1 0:00:02 --> 00:00:08 Remember, earlier in the course 2 00:00:04 --> 00:00:10 we measured the average speed of a bullet 3 00:00:07 --> 00:00:13 which we fired from a rifle. 4 00:00:09 --> 00:00:15 That was because we had the ability of very fast timing. 5 00:00:14 --> 00:00:20 In the old days, fast timing was not possible 6 00:00:18 --> 00:00:24 and people measured the speed of bullets in a very delicate way 7 00:00:23 --> 00:00:29 and all the tools that we have learned we can apply now 8 00:00:28 --> 00:00:34 to this device which we call the ballistic pendulum. 9 00:00:32 --> 00:00:38 10 00:00:38 --> 00:00:44 We have a pendulum with a very heavy object 11 00:00:42 --> 00:00:48 hanging here at the end-- I call it the block. 12 00:00:45 --> 00:00:51 You see it here. 13 00:00:48 --> 00:00:54 And this pendulum has length L. 14 00:00:50 --> 00:00:56 Ours is about one meter. 15 00:00:52 --> 00:00:58 I will give you the exact numbers later. 16 00:00:54 --> 00:01:00 And we have a bullet of mass little m 17 00:00:58 --> 00:01:04 and the bullet comes in with velocity v. 18 00:01:02 --> 00:01:08 It gets completely absorbed, sticks in there. 19 00:01:06 --> 00:01:12 It's a completely inelastic collision, 20 00:01:09 --> 00:01:15 and the pendulum will then pick up the velocity v prime 21 00:01:14 --> 00:01:20 with the bullet inside. 22 00:01:16 --> 00:01:22 The bullet is somewhere here. 23 00:01:18 --> 00:01:24 24 00:01:21 --> 00:01:27 Momentum is conserved, so we clearly have that, 25 00:01:25 --> 00:01:31 m v equals (m plus M) times v prime. 26 00:01:35 --> 00:01:41 So if you could measure v prime, 27 00:01:37 --> 00:01:43 then you could measure the speed of the bullet, 28 00:01:41 --> 00:01:47 which is v. 29 00:01:42 --> 00:01:48 How do we measure v prime? 30 00:01:45 --> 00:01:51 Well, we wait for the pendulum to come to a halt, 31 00:01:51 --> 00:01:57 let's say here, when the speed is zero. 32 00:01:56 --> 00:02:02 When it was here, it had a speed v prime. 33 00:02:02 --> 00:02:08 And we know that there was kinetic energy here-- 34 00:02:07 --> 00:02:13 no gravitational potential energy. 35 00:02:11 --> 00:02:17 I can call this level U = 0, 36 00:02:14 --> 00:02:20 but right here, if this difference in height is h, 37 00:02:18 --> 00:02:24 then all the kinetic energy has been converted 38 00:02:22 --> 00:02:28 to gravitational potential energy. 39 00:02:25 --> 00:02:31 So we apply the theorem, the work-energy theorem, 40 00:02:31 --> 00:02:37 or you could say... and it's equally valid, you could say 41 00:02:34 --> 00:02:40 we applied the conservation of mechanical energy. 42 00:02:37 --> 00:02:43 And so this kinetic energy, 43 00:02:40 --> 00:02:46 which is one-half (m plus M) times v prime squared 44 00:02:45 --> 00:02:51 is now converted exclusively to gravitational potential energy, 45 00:02:52 --> 00:02:58 which equals (m plus M) times g times that h. 46 00:02:57 --> 00:03:03 And we lose our (m plus M), 47 00:03:00 --> 00:03:06 so v prime would be the square root of 2gh. 48 00:03:05 --> 00:03:11 And so all you would have to measure h, 49 00:03:07 --> 00:03:13 and then you know v prime, 50 00:03:08 --> 00:03:14 and if you know v prime, you know the speed of the bullet. 51 00:03:11 --> 00:03:17 But life is not that simple. 52 00:03:13 --> 00:03:19 It's very difficult to measure h, 53 00:03:16 --> 00:03:22 and I can make you see that. 54 00:03:18 --> 00:03:24 Suppose this angle-- angle theta-- when it comes to a halt 55 00:03:24 --> 00:03:30 is only two degrees. 56 00:03:26 --> 00:03:32 57 00:03:29 --> 00:03:35 Then h, which is L times (1 minus cosine theta), 58 00:03:34 --> 00:03:40 is only 0.6 millimeters 59 00:03:37 --> 00:03:43 for the dimensions that I have chosen here 60 00:03:43 --> 00:03:49 for a length of one meter. 61 00:03:45 --> 00:03:51 And you can't even see it-- it's invisible-- 62 00:03:47 --> 00:03:53 let alone that you can measure it to any degree of accuracy. 63 00:03:51 --> 00:03:57 So what are we going to do now? 64 00:03:55 --> 00:04:01 Well, we are going to not measure h, 65 00:03:59 --> 00:04:05 but we are going to measure x. 66 00:04:02 --> 00:04:08 I call this x = 0. 67 00:04:07 --> 00:04:13 And here, when the pendulum comes to a halt, I call that x. 68 00:04:13 --> 00:04:19 For a two-degree angle, 69 00:04:16 --> 00:04:22 x is approximately 3½ centimeters. 70 00:04:19 --> 00:04:25 It can easily be checked by you, of course. 