English subtitles for clip: File:Prof Walter Lewin Elastic Collisions.ogv
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1 00:00:01,000 --> 00:00:05,000 All right, last time we talked exclusively 2 00:00:05,000 --> 00:00:07,000 about completely inelastic collisions. 3 00:00:07,000 --> 00:00:09,000 Today I will talk 4 00:00:09,000 --> 00:00:12,000 about collisions in more general terms. 5 00:00:12,000 --> 00:00:15,000 Let's take a one-dimensional case. 6 00:00:15,000 --> 00:00:22,000 We have here m1 and we have here m2, 7 00:00:22,000 --> 00:00:29,000 and to make life a little easy, we'll make v2 zero 8 00:00:29,000 --> 00:00:35,000 and this particle has velocity v1. 9 00:00:35,000 --> 00:00:44,000 After the collision, m2 has a velocity v2 prime, 10 00:00:44,000 --> 00:00:52,000 and m1, let it have a velocity v1 prime. 11 00:00:52,000 --> 00:00:54,000 I don't even know whether it's in this direction 12 00:00:54,000 --> 00:00:55,000 or whether it is in that direction. 13 00:00:55,000 --> 00:00:59,000 You will see that either one is possible. 14 00:00:59,000 --> 00:01:02,000 To find v1 prime and to find v2 prime, 15 00:01:02,000 --> 00:01:06,000 it's clear that you now need two equations. 16 00:01:06,000 --> 00:01:11,000 And if there is no net external force on the system as a whole 17 00:01:11,000 --> 00:01:15,000 during the collisions, then momentum is conserved. 18 00:01:15,000 --> 00:01:17,000 And so you can write down 19 00:01:17,000 --> 00:01:27,000 that m1 v1 must be m1 v1 prime plus m2 v2 prime. 20 00:01:27,000 --> 00:01:29,000 Now, you may want to put arrows over there 21 00:01:29,000 --> 00:01:31,000 to indicate that these are vectors, 22 00:01:31,000 --> 00:01:33,000 but since it's a one-dimensional case, 23 00:01:33,000 --> 00:01:34,000 you can leave the arrows off 24 00:01:34,000 --> 00:01:36,000 and the signs will then automatically take care 25 00:01:36,000 --> 00:01:38,000 of the direction. 26 00:01:38,000 --> 00:01:41,000 If you call this plus, then if you get a minus sign, 27 00:01:41,000 --> 00:01:46,000 you know that the velocity is in the opposite direction. 28 00:01:46,000 --> 00:01:52,000 So now we need a second equation. 29 00:01:52,000 --> 00:01:54,000 Now, in physics 30 0:01:54,000 --> 00:01:57,000 we do believe very strongly in the conservation of energy, 31 00:01:57,000 --> 00:02:01,000 not necessarily in the conservation of kinetic energy. 32 00:02:01,000 --> 00:02:04,000 As you have seen last time, you can destroy kinetic energy. 33 00:02:04,000 --> 00:02:07,000 But somehow we believe that if you destroy energy, 34 00:02:07,000 --> 00:02:09,000 it must come out in some other form, 35 00:02:09,000 --> 00:02:12,000 and you cannot create energy out of nothing. 36 00:02:12,000 --> 00:02:15,000 And in the case of the completely inelastic collisions 37 00:02:15,000 --> 00:02:16,000 that we have seen last time, 38 00:02:16,000 --> 00:02:19,000 we lost kinetic energy, which was converted to heat. 39 00:02:19,000 --> 00:02:21,000 There was internal friction. 40 00:02:21,000 --> 00:02:23,000 When the car wreck plowed into each other, 41 00:02:23,000 --> 00:02:26,000 there was internal friction-- no external friction-- 42 00:02:26,000 --> 00:02:29,000 and that took out kinetic energy. 43 00:02:29,000 --> 00:02:35,000 And so, in its most general form, you can write down 44 00:02:35,000 --> 00:02:39,000 that the kinetic energy before the collision 45 00:02:39,000 --> 00:02:40,000 plus some number Q 46 00:02:40,000 --> 00:02:43,000 equals the kinetic energy after the collision. 