the square root of 113."},{"Start":"19:49.020 ","End":"19:54.060","Text":"Now let\u0027s work out the size of our v_2 vector, t=3."},{"Start":"19:54.060 ","End":"19:57.450","Text":"Again, Pythagoras, so the I component squared,"},{"Start":"19:57.450 ","End":"20:01.680","Text":"so negative 10^2 is 100 plus the j component squared,"},{"Start":"20:01.680 ","End":"20:05.325","Text":"so 7 squared is 49."},{"Start":"20:05.325 ","End":"20:12.290","Text":"That is simply going to be equal to the square root of 149."},{"Start":"20:12.290 ","End":"20:17.270","Text":"Now let\u0027s work out what cosine of Alpha is."},{"Start":"20:17.270 ","End":"20:22.160","Text":"The cosine of the angle between our vectors v_1 and v_2 at t=3."},{"Start":"20:22.160 ","End":"20:24.995","Text":"We\u0027ll have v_1 at t=3."},{"Start":"20:24.995 ","End":"20:28.460","Text":"product with v_2 at t=3."},{"Start":"20:28.460 ","End":"20:34.709","Text":"We\u0027ll have negative 7 multiplied by negative"},{"Start":"20:34.709 ","End":"20:40.350","Text":"10 plus our j components multiplied 1 by another,"},{"Start":"20:40.350 ","End":"20:46.555","Text":"sorry, plus 8 multiplied by 7."},{"Start":"20:46.555 ","End":"20:53.720","Text":"Then that\u0027s going to be divided by the size of the 2 vectors multiplied by each other."},{"Start":"20:53.720 ","End":"21:01.015","Text":"That\u0027s the square root of 113 multiplied by the square root of 149."},{"Start":"21:01.015 ","End":"21:06.765","Text":"We\u0027ll have negative 7 multiplied by negative 10 is 70"},{"Start":"21:06.765 ","End":"21:12.680","Text":"plus 8 times 7 is 42 divided by this denominator."},{"Start":"21:12.680 ","End":"21:15.200","Text":"I\u0027m just going to plug that into my calculator."},{"Start":"21:15.200 ","End":"21:21.475","Text":"I\u0027ll get the cosine of Alpha is approximately equal to 0.97."},{"Start":"21:21.475 ","End":"21:25.715","Text":"Now, in order to find what my Alpha is equal to,"},{"Start":"21:25.715 ","End":"21:29.570","Text":"my Alpha will be equal to cosine to the minus 1,"},{"Start":"21:29.570 ","End":"21:33.395","Text":"or I cos of 0.97,"},{"Start":"21:33.395 ","End":"21:36.635","Text":"which is going to be approximately equal to,"},{"Start":"21:36.635 ","End":"21:42.215","Text":"once I plug this into the calculator, 13.82 degrees."},{"Start":"21:42.215 ","End":"21:46.910","Text":"That\u0027s the answer to question number 3,"},{"Start":"21:46.910 ","End":"21:49.770","Text":"and that is the end of this lesson."}],"ID":10791}],"Thumbnail":null,"ID":5404}]

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