71 00:04:23 --> 00:04:29 So you get a huge displacement in this direction compared to h. 72 00:04:28 --> 00:04:34 73 00:04:29 --> 00:04:35 If you use small angle approximation-- 74 00:04:32 --> 00:04:38 and you better believe that two degrees is very small-- 75 00:04:35 --> 00:04:41 then you can prove, which is purely geometrical mathematics-- 76 00:04:40 --> 00:04:46 and I leave you with that proof-- 77 00:04:42 --> 00:04:48 that this is approximately x squared divided by 2L. 78 00:04:48 --> 00:04:54 I want you to prove that. 79 00:04:49 --> 00:04:55 You take the expansion of the cosine, 80 00:04:52 --> 00:04:58 the theories of the Taylor series, 81 00:04:54 --> 00:05:00 and you cut it off somewhere, 82 00:04:56 --> 00:05:02 and this is not so difficult to prove. 83 00:04:58 --> 00:05:04 In other words, v prime squared, 84 00:05:03 --> 00:05:09 which is 2gh, can now be replaced 85 00:05:07 --> 00:05:13 by approximately 2g times x squared divided by 2L 86 00:05:14 --> 00:05:20 which is g times x squared divided by L. 87 00:05:18 --> 00:05:24 88 00:05:20 --> 00:05:26 And so the velocity of the bullet, v, 89 00:05:28 --> 00:05:34 which is (m plus M) divided by m-- 90 00:05:34 --> 00:05:40 I bring the m down there-- times v prime, 91 00:05:38 --> 00:05:44 but v prime is now the square root of this, 92 00:05:42 --> 00:05:48 so I get an x here times the square root of g over L. 93 00:05:46 --> 00:05:52 And what you see now, but in a very clever way, 94 00:05:49 --> 00:05:55 by getting x in here, we can now do a quite accurate measurement 95 00:05:53 --> 00:05:59 of the speed of this bullet, 96 00:05:55 --> 00:06:01 because we can measure x with a fair accuracy-- 97 00:05:57 --> 00:06:03 maybe an uncertainty of only one or two millimeters in x 98 00:06:01 --> 00:06:07 out of the 3½ or four or five centimeters. 99 00:06:03 --> 00:06:09 100 00:06:07 --> 00:06:13 We're going to measure the speed of such a bullet-- 101 00:06:10 --> 00:06:16 we did that before-- 102 00:06:11 --> 00:06:17 to get a number which is not too different. 103 00:06:14 --> 00:06:20 I think we got something like 200 to 250 meters per second. 104 00:06:18 --> 00:06:24 I will give you the input for this experiment. 105 00:06:22 --> 00:06:28 The bullet mass is 2 plus or minus 0.2 grams. 106 00:06:28 --> 00:06:34 I apologize that I don't give you all the numbers 107 00:06:30 --> 00:06:36 in mks units. 108 00:06:32 --> 00:06:38 You will have to convert them, of course, to mks units. 109 00:06:35 --> 00:06:41 The length of the pendulum 110 00:06:36 --> 00:06:42 is 1.13 meters plus or minus two centimeters-- 111 00:06:44 --> 00:06:50 we're not certain-- 112 00:06:46 --> 00:06:52 with an accuracy of about two centimeters. 113 00:06:49 --> 00:06:55 And the mass of that block, which is huge, 114 00:06:52 --> 00:06:58 I believe is 3,200 grams, 3.2 kilograms, 115 00:06:58 --> 00:07:04 with an uncertainty of about two grams. 116 00:07:04 --> 00:07:10 So we have a ten percent uncertainty in the mass, 117 00:07:11 --> 00:07:17 we have a two percent uncertainty in the length, 118 00:07:15 --> 00:07:21 and we have a negligible uncertainty 119 00:07:17 --> 00:07:23 in the mass of the block-- that's negligibly small. 120 00:07:21 --> 00:07:27 So if I want to know now what the velocity is 121 00:07:24 --> 00:07:30 of the speed of the bullet, 122 00:07:26 --> 00:07:32 I can calculate what (m plus M) divided by m is 123 00:07:29 --> 00:07:35 and I can calculate the square root of g over L, 124 00:07:33 --> 00:07:39 and with those numbers 125 00:07:34 --> 00:07:40 I find 4.7 times ten to the third times x. 126 00:07:37 --> 00:07:43 We have to measure x. 127 00:07:39 --> 00:07:45 128 00:07:40 --> 00:07:46 Now if you look at the uncertainties 129 00:07:43 --> 00:07:49 in the whole thing, we're going to measure x, 130 00:07:45 --> 00:07:51 which may not be 3½ centimeters, it may be four or five-- 131 00:07:49 --> 00:07:55 but the uncertainty in our measurement 132 00:07:51 --> 00:07:57 will probably be one or two millimeters. 133 00:07:53 --> 00:07:59 Let's say it's 0.2 centimeters. 134 00:07:55 --> 00:08:01 So this is not our real number yet. 135 00:07:58 --> 00:08:04 So that would be an uncertainty of about two out of 50, 136 00:08:01 --> 00:08:07 is four out of a hundred, is four percent. 137 00:08:04 --> 00:08:10 This is four percent, roughly four percent. 138 00:08:07 --> 00:08:13 139 00:08:08 --> 00:08:14 So I would say we get the final speed of the bullet 140 00:08:13 --> 00:08:19 to an accuracy of about 15% 141 00:08:16 --> 00:08:22 if we combine all the uncertainties. 142 00:08:18 --> 00:08:24 143 00:08:20 --> 00:08:26 So let's give it a shot, no pun implied. 144 00:08:25 --> 00:08:31 And we'll... we can make you see 145 00:08:29 --> 00:08:35 the pendulum right there very shortly. 146 00:08:34 --> 00:08:40 And it is very cleverly designed. 147 00:08:38 --> 00:08:44 When the pendulum starts to swing, 148 00:08:42 --> 00:08:48 it's going to move a small object. 