47 00:02:43,000 --> 00:02:46,000 And if you know Q, then you have a second equation, 48 00:02:46,000 --> 00:02:51,000 and then you can solve for v1 prime and for v2 prime. 49 00:02:51,000 --> 00:02:58,000 If Q is larger than zero, then you have gained kinetic energy. 50 00:02:58,000 --> 00:03:00,000 That is possible; we did that last time. 51 00:03:00,000 --> 00:03:03,000 We had two cars which were connected by a spring, 52 00:03:03,000 --> 00:03:05,000 and we burned the wire 53 00:03:05,000 --> 00:03:08,000 and each went off in the opposite direction. 54 00:03:08,000 --> 00:03:09,000 There was no kinetic energy 55 00:03:09,000 --> 00:03:11,000 before... if you want to call it the collision, 56 00:03:11,000 --> 00:03:13,000 but there was kinetic energy afterwards. 57 00:03:13,000 --> 00:03:15,000 That was the potential energy of the spring 58 00:03:15,000 --> 00:03:17,000 that was converted into kinetic energy. 59 00:03:17,000 --> 00:03:19,000 So Q can be larger than zero. 60 00:03:19,000 --> 00:03:24,000 We call that a superelastic collision. 61 00:03:24,000 --> 00:03:26,000 It could be an explosion. 62 00:03:26,000 --> 00:03:28,000 That's a superelastic collision. 63 00:03:28,000 --> 00:03:31,000 And then there is the possibility that Q equals zero, 64 00:03:31,000 --> 00:03:32,000 a very special case. 65 00:03:32,000 --> 00:03:34,000 We will deal with that today, 66 00:03:34,000 --> 00:03:37,000 and we call that an elastic collision. 67 00:03:37,000 --> 00:03:41,000 I will often call it a completely elastic collision, 68 00:03:41,000 --> 00:03:44,000 which is really not necessary. 69 00:03:44,000 --> 00:03:47,000 "Elastic" itself already means Q is zero. 70 00:03:47,000 --> 00:03:49,000 And then there is a case-- 71 00:03:49,000 --> 00:03:52,000 of which we have seen several examples last time-- 72 00:03:52,000 --> 00:03:55,000 of inelastic collisions, when you lose kinetic energy, 73 00:03:55,000 --> 00:04:01,000 so this is an inelastic collision. 74 00:04:01,000 --> 00:04:03,000 And so, if you know what Q is, 75 00:04:03,000 --> 00:04:05,000 then you can solve these equations. 76 00:04:05,000 --> 00:04:07,000 Whenever Q is less than zero, 77 00:04:07,000 --> 00:04:10,000 whenever you lose kinetic energy, 78 00:04:10,000 --> 00:04:14,000 the loss, in general, goes into heat. 79 00:04:14,000 --> 00:04:17,000 Now I want to continue a case 80 00:04:17,000 --> 00:04:22,000 whereby I have a completely elastic collision. 81 00:04:22,000 --> 00:04:24,000 So Q is zero. 82 00:04:24,000 --> 00:04:28,000 Momentum is conserved, because there was no net external force, 83 00:04:28,000 --> 00:04:31,000 so now kinetic energy is also conserved. 84 00:04:31,000 --> 00:04:35,000 And so I can write down now one-half m1 v1 squared-- 85 00:04:35,000 --> 00:04:38,000 that was the kinetic energy before the collision-- 86 00:04:38,000 --> 00:04:42,000 must be the kinetic energy after the collision 87 00:04:42,000 --> 00:04:45,000 one-half m1 v1 prime squared 88 00:04:45,000 --> 00:04:51,000 plus one-half m2 v2 prime squared. 89 00:04:51,000 --> 00:04:53,000 This is my equation number one, 90 00:04:53,000 --> 00:04:55,000 and this is my equation number two. 91 00:04:55,000 --> 00:04:57,000 And they can be solved; you can solve them. 92 00:04:57,000 --> 00:04:59,000 They are solved in your book. 93 00:04:59,000 --> 00:05:01,000 I will simply give you the results, 94 00:05:01,000 --> 00:05:03,000 because the results are very interesting to play with. 95 00:05:03,000 --> 00:05:05,000 That's what we will be doing today. 