WEBVTT 00:00:00.000 --> 00:00:03.160 This is College Physics Answers with Shaun Dychko. 00:00:04.100 --> 00:00:07.260 A ball is dropped from a height of 1.5 meters 00:00:07.260 --> 00:00:11.780 and it achieves some maximum velocity which we'll call v 1— 00:00:11.780 --> 00:00:13.620 just before it hits the ground here— 00:00:13.700 --> 00:00:16.800 and we'll figure out what that velocity is in part (a). 00:00:17.180 --> 00:00:18.940 And then it rebounds 00:00:18.940 --> 00:00:21.960 with a velocity which we'll call v 1 prime 00:00:21.960 --> 00:00:26.660 keeping the subscript 1 because it's a velocity at the same position here 00:00:26.760 --> 00:00:28.700 but this velocity is different because 00:00:28.720 --> 00:00:33.400 the bounce is not perfectly efficient and so some energy is lost 00:00:33.400 --> 00:00:39.380 and so this rebound velocity, v 1 prime, is going to be less than v 1 was 00:00:39.380 --> 00:00:41.380 when it was going downwards. 00:00:42.080 --> 00:00:46.160 And then it will reach some maximum height of 1.10 meters 00:00:46.160 --> 00:00:50.520 which we'll call position 2 and it will have some velocity 2 00:00:50.520 --> 00:00:54.840 of zero there as everything does when it reaches a maximum height, 00:00:54.840 --> 00:00:57.180 velocity is always zero at the very maximum 00:00:57.180 --> 00:01:01.600 of an arc or of something just going through the air. 00:01:02.180 --> 00:01:06.320 So we find v 1 going downwards here in part (a); 00:01:06.320 --> 00:01:09.720 in part (b), we find v 1 prime, the rebound velocity; 00:01:09.720 --> 00:01:13.440 in part (c), we find the acceleration of this tennis ball 00:01:13.440 --> 00:01:15.640 when it's in contact with the ground 00:01:15.640 --> 00:01:21.380 and then in part (d), we figure out how much the tennis ball is compressed. 00:01:21.520 --> 00:01:26.060 So it starts with some spherical diameter here 00:01:26.300 --> 00:01:28.360 and gets compressed into this ellipse 00:01:28.420 --> 00:01:31.520 and we'll figure out how much that compression is. 00:01:32.040 --> 00:01:33.240 Okay. 00:01:34.400 --> 00:01:37.920 So we are going to figure out what is velocity v 1 then 00:01:37.920 --> 00:01:42.020 and we know that v 1 squared equals v initial squared 00:01:42.020 --> 00:01:44.740 minus 2 times acceleration due to gravity 00:01:44.740 --> 00:01:46.740 multiplied by the final height 00:01:46.740 --> 00:01:49.860 minus the initial height—that's equation 77— 00:01:50.820 --> 00:01:56.900 and we are gonna solve for v 1 by taking the square root of both sides 00:01:56.900 --> 00:02:00.400 and that is the plus or negative square root 00:02:00.740 --> 00:02:05.240 and we have to use our knowledge of the physical situation 00:02:05.240 --> 00:02:08.280 to decide whether it should be positive or negative. 00:02:08.500 --> 00:02:10.720 Now because the tennis ball is moving down 00:02:10.720 --> 00:02:13.480 and we usually take upwards to be the positive direction 00:02:13.480 --> 00:02:17.080 that means this velocity is going to be negative. 00:02:17.960 --> 00:02:19.660 So we choose the negative here 00:02:19.840 --> 00:02:23.320 and we have the negative square root of 0 meters per second squared— 00:02:23.320 --> 00:02:25.420 that's v naught is zero, it was dropped 00:02:25.520 --> 00:02:29.020 and so whenever the question says the word 'dropped' that means 00:02:29.020 --> 00:02:32.160 we are given no initial velocity— 00:02:32.580 --> 00:02:36.380 minus 2 times acceleration due to gravity, 9.80 meters per second squared, 00:02:36.380 --> 00:02:40.760 multiplied by the final height which is 0 meters at the ground level, y 1, 00:02:41.060 --> 00:02:44.720 minus the initial height of 1.50 meters 00:02:44.720 --> 00:02:48.820 and this works out to negative 5.42 meters per second. 