149 00:08:46 --> 00:08:52 150 00:08:53 --> 00:08:59 There you have it. 151 00:08:54 --> 00:09:00 I think I can turn this on again. 152 00:08:57 --> 00:09:03 I think that's fine. 153 00:08:59 --> 00:09:05 There is a very small wiper, which is this black wiper, 154 00:09:03 --> 00:09:09 and as the pendulum swings-- 155 00:09:05 --> 00:09:11 this is five centimeters, this is ten centimeters-- 156 00:09:09 --> 00:09:15 the wiper will stay at the largest extension 157 00:09:12 --> 00:09:18 when the pendulum swings back. 158 00:09:15 --> 00:09:21 Okay. 159 00:09:17 --> 00:09:23 When we fire bullets, it's always a little risky. 160 00:09:21 --> 00:09:27 I have here the... the bolt. 161 00:09:26 --> 00:09:32 162 00:09:29 --> 00:09:35 Put the bolt in. 163 00:09:30 --> 00:09:36 The bullet's in my pocket-- there's one. 164 00:09:34 --> 00:09:40 That's right. 165 00:09:35 --> 00:09:41 I take the bullet. 166 00:09:37 --> 00:09:43 And I cock the gun. 167 00:09:40 --> 00:09:46 Everything done right, yes? 168 00:09:43 --> 00:09:49 So you can look there and then see the swing 169 00:09:45 --> 00:09:51 of this very massive block, 170 00:09:47 --> 00:09:53 and the bullet will get absorbed in that block. 171 00:09:50 --> 00:09:56 172 00:09:52 --> 00:09:58 You ready for this? 173 00:09:54 --> 00:10:00 Three, two, one (fires ), zero. 174 00:09:57 --> 00:10:03 I would say 5.2 centimeters seems to be about right, 175 00:10:04 --> 00:10:10 so x observed is about 5.2 centimeters, 176 00:10:11 --> 00:10:17 and we know the uncertainty of the 15% already, 177 00:10:16 --> 00:10:22 so I have to calculate now the 4.7 times ten to the third. 178 00:10:22 --> 00:10:28 4.7 exponent third, multiplied by... 179 00:10:25 --> 00:10:31 I go mks, of course, so that this .052, 180 00:10:29 --> 00:10:35 and that is 244 meters per second. 181 00:10:32 --> 00:10:38 I remember last time we had something very similar. 182 00:10:36 --> 00:10:42 So the speed of the bullet is about 244 meters per second. 183 00:10:46 --> 00:10:52 It's a little under the speed of sound, 184 00:10:48 --> 00:10:54 which is 340 meters per second, 185 00:10:49 --> 00:10:55 and we came to the same conclusion last time. 186 00:10:52 --> 00:10:58 187 00:10:56 --> 00:11:02 All right. 188 00:10:57 --> 00:11:03 189 00:11:05 --> 00:11:11 We have kinetic energy in the bullet 190 00:11:07 --> 00:11:13 before the bullet hit the block, 191 00:11:09 --> 00:11:15 and you can calculate how much that is, 192 00:11:12 --> 00:11:18 because you know the speed now and you know the mass-- 193 00:11:16 --> 00:11:22 one-half mv squared. 194 00:11:18 --> 00:11:24 You can also calculate how much kinetic energy there is 195 00:11:22 --> 00:11:28 when this bullet is absorbed in here. 196 00:11:24 --> 00:11:30 That's very easy-- 197 00:11:26 --> 00:11:32 that's one-half times the total mass, (m plus M), 198 00:11:29 --> 00:11:35 times v prime squared, which you also know now. 199 00:11:33 --> 00:11:39 And so you will see then, perhaps to your surprise, 200 00:11:37 --> 00:11:43 that if you compare the two, 201 00:11:40 --> 00:11:46 that 99.94% of all available kinetic energy 202 00:11:43 --> 00:11:49 before the collision was destroyed, 203 00:11:46 --> 00:11:52 and therefore was converted to heat. 204 00:11:48 --> 00:11:54 That happened, of course... the heat was produced in that block. 205 00:11:52 --> 00:11:58 206 00:11:54 --> 00:12:00 Now I'm going to change to the concept of impulse. 207 00:11:58 --> 00:12:04 It's not completely unrelated to what we just did. 208 00:12:02 --> 00:12:08 An impulse is giving someone a kick, that's what an impulse is. 209 00:12:08 --> 00:12:14 Our bullet gave an impulse to this block, it gave it a kick. 210 00:12:13 --> 00:12:19 211 00:12:14 --> 00:12:20 Impulse. 212 00:12:18 --> 00:12:24 Impulse is a vector, 213 00:12:20 --> 00:12:26 and it is defined as the integral of F dt 214 00:12:25 --> 00:12:31 during a certain amount of time-- 215 00:12:29 --> 00:12:35 let's say from zero to delta t. 216 00:12:35 --> 00:12:41 Now, F equals ma, which is also dp/dt-- 217 00:12:43 --> 00:12:49 we have seen this now several times-- 218 00:12:46 --> 00:12:52 the rate of change of momentum-- 219 00:12:48 --> 00:12:54 and so I can substitute that in here, 220 00:12:52 --> 00:12:58 and so I find then the integral 221 00:12:56 --> 00:13:02 from zero to delta t of dp/dt dt, 222 00:13:00 --> 00:13:06 and that makes me move to the domain of momenta, 223 00:13:06 --> 00:13:12 so I have now simply the integral over dp 224 00:13:10 --> 00:13:16 from some initial momentum, pi, 225 00:13:14 --> 00:13:20 to some final momentum, pf. 226 00:13:17 --> 00:13:23 And so that is simply 227 00:13:20 --> 00:13:26 the final momentum minus the initial momentum. 