96 00:05:05,000 --> 00:05:19,000 v1 prime will be m1 minus m2 divided by m1 plus m2 times v1 97 00:05:19,000 --> 00:05:31,000 and v2 prime will be 2 m1 divided by m1 plus m2 times v1. 98 00:05:31,000 --> 00:05:34,000 The first thing that you already see right away 99 00:05:34,000 --> 00:05:38,000 is that v2 prime is always in the same direction as v1. 100 00:05:38,000 --> 00:05:39,000 That's completely obvious, 101 00:05:39,000 --> 00:05:43,000 because the second object was standing still, remember? 102 00:05:43,000 --> 00:05:46,000 So if you plow something into the second object, 103 00:05:46,000 --> 00:05:48,000 they obviously continue in that direction. 104 00:05:48,000 --> 00:05:49,000 That's clear. 105 00:05:49,000 --> 00:05:52,000 So you see you can never have a sign reversal here. 106 00:05:52,000 --> 00:05:55,000 Here, however, you can have a sign reversal. 107 00:05:55,000 --> 00:05:57,000 If you bounce a ping-pong ball off a billiard ball, 108 00:05:57,000 --> 00:05:59,000 the ping-pong ball will come back 109 00:05:59,000 --> 00:06:01,000 and this one becomes negative, 110 00:06:01,000 --> 00:06:04,000 whereas if you plow a billiard ball into a ping-pong ball, 111 00:06:04,000 --> 00:06:05,000 it will go forward. 112 00:06:05,000 --> 00:06:08,000 And so this can both be negative and can be positive 113 00:06:08,000 --> 00:06:11,000 depending upon whether the upstairs is negative 114 00:06:11,000 --> 00:06:13,000 or positive. 115 00:06:13,000 --> 00:06:16,000 So this is the result which holds under three conditions: 116 00:06:16,000 --> 00:06:20,000 that the kinetic energy is conserved, so Q is zero; 117 00:06:20,000 --> 00:06:21,000 that momentum is conserved; 118 00:06:21,000 --> 00:06:27,000 and that v2 before the collision equals zero. 119 00:06:27,000 --> 00:06:33,000 Let's look at three interesting cases whereby we go to extremes. 120 00:06:33,000 --> 00:06:35,000 And let's first take the case 121 00:06:35,000 --> 00:06:40,000 that m1 is much, much larger than m2. 122 00:06:40,000 --> 00:06:43,000 m1 is much, much larger than m2. 123 00:06:43,000 --> 00:06:50,000 Another way of thinking about that is that let m2 go to zero. 124 00:06:50,000 --> 00:06:53,000 Extreme case, the limiting case. 125 00:06:53,000 --> 00:06:56,000 So it's like having a bowling ball 126 00:06:56,000 --> 00:06:59,000 that you collide with a ping-pong ball. 127 00:06:59,000 --> 00:07:05,000 If you look at that equation when m2 goes to zero-- 128 00:07:05,000 --> 00:07:12,000 this is zero, this is zero-- notice that v1 prime equals v1. 129 00:07:12,000 --> 00:07:14,000 That is completely intuitive. 130 00:07:14,000 --> 00:07:17,000 If a bowling ball collides with a ping-pong ball 131 00:07:17,000 --> 00:07:19,000 the bowling ball doesn't even see the ping-pong ball. 132 00:07:19,000 --> 00:07:22,000 It continues its route as if nothing happened. 133 00:07:22,000 --> 00:07:23,000 That's exactly what you see. 134 00:07:23,000 --> 00:07:28,000 After the collision, the bowling ball continues unaltered. 135 00:07:28,000 --> 00:07:30,000 What is v2 prime? 136 00:07:30,000 --> 00:07:32,000 That is not so intuitive. 137 00:07:32,000 --> 00:07:38,000 If you substitute in there m2 equals zero, 138 00:07:38,000 --> 00:07:46,000 then you get plus 2 v1-- not obvious at all, plus 2 v1. 