00:02:50.360 --> 00:02:51.580 In part (b), 00:02:51.580 --> 00:02:55.440 we want to know the rebound velocity, v 1 prime, 00:02:56.240 --> 00:03:01.980 and we know that v 2 squared, the velocity at the very top of its rebound 00:03:01.980 --> 00:03:04.260 when it's at a height of 1.10 meters, 00:03:04.260 --> 00:03:08.680 v 2 squared equals the initial velocity, v 1 prime squared, 00:03:08.680 --> 00:03:13.720 minus 2g times the final height of y 2 minus the initial height of y 1 00:03:13.800 --> 00:03:18.020 so it gets to this final height, y 2, and it started down here at y 1. 00:03:20.560 --> 00:03:30.000 And we are gonna solve for v 1 prime by adding 2g y 2 minus y 1 to both sides 00:03:33.040 --> 00:03:37.080 and this v 2 is 0 so we are just gonna make that term disappear 00:03:37.080 --> 00:03:39.800 and switching the sides around, we have v 1 prime 00:03:39.800 --> 00:03:43.420 on the left here and we have also taken the square root of both sides 00:03:43.420 --> 00:03:48.380 and so that's why we have a square root of 2g times y 2 minus y 1 here. 00:03:49.520 --> 00:03:53.940 And we'll take the positive now because we can see that 00:03:53.940 --> 00:03:55.940 the rebound is going upwards 00:03:55.940 --> 00:03:57.900 so v 1 prime is positive. 00:03:59.160 --> 00:04:02.580 So we have positive square root of 2 times acceleration due to gravity 00:04:02.580 --> 00:04:05.020 multiplied by the final height of 1.10 meters 00:04:05.020 --> 00:04:07.680 minus the initial height at the ground of 0 meters 00:04:07.680 --> 00:04:12.640 giving us a velocity, v 1 prime, of 4.64 meters per second. 00:04:13.720 --> 00:04:17.440 The acceleration of the tennis ball, while it's in contact with the ground 00:04:17.440 --> 00:04:22.580 is the final velocity, which is the rebound velocity, 00:04:22.700 --> 00:04:27.500 minus the initial downward velocity just before it contacted the ground 00:04:27.740 --> 00:04:30.960 divided by the time it spends in contact with the ground 00:04:30.960 --> 00:04:33.660 which we are told is 3.5 milliseconds. 00:04:34.420 --> 00:04:38.460 So we have 4.6433 meters per second— rebound velocity— 00:04:38.500 --> 00:04:40.180 minus the negative— 00:04:40.180 --> 00:04:43.600 and it's important to notice that there's a minus from the formula 00:04:43.680 --> 00:04:49.940 and then a negative for the velocity v 1 together is gonna make a positive— 00:04:50.860 --> 00:04:56.080 5.4222 meters per second divided by 3.50 milliseconds written as times 00:04:56.080 --> 00:04:58.520 10 to the minus 3 seconds giving us 00:04:58.520 --> 00:05:02.040 2.88 times 10 to the 3 meters per second squared 00:05:02.040 --> 00:05:04.040 and that's a positive acceleration. 00:05:05.600 --> 00:05:06.740 Okay. 00:05:07.200 --> 00:05:08.660 So in part (d), 00:05:08.780 --> 00:05:11.440 we are going to figure out how much the tennis ball is compressed 00:05:11.440 --> 00:05:13.520 while it's in contact with the ground. 