228 00:13:26 --> 00:13:32 So what an impulse does, it changes the momentum. 229 00:13:33 --> 00:13:39 There is a force that acts on something 230 00:13:35 --> 00:13:41 for a short amount of time-- 231 00:13:38 --> 00:13:44 could be a little longer, as you will see with rockets-- 232 00:13:42 --> 00:13:48 and that gives it a change in momentum. 233 00:13:44 --> 00:13:50 If we have an object that we drop on the floor, 234 00:13:48 --> 00:13:54 so we have an object, mass m, and we drop it on the floor 235 00:13:52 --> 00:13:58 and we let it fall over a distance h, 236 00:13:55 --> 00:14:01 then it's going to hit the floor with a certain speed-- 237 00:14:00 --> 00:14:06 we know it's down, the velocity, 238 00:14:03 --> 00:14:09 and that equals the square root of 2gh. 239 00:14:07 --> 00:14:13 240 00:14:08 --> 00:14:14 If this were a completely elastic collision, 241 00:14:11 --> 00:14:17 which depends, of course, on the quality of the object, 242 00:14:15 --> 00:14:21 and it depends on the quality of the floor-- 243 00:14:18 --> 00:14:24 maybe a super ball on marble 244 00:14:20 --> 00:14:26 would be almost completely elastic, 245 00:14:23 --> 00:14:29 then the ball would bounce back with that same velocity. 246 00:14:29 --> 00:14:35 And if that were indeed a completely elastic collision, 247 00:14:32 --> 00:14:38 then you can see 248 00:14:34 --> 00:14:40 that the impulse that the ball is given to-- 249 00:14:39 --> 00:14:45 that is given to the ball as the ball hits the floor-- 250 00:14:44 --> 00:14:50 the floor is giving an impulse to the ball, 251 00:14:47 --> 00:14:53 and that impulse equals 2mv. 252 00:14:50 --> 00:14:56 The ball changes its momentum. 253 00:14:53 --> 00:14:59 It was first mv in this direction, 254 00:14:56 --> 00:15:02 and now it's mv in this direction, 255 00:14:58 --> 00:15:04 so the change is 2mv. 256 00:15:00 --> 00:15:06 So an impulse is given to the ball. 257 00:15:02 --> 00:15:08 Now, if the collision were completely inelastic, 258 00:15:07 --> 00:15:13 then the ball would just... say, like a tomato, 259 00:15:10 --> 00:15:16 I throw a tomato on the floor, it goes (splat ). 260 00:15:12 --> 00:15:18 No speed anymore when it hits the ground, 261 00:15:15 --> 00:15:21 then, of course, the impulse would only be mv, 262 00:15:18 --> 00:15:24 because then it doesn't come back up, 263 00:15:20 --> 00:15:26 so there is no... the change in momentum is then smaller. 264 00:15:24 --> 00:15:30 265 00:15:25 --> 00:15:31 We have here two balls that look alike. 266 00:15:29 --> 00:15:35 They have a mass of 0.1 kilogram, 267 00:15:34 --> 00:15:40 so m equals 0.1 kilogram, 268 00:15:39 --> 00:15:45 and I will drop them from a height of about 1½ meters, 269 00:15:43 --> 00:15:49 and that gives them a speed when they hit the floor 270 00:15:47 --> 00:15:53 of about 5½ meters per second. 271 00:15:51 --> 00:15:57 And so the momentum change is 2mv, 272 00:15:55 --> 00:16:01 so the impulse equals 2mv, is about 1.1, 273 00:16:02 --> 00:16:08 and that would be kilograms-meters per second. 274 00:16:09 --> 00:16:15 And that means that if the collision time 275 00:16:13 --> 00:16:19 is delta t seconds, 276 00:16:15 --> 00:16:21 that the average force acting upon this ball 277 00:16:18 --> 00:16:24 during the collision with the floor 278 00:16:21 --> 00:16:27 equals the impulse divided by delta t, 279 00:16:26 --> 00:16:32 because remember, that was our definition of impulse. 280 00:16:29 --> 00:16:35 So if we know the impulse, 281 00:16:30 --> 00:16:36 we get a feeling for the average force. 282 00:16:33 --> 00:16:39 And for the ball that I will drop on the floor, 283 00:16:35 --> 00:16:41 we have done fast photography. 284 00:16:37 --> 00:16:43 I will show you some results of the fast photography 285 00:16:40 --> 00:16:46 with a different ball, but nevertheless, we did it 286 00:16:42 --> 00:16:48 with the ball that I will drop on the floor very shortly-- 287 00:16:45 --> 00:16:51 which is this one-- 288 00:16:46 --> 00:16:52 that impact time is only two milliseconds. 289 00:16:48 --> 00:16:54 It's hard to believe that in two milliseconds 290 00:16:50 --> 00:16:56 the entire collision occurs. 291 00:16:52 --> 00:16:58 And so if you substitute in here now two milliseconds, 292 00:16:55 --> 00:17:01 then you get for the average force 550 newtons. 293 00:17:01 --> 00:17:07 Just imagine, this ball has a mass of 0.1 kilogram, 294 00:17:06 --> 00:17:12 the weight is one newton, 295 00:17:07 --> 00:17:13 and during the impact, it weighs 550 times more. 296 00:17:12 --> 00:17:18 What an incredible weight increase! 