WEBVTT 00:00:00.000 --> 00:00:02.860 This is Giancoli Answers with Mr. Dychko. 00:00:05.013 --> 00:00:07.053 We have that the centripetal force 00:00:07.050 --> 00:00:09.306 on the satellite is provided by gravity. 00:00:09.613 --> 00:00:12.453 So, m v squared over r equals 00:00:12.693 --> 00:00:15.693 g times mass of the satellite times mass of the Earth over 00:00:15.690 --> 00:00:18.346 distance from the center of the Earth to the satellite squared. 00:00:19.020 --> 00:00:20.586 Now, the distance from the center of the Earth 00:00:20.580 --> 00:00:22.426 is gonna be the radius of the Earth 00:00:22.640 --> 00:00:25.213 plus this distance above the surface of the Earth. 00:00:25.320 --> 00:00:29.266 So, we'll call that delta r 1 for the satellite 1, the close one, 00:00:29.426 --> 00:00:33.000 and that's 7500 kilometers, above the surface. 00:00:33.000 --> 00:00:36.386 And then delta r 2 is 15000 kilometers, above the surface. 00:00:37.040 --> 00:00:40.346 And we can do some algebra on this to solve for 00:00:40.340 --> 00:00:41.906 what the speed should be. 00:00:42.453 --> 00:00:44.973 We can multiply both sides by r, 00:00:45.426 --> 00:00:47.813 and that turns this into r to the power of 1. 00:00:48.080 --> 00:00:50.586 And divide both sides by mass of the satellite 00:00:50.580 --> 00:00:53.053 and then take the square root of both sides to solve for v. 00:00:53.093 --> 00:00:54.880 And so we have v, for the close satellite, is 00:00:54.880 --> 00:00:58.160 square root G 00:01:02.533 --> 00:01:04.746 Well, it looks like we made a slight mistake here 00:01:04.740 --> 00:01:07.093 but it doesn't actually affect the answer which is interesting. 00:01:07.490 --> 00:01:10.413 So, this mass of the satellite disappears 00:01:10.410 --> 00:01:12.800 because it ends up canceling down here. 00:01:12.946 --> 00:01:13.826 Anyway. 00:01:13.820 --> 00:01:16.640 So, we have this is correct. 00:01:18.973 --> 00:01:20.920 We have, in the first case, 00:01:20.920 --> 00:01:23.026 the speed of the satellite is square root G 00:01:23.020 --> 00:01:24.426 mass of the Earth divided by 00:01:24.420 --> 00:01:27.173 the total distance from the center of the Earth and the satellite. 00:01:27.170 --> 00:01:30.093 So that's radius of the Earth plus this distance above the surface. 00:01:30.200 --> 00:01:33.426 And then, for satellite 2, it's the same idea with delta r 2. 00:01:34.120 --> 00:01:37.160 And then taking the ratio of those velocities, 00:01:37.413 --> 00:01:40.066 you go v 1 divided by v 2. 00:01:40.253 --> 00:01:44.186 We can think of the square root as, on the top and on the bottom, 00:01:44.933 --> 00:01:50.120 separately and then it becomes more clear that this thing cancels with this thing 00:01:50.120 --> 00:01:56.080 although neither of them contain the mass of the satellite as it turns out but either way. 00:01:58.160 --> 00:02:01.866 So, we are doing this division 00:02:01.860 --> 00:02:04.773 of a fraction is the same as multiplying by its reciprocal 00:02:04.770 --> 00:02:06.520 is what else is going on here. 00:02:06.520 --> 00:02:10.333 So, dividing by v 2 is the same as multiplying by 00:02:10.466 --> 00:02:12.040 the reciprocal of v 2. 00:02:12.040 --> 00:02:17.186 So multiplying by the denominator put on top and the 00:02:17.180 --> 00:02:19.520 numerator is put on the bottom. 00:02:20.940 --> 00:02:26.586 And then these square root G mass of the Earth factors, cancel 00:02:26.813 --> 00:02:29.106 and we are left with the ratio of the velocities 00:02:29.293 --> 00:02:31.866 is the same as the square root of 00:02:32.160 --> 00:02:35.360 the distance from the center of the Earth, 00:02:36.066 --> 00:02:37.106 for each of these. 00:02:37.306 --> 00:02:38.400 So square root of 00:02:38.560 --> 00:02:41.626 6.38 times 10 to the 6 meters—radius of the Earth—plus 00:02:41.880 --> 00:02:44.773 the distance above the surface of the Earth for the 00:02:45.400 --> 00:02:49.120 far satellite, which is 15000 times 10 to the 3 meters 00:02:49.333 --> 00:02:51.826 divided by 6.38 times 10 to the 6 00:02:52.040 --> 00:02:54.786 plus 7500 times 10 to the 3 meters 00:02:54.780 --> 00:02:56.306 and that gives 1.2. 00:02:56.426 --> 00:02:58.200 So, the close satellite is 00:02:58.200 --> 00:03:00.880 1.2 times as fast as the far satellite 00:03:00.880 --> 00:03:02.533 because we could rewrite this as 00:03:02.533 --> 00:03:05.520 v 1 equals 1.2 times v 2. 00:03:06.600 --> 00:03:10.893 And so there's an algebraic way of saying that, 00:03:11.213 --> 00:03:14.693 or arithmetic way of saying that v 1 is faster than v 2. 00:03:14.690 --> 00:03:16.880 But you can also think of it in a different way. 00:03:16.880 --> 00:03:19.386 You can say the close satellite has to be faster because 00:03:19.533 --> 00:03:23.280 since the acceleration due to gravity is greater here, closer to the Earth, 00:03:23.506 --> 00:03:25.973 but it means, it has more of a tendency to 00:03:25.970 --> 00:03:28.360 go down; it's falling faster, 00:03:28.360 --> 00:03:30.933 and so it has to scoot to the side faster 00:03:30.930 --> 00:03:32.920 in order to avoid colliding with the Earth. 00:03:32.920 --> 00:03:34.866 That's another way to think about it.