00:05:13.820 --> 00:05:19.180 So we consider this initial moment when it just begins to contact the ground 00:05:19.180 --> 00:05:20.520 and at this point, 00:05:20.880 --> 00:05:23.940 we are gonna consider this very top point of the ball 00:05:23.940 --> 00:05:29.380 and we'll follow it through to the point where the ball is fully compressed 00:05:29.460 --> 00:05:31.840 and at this point, 00:05:31.840 --> 00:05:37.300 when the ball is compressed, this point at the top of the ball will have zero velocity 00:05:38.020 --> 00:05:42.920 whereas initially, it has a velocity of negative 5.4222 meters per second 00:05:43.140 --> 00:05:49.140 and I'm calling this v naught, which is sort of a relabelling 00:05:49.140 --> 00:05:52.100 of what I called v 1 before 00:05:52.100 --> 00:05:57.140 but let's reset our subscripts here 00:05:57.140 --> 00:06:00.640 and call this v naught so that we can write our formulas 00:06:00.640 --> 00:06:02.700 the way we are used to seeing them. 00:06:03.140 --> 00:06:04.140 Okay. 00:06:04.540 --> 00:06:08.960 So this point has a velocity v naught downwards 00:06:08.960 --> 00:06:10.980 and it has an acceleration upwards 00:06:10.980 --> 00:06:14.540 due to being in contact with the ground 00:06:14.540 --> 00:06:18.280 and that acceleration is what we calculated in part (c). 00:06:19.280 --> 00:06:24.240 And then despite the acceleration upwards, it is nevertheless still going down because 00:06:24.240 --> 00:06:27.140 the acceleration has not been applied for long enough to 00:06:27.140 --> 00:06:28.660 reverse the direction of this point 00:06:28.700 --> 00:06:31.400 and so it's continuing down despite an acceleration upwards 00:06:31.400 --> 00:06:33.300 but it's slowing down very rapidly 00:06:33.300 --> 00:06:36.440 to the point where it finally gets to a velocity of zero 00:06:36.740 --> 00:06:39.740 and we want to know what is the difference 00:06:39.740 --> 00:06:43.500 between these two positions of the top of the ball; 00:06:43.560 --> 00:06:44.720 that will be our compression, 00:06:44.720 --> 00:06:49.560 that is this initial height of y naught minus the final height of y. 00:06:50.540 --> 00:06:54.640 So we have equation 54 which tells us that the final velocity squared equals 00:06:54.640 --> 00:06:58.360 the initial velocity squared plus 2 times acceleration 00:06:58.360 --> 00:07:02.560 times the final position minus the initial position 00:07:03.100 --> 00:07:05.660 and we are gonna solve for y minus y naught 00:07:05.920 --> 00:07:10.260 and we'll do that by subtracting v naught squared from both sides 00:07:10.260 --> 00:07:14.160 and then also noticing this term is zero because the final velocity is zero 00:07:14.160 --> 00:07:18.340 and we have, after switching the sides around, 00:07:18.340 --> 00:07:21.980 we have 2a times y minus y naught equals negative v naught squared. 00:07:22.300 --> 00:07:26.320 Then we are gonna divide both sides by negative 2a 00:07:27.320 --> 00:07:31.380 and so these negative make a positive so that's positive v naught squared over 2a 00:07:31.380 --> 00:07:33.400 and on the left side, 00:07:33.400 --> 00:07:36.140 this negative is gonna switch the signs of these two terms around 00:07:36.140 --> 00:07:37.880 so that we have y naught minus y 00:07:37.960 --> 00:07:42.220 and that's a bit nicer to write it this way because 00:07:42.220 --> 00:07:45.920 we know that that's going to give us a positive number 00:07:45.920 --> 00:07:51.100 and it's just nice to express our compression as a positive amount, a magnitude. 00:07:51.780 --> 00:07:55.360 So we have y naught minus y equals v naught squared over 2a 00:07:55.460 --> 00:07:59.940 so that's negative 5.4222 meters per second squared divided by 00:07:59.940 --> 00:08:02.580 2 times the acceleration that we calculated in part (c) 00:08:02.620 --> 00:08:08.020 and that gives 0.0051116 meters which is 5.11 millimeters. 00:08:08.100 --> 00:08:10.220 So it compresses about half a centimeter 00:08:10.280 --> 00:08:12.460 while it's in contact with the ground.