297 00:17:14 --> 00:17:20 And the average acceleration 298 00:17:16 --> 00:17:22 that it experiences during the impact is 550 times g. 299 00:17:22 --> 00:17:28 People play tennis, 300 00:17:24 --> 00:17:30 and they have speeds of hundreds of miles per hour-- 301 00:17:27 --> 00:17:33 speeds are way higher than we have here, ten times higher-- 302 00:17:30 --> 00:17:36 and so the weight increase is even more. 303 00:17:36 --> 00:17:42 Now, if the collision were completely inelastic, 304 00:17:40 --> 00:17:46 so that if it were a tomato or an egg, 305 00:17:44 --> 00:17:50 and this one wouldn't come up, 306 00:17:45 --> 00:17:51 the average force would still be approximately the same, 307 00:17:49 --> 00:17:55 the reason being that the impulse will be half. 308 00:17:53 --> 00:17:59 But if the impact time were also half 309 00:17:56 --> 00:18:02 and the impulse were half, then, of course, the force... 310 00:17:59 --> 00:18:05 the average force will be the same-- 311 00:18:01 --> 00:18:07 very high, but for a shorter amount of time. 312 00:18:04 --> 00:18:10 313 00:18:06 --> 00:18:12 So I want to show... 314 00:18:07 --> 00:18:13 oh, I first want to show you now the... these two balls. 315 00:18:09 --> 00:18:15 One is almost complete elastic collision with the floor. 316 00:18:14 --> 00:18:20 Whether this is a complete elastic collision 317 00:18:16 --> 00:18:22 depends not only on this ball-- whether it is a super ball-- 318 00:18:19 --> 00:18:25 it also depends on the condition of the floor. 319 00:18:21 --> 00:18:27 This is not a very good floor, this is not marble. 320 00:18:23 --> 00:18:29 So when I drop this one, 321 00:18:25 --> 00:18:31 it doesn't come up to this point here. 322 00:18:27 --> 00:18:33 So it's not a completely elastic collision, 323 00:18:30 --> 00:18:36 but it bounces pretty much. 324 00:18:31 --> 00:18:37 So it is somewhere in completely inelastic 325 00:18:34 --> 00:18:40 and completely elastic. 326 00:18:36 --> 00:18:42 It's not bad, right? 327 00:18:38 --> 00:18:44 It's not bad. 328 00:18:39 --> 00:18:45 Now this one. 329 00:18:41 --> 00:18:47 Watch it. 330 00:18:42 --> 00:18:48 (splat ) 331 00:18:43 --> 00:18:49 Looks alike, but it ain't. 332 00:18:46 --> 00:18:52 This one is completely inelastic. 333 00:18:47 --> 00:18:53 It goes to the floor and it goes clunk. 334 00:18:51 --> 00:18:57 You see a small bounce, but that's it. 335 00:18:53 --> 00:18:59 And so the impact times are very short-- 336 00:18:59 --> 00:19:05 two milliseconds in the case of the one that bounces back, 337 00:19:03 --> 00:19:09 one millisecond in the case of the one that went clunk, 338 00:19:06 --> 00:19:12 and their average weight 339 00:19:07 --> 00:19:13 is about 550 times their normal weight. 340 00:19:11 --> 00:19:17 341 00:19:13 --> 00:19:19 I'd like to show you fast photography 342 00:19:16 --> 00:19:22 on not this very same ball, 343 00:19:18 --> 00:19:24 but on another one that we did. 344 00:19:20 --> 00:19:26 Let me take this... this out. 345 00:19:23 --> 00:19:29 And that is a ball that comes down 346 00:19:27 --> 00:19:33 with a speed of four meters per second. 347 00:19:31 --> 00:19:37 And each frame is one millisecond, 348 00:19:34 --> 00:19:40 so you will see a ruler, 349 00:19:36 --> 00:19:42 and the ruler indicates... has marks in centimeters, 350 00:19:39 --> 00:19:45 and so you will see it go... 351 00:19:41 --> 00:19:47 in four milliseconds it will go one centimeter, 352 00:19:45 --> 00:19:51 so it has a speed of 2½ meters per second. 353 00:19:47 --> 00:19:53 It will hit the floor, 354 00:19:48 --> 00:19:54 and then we can count the number of milliseconds 355 00:19:50 --> 00:19:56 that it takes contact and going back up again. 356 00:19:54 --> 00:20:00 It's not going to be two milliseconds, 357 00:19:56 --> 00:20:02 it's a little longer, but, again, impressively short. 358 00:20:00 --> 00:20:06 All right, we're going to make it rather dark 359 00:20:03 --> 00:20:09 in order to get decent quality. 360 00:20:06 --> 00:20:12 I'm going to turn five of these off 361 00:20:10 --> 00:20:16 and I'm going to set the TV at two, 362 00:20:14 --> 00:20:20 and now I will start this. 363 00:20:17 --> 00:20:23 There we go. 364 00:20:20 --> 00:20:26 Let's hope that that will go. 365 00:20:22 --> 00:20:28 Okay, there comes the ball down. 366 00:20:25 --> 00:20:31 These marks are in centimeters. 367 00:20:30 --> 00:20:36 So there's a ruler in centimeters. 368 00:20:34 --> 00:20:40 This is one centimeter. 369 00:20:35 --> 00:20:41 I'll rewind a little, because we were a little too late. 370 00:20:41 --> 00:20:47 Okay, let's start again. 371 00:20:43 --> 00:20:49 So watch when it passes this mark-- one, two, three-- 372 00:20:49 --> 00:20:55 you see four milliseconds for about one centimeter. 