139 00:07:46,000 --> 00:07:48,000 It's not something I even want you to see; 140 00:07:48,000 --> 00:07:49,000 I can't see it either. 141 00:07:49,000 --> 00:07:50,000 I'll do a demonstration. 142 00:07:50,000 --> 00:07:52,000 You can see that it really happens. 143 00:07:52,000 --> 00:07:55,000 So, now you take a bowling ball 144 00:07:55,000 --> 00:07:59,000 and you collide the bowling ball with the ping-pong ball 145 00:07:59,000 --> 00:08:04,000 and the ping-pong ball will get a velocity 2 v1-- 146 00:08:04,000 --> 00:08:05,000 not more, not less-- 147 00:08:05,000 --> 00:08:09,000 and the bowling ball continues at the same speed. 148 00:08:09,000 --> 00:08:14,000 Now let's take a case whereby m1 equals much, much less than m2; 149 00:08:14,000 --> 00:08:18,000 in other words, in the limiting case, m1 goes to zero. 150 00:08:18,000 --> 00:08:20,000 And we substitute that in here. 151 00:08:20,000 --> 00:08:24,000 So m1 goes to zero, so this is zero 152 00:08:24,000 --> 00:08:29,000 and so you see v1 prime equals minus v1. 153 00:08:29,000 --> 00:08:33,000 v1 prime equals minus v1, completely obvious. 154 00:08:33,000 --> 00:08:36,000 The ping-pong ball bounces off the bowling ball 155 00:08:36,000 --> 00:08:37,000 and it just bounces back. 156 00:08:37,000 --> 00:08:39,000 And this is what you see. 157 00:08:39,000 --> 00:08:41,000 And the bowling ball doesn't do anything, 158 00:08:41,000 --> 00:08:46,000 because m1 goes to zero, so v2 prime goes to zero. 159 00:08:46,000 --> 00:08:48,000 […] 160 00:08:48,000 --> 00:08:50,000 So that's very intuitive. 161 00:08:50,000 --> 00:08:55,000 And now we have a very cute case that m1 equals m2. 162 00:08:55,000 --> 00:08:58,000 And when you substitute that in here-- 163 00:08:58,000 --> 00:09:03,000 when m1 equals m2-- v1 prime becomes zero. 164 00:09:03,000 --> 00:09:10,000 So the first one stops with v2 prime becomes v1. 165 00:09:10,000 --> 00:09:13,000 If m1 equals m2, 166 00:09:13,000 --> 00:09:15,000 you have two downstairs here and two upstairs 167 00:09:15,000 --> 00:09:21,000 and you see that v2 prime equals v1. 168 00:09:21,000 --> 00:09:22,000 And that is a remarkable case-- 169 00:09:22,000 --> 00:09:25,000 you've all seen that, you've all played with Newton's cradle. 170 00:09:25,000 --> 00:09:26,000 You have two billiard balls. 171 00:09:26,000 --> 00:09:29,000 One is still and the other one bangs on it. 172 00:09:29,000 --> 00:09:30,000 The first one stops 173 00:09:30,000 --> 00:09:33,000 and the second one takes off with the speed of the first. 174 00:09:33,000 --> 00:09:34,000 An amazing thing. 175 00:09:34,000 --> 00:09:36,000 We've all seen it. 176 00:09:36,000 --> 00:09:37,000 I presume you have all seen it. 177 00:09:37,000 --> 00:09:39,000 Most people do this with pendulums 178 00:09:39,000 --> 00:09:42,000 where they bounce these balls against each other. 179 00:09:42,000 --> 00:09:44,000 I will do it here with a model 180 00:09:44,000 --> 00:09:46,000 that you can see a little easier. 181 00:09:46,000 --> 00:09:48,000 I have here billiard balls, 182 00:09:48,000 --> 00:09:50,000 and if I bounce this one on this one, 183 00:09:50,000 --> 00:09:52,000 then we have case number three. 184 00:09:52,000 --> 00:09:54,000 Then you see this one stands still 185 00:09:54,000 --> 00:09:58,000 and this one takes over the speed-- quite amazing. 186 00:09:58,000 --> 00:10:00,000 Every time I see this, I love it.