373 00:20:52 --> 00:20:58 So that's 2½ meters per second. 374 00:20:55 --> 00:21:01 And now we'll count the number of milliseconds at impact. 375 00:20:57 --> 00:21:03 376 00:21:00 --> 00:21:06 One, two, three, four, five, six, and it's off. 377 00:21:06 --> 00:21:12 About six, maybe seven milliseconds. 378 00:21:08 --> 00:21:14 379 00:21:10 --> 00:21:16 And this is no special ball. 380 00:21:12 --> 00:21:18 These impact times are amazingly short. 381 00:21:15 --> 00:21:21 382 00:21:32 --> 00:21:38 Now I have something very special for you-- 383 00:21:35 --> 00:21:41 something really special, 384 00:21:38 --> 00:21:44 something that has kept me awake-- 385 00:21:40 --> 00:21:46 a lot of things keep me awake in physics, 386 00:21:42 --> 00:21:48 and not only in physics... 387 00:21:44 --> 00:21:50 (scattered laughter ) 388 00:21:45 --> 00:21:51 but this... but this is very special. 389 00:21:48 --> 00:21:54 This is very special. 390 00:21:49 --> 00:21:55 I have here a basketball. 391 00:21:52 --> 00:21:58 (ball bouncing ) 392 00:21:56 --> 00:22:02 Not completely elastic, but not bad. 393 00:21:58 --> 00:22:04 A tennis ball-- not completely elastic, but not bad. 394 00:22:03 --> 00:22:09 Now I'm going to drop them together vertically down, 395 00:22:07 --> 00:22:13 and then this ball will bounce up somehow. 396 00:22:12 --> 00:22:18 And so the question that I have for you is, 397 00:22:14 --> 00:22:20 do you think if I drop it from this height 398 00:22:17 --> 00:22:23 that this tennis ball will sort of come up 399 00:22:19 --> 00:22:25 at most to this height, 400 00:22:21 --> 00:22:27 or do you think it will be lower, 401 00:22:23 --> 00:22:29 or do you think it will be higher? 402 00:22:25 --> 00:22:31 So use your intuition. 403 00:22:26 --> 00:22:32 (chuckling ): In the worst case it can be... it can be wrong-- 404 00:22:30 --> 00:22:36 he is already pushing his finger up-- 405 00:22:32 --> 00:22:38 so what do you think? 406 00:22:33 --> 00:22:39 Will the tennis ball reach about the same height? 407 00:22:36 --> 00:22:42 Who is in favor of that? 408 00:22:38 --> 00:22:44 Who is in favor of higher? 409 00:22:40 --> 00:22:46 410 00:22:41 --> 00:22:47 Wow. 411 00:22:42 --> 00:22:48 Who is in favor of alot higher? 412 00:22:45 --> 00:22:51 Okay. 413 00:22:46 --> 00:22:52 Okay, I'll try it. 414 00:22:47 --> 00:22:53 Now, I cannot guarantee you that this ball will go straight up 415 00:22:51 --> 00:22:57 after the impact, because clearly that's impossible. 416 00:22:54 --> 00:23:00 That has zero chance, 417 00:22:55 --> 00:23:01 so it probably go up in some direction. 418 00:22:58 --> 00:23:04 But you will see the effect that I had in mind. 419 00:23:01 --> 00:23:07 420 00:23:03 --> 00:23:09 So there we go. 421 00:23:06 --> 00:23:12 (balls bounce ) 422 00:23:07 --> 00:23:13 And you see that, indeed, that tennis ball... 423 00:23:13 --> 00:23:19 goesway higher. 424 00:23:14 --> 00:23:20 I'll try it once more to see 425 00:23:15 --> 00:23:21 whether I can get it to go up a little bit more vertically, 426 00:23:18 --> 00:23:24 but that is very difficult. 427 00:23:20 --> 00:23:26 It goes way higher, 428 00:23:21 --> 00:23:27 and this is something that you should be able to calculate, 429 00:23:24 --> 00:23:30 and you can, and you will. 430 00:23:26 --> 00:23:32 Believe me, it's part of assignment number six. 431 00:23:28 --> 00:23:34 You haven't seen it yet. 432 00:23:30 --> 00:23:36 433 00:23:31 --> 00:23:37 There we go. 434 00:23:33 --> 00:23:39 (ball bounces ) 435 00:23:34 --> 00:23:40 Oh, boy, that was better. 436 00:23:35 --> 00:23:41 437 00:23:37 --> 00:23:43 Well... oh! 438 00:23:40 --> 00:23:46 Do any one of you know 439 00:23:41 --> 00:23:47 approximately how much higher it goes, 440 00:23:43 --> 00:23:49 if this ball has way higher mass than this one? 441 00:23:46 --> 00:23:52 Of course, the mass ratio comes into it. 442 00:23:48 --> 00:23:54 Any idea? 443 00:23:49 --> 00:23:55 Twice as high? 444 00:23:50 --> 00:23:56 You'll be surprised when you do assignment six. 445 00:23:54 --> 00:24:00 Much higher. 446 00:23:56 --> 00:24:02 Okay, in fact, you could even see it here all right 447 00:23:59 --> 00:24:05 that it was quite a bit higher than twice. 448 00:24:01 --> 00:24:07 Great... a great experiment, 449 00:24:03 --> 00:24:09 and it's something you can do yourself in your dormitory. 450 00:24:09 --> 00:24:15 Now I want to discuss in the remaining time about rockets. 451 00:24:13 --> 00:24:19 A rocket experience an impulse from the engine, 452 00:24:17 --> 00:24:23 and that changes the momentum of the rocket. 453 00:24:21 --> 00:24:27 But before we go into the details of the rocket, 454 00:24:26 --> 00:24:32 think about, again, this idea of throwing objects on the floor. 455 00:24:31 --> 00:24:37 And let's turn to tomatoes. 456 00:24:33 --> 00:24:39 I want tomatoes, because I want a complete inelastic collision, 457 00:24:37 --> 00:24:43 so tomatoes hit the floor, and it's not that I... 458 00:24:40 --> 00:24:46 I'm not only going to throw one tomato on the floor, 459 00:24:44 --> 00:24:50 but I'm very angry today, 460 00:24:46 --> 00:24:52 I'm going to throw a lot of these tomatoes on the floor-- 461 00:24:50 --> 00:24:56 n, as in Nancy, tomatoes on the floor. 462 00:24:52 --> 00:24:58 If one tomato hits the floor, 463 00:24:55 --> 00:25:01 the change of momentum is mv, if m is the mass of the tomato. 464 00:25:00 --> 00:25:06 But I'm going to throw n on the floor, 465 00:25:03 --> 00:25:09 so the change of momentum is n, as in Nancy, 466 00:25:06 --> 00:25:12 times the mass of the tomato times v. 467 00:25:09 --> 00:25:15 And this is the number of kilograms per second 468 00:25:13 --> 00:25:19 of tomatoes that I throw on the floor. 469 00:25:17 --> 00:25:23 So this is a change of momentum. 470 00:25:20 --> 00:25:26 And so this equals delta p divided by delta t, 471 00:25:26 --> 00:25:32 and that's an average force. 472 00:25:28 --> 00:25:34 473 00:25:31 --> 00:25:37 So the floor will experience a force in down direction. 474 00:25:36 --> 00:25:42 Of course that was also the case here 475 00:25:40 --> 00:25:46 when the ball experienced a force up, 476 00:25:43 --> 00:25:49 which we calculated here. 477 00:25:45 --> 00:25:51 The floor experienced, of course, 478 00:25:47 --> 00:25:53 the same force in down direction-- 479 00:25:50 --> 00:25:56 action equals minus reaction-- Newton's third law. 480 00:25:53 --> 00:25:59 So the force experiences... the floor experiences a force 481 00:25:57 --> 00:26:03 in down direction. 482 00:25:58 --> 00:26:04 And I can write it down in a somewhat more civilized form: 483 00:26:04 --> 00:26:10 F equals dm/dt times the velocity, 484 00:26:07 --> 00:26:13 if the velocity of the tomatoes that hit the floor is constant. 485 00:26:14 --> 00:26:20 And we're going to apply this to rockets, 486 00:26:16 --> 00:26:22 whereby the exhaust out of rockets 487 00:26:18 --> 00:26:24 has, relative to the rocket, a constant speed. 488 00:26:20 --> 00:26:26 And this is then the number of kilograms per second 489 00:26:25 --> 00:26:31 that I throw on the floor. 490 00:26:29 --> 00:26:35 This is very real. 491 00:26:30 --> 00:26:36 I can throw these tomatoes on a bathroom scale, 492 00:26:33 --> 00:26:39 and if I threw four kilograms per second on a bathroom scale, 493 00:26:37 --> 00:26:43 and they hit the bathroom scale with five meters per second, 494 00:26:41 --> 00:26:47 you better believe it that you will see that the bathroom scale 495 00:26:45 --> 00:26:51 will reach an average force 496 00:26:46 --> 00:26:52 of about 20 newtons-- four times five. 497 00:26:49 --> 00:26:55 If it weren't tomatoes, 498 00:26:51 --> 00:26:57 but if they were super balls which would bounce up, 499 00:26:53 --> 00:26:59 then the momentum change would be double, 500 00:26:56 --> 00:27:02 and so the bathroom scale would indicate 40 newtons. 501 00:26:59 --> 00:27:05 So this is a real thing, it's a real force that you can record. 502 00:27:02 --> 00:27:08 503 00:27:04 --> 00:27:10 Now I'm going to be a little bit unpleasant to you. 504 00:27:08 --> 00:27:14 I'm going to throw rotten tomatoes at you. 505 00:27:11 --> 00:27:17 506 00:27:14 --> 00:27:20 Here you are, and I have here a tomato, 507 00:27:18 --> 00:27:24 and this tomato has zero speed to start with. 508 00:27:23 --> 00:27:29 509 00:27:26 --> 00:27:32 Let me make you a little bigger, 510 00:27:28 --> 00:27:34 otherwise I won't even... I won't even hit you. 511 00:27:30 --> 00:27:36 512 00:27:33 --> 00:27:39 So this is you. 513 00:27:34 --> 00:27:40 514 00:27:36 --> 00:27:42 And so I give this tomato a certain velocity, v of x. 515 00:27:42 --> 00:27:48 The tomato hits you, (splat ). 516 00:27:46 --> 00:27:52 Maybe it stays there, it's possible. 517 00:27:48 --> 00:27:54 Maybe (squish ), it will drip down. 518 00:27:51 --> 00:27:57 But in any case, the velocity in the x direction is gone. 519 00:27:58 --> 00:28:04 So it hits you with a velocity vx and then v of x equals zero. 520 00:28:05 --> 00:28:11 And it may make a mess. 521 00:28:07 --> 00:28:13 522 00:28:09 --> 00:28:15 You will experience a force. 523 00:28:10 --> 00:28:16 If I keep throwing these tomatoes at you all the time... 524 00:28:14 --> 00:28:20 and this is the force that you will experience-- 525 00:28:17 --> 00:28:23 and that force is, of course, in this direction. 526 00:28:20 --> 00:28:26 527 00:28:25 --> 00:28:31 You've got all these tomatoes, and you feel that as a force. 528 00:28:28 --> 00:28:34 But now look at the symmetry of the problem. 529 00:28:32 --> 00:28:38 Here the velocity goes from vx to zero. 530 00:28:35 --> 00:28:41 But I, throwing the tomatoes, 531 00:28:38 --> 00:28:44 have to increase the velocity from zero to vx. 532 00:28:42 --> 00:28:48 So for obvious reasons, 533 00:28:44 --> 00:28:50 I must then feel a force in this direction-- 534 00:28:47 --> 00:28:53 think of it as a recoil when you fire a bullet. 535 00:28:51 --> 00:28:57 So I experience exactly the same force, but in this direction, 536 00:28:55 --> 00:29:01 and that now is the idea behind a rocket. 537 00:28:58 --> 00:29:04 A rocket is spewing out tomatoes-- 538 00:29:01 --> 00:29:07 well, not quite tomatoes-- 539 00:29:03 --> 00:29:09 it's spewing out hot gas in this direction, 540 00:29:06 --> 00:29:12 and then the rocket will experience a force 541 00:29:10 --> 00:29:16 in that direction. 542 00:29:12 --> 00:29:18 That is the basic concept behind a rocket. 543 00:29:15 --> 00:29:21 And the higher the speed of the gas that it throws out-- 544 00:29:20 --> 00:29:26 the higher that velocity-- 545 00:29:23 --> 00:29:29 the more kilograms per second it spits out, 546 00:29:27 --> 00:29:33 the higher dm/dt, the higher will be the force on the rocket, 547 00:29:31 --> 00:29:37 and this force on the rocket is called the thrust of the rocket. 548 00:29:36 --> 00:29:42 549 00:29:39 --> 00:29:45 So if we have a rocket in space-- 550 00:29:43 --> 00:29:49 here's the rocket-- 551 00:29:49 --> 00:29:55 and the rocket is spewing out gas with a velocity, u, 552 00:29:56 --> 00:30:02 which is fixed relative to the rocket-- 553 00:30:00 --> 00:30:06 it's a burning of chemical energy. 554 00:30:02 --> 00:30:08 Chemicals are burned, it comes out with a certain speed, 555 00:30:07 --> 00:30:13 and the rocket will then experience a force, 556 00:30:11 --> 00:30:17 which we call the thrust, 557 00:30:13 --> 00:30:19 and that is given by this equation. 558 00:30:15 --> 00:30:21 If you know how many kilograms per second are spewed out, 559 00:30:18 --> 00:30:24 and you know what the velocity is, which I've called u here, 560 00:30:21 --> 00:30:27 this will tell you what the thrust is of that rocket. 561 00:30:24 --> 00:30:30 If we take the case of the Saturn rockets 562 00:30:29 --> 00:30:35 that were used for the landing on the Moon... 563 00:30:34 --> 00:30:40 564 00:30:36 --> 00:30:42 Saturn. 565 00:30:37 --> 00:30:43 566 00:30:40 --> 00:30:46 For the Saturn rockets, 567 00:30:42 --> 00:30:48 the speed u was about 2½ kilometers per second. 568 00:30:48 --> 00:30:54 So the gas came out with 2½ kilometers per second 569 00:30:51 --> 00:30:57 relative to the rocket, 570 00:30:52 --> 00:30:58 and the rate at which this gas was coming out is phenomenal-- 571 00:30:56 --> 00:31:02 15 tons of material per second. 572 00:31:01 --> 00:31:07 dm/dt is about 15,000 kilograms per second-- 573 00:31:07 --> 00:31:13 it's almost unimaginable. 574 00:31:11 --> 00:31:17 And that would give it then a thrust 575 00:31:16 --> 00:31:22 of about 35 million newtons, 576 00:31:18 --> 00:31:24 which, of course, was higher than the weight of the rocket, 577 00:31:24 --> 00:31:30 otherwise the rocket would never go up. 578 00:31:28 --> 00:31:34 An incredible thrust. 579 00:31:29 --> 00:31:35 I have also a nice problem for you in assignment six 580 00:31:32 --> 00:31:38 with the Saturn rockets. 581 00:31:34 --> 00:31:40 You'll see these numbers again. 582 00:31:36 --> 00:31:42 These are rounded off. 583 00:31:37 --> 00:31:43 584 00:31:39 --> 00:31:45 So rockets obtain an impulse from their engines, 585 00:31:43 --> 00:31:49 a force acts upon the rocket for a certain amount of time-- 586 00:31:47 --> 00:31:53 we call that the burn time-- 587 00:31:50 --> 00:31:56 but as they burn the fuel, the mass of the rocket goes down, 588 00:31:55 --> 00:32:01 because the fuel leaves. 589 00:31:57 --> 00:32:03 And therefore the acceleration during the burn goes up, 590 00:32:01 --> 00:32:07 because the mass goes down. 591 00:32:03 --> 00:32:09 And this makes it somewhat complicated 592 00:32:06 --> 00:32:12 to derive the velocity change 593 00:32:09 --> 00:32:15 that you get during the burn, during this impulse. 594 00:32:14 --> 00:32:20 It's done in your book-- Ohanian. 595 00:32:17 --> 00:32:23 I find that derivation a little bit complicated. 596 00:32:22 --> 00:32:28 I looked in other books, 597 00:32:24 --> 00:32:30 and I found one in the book by Tipler on physics, 598 00:32:28 --> 00:32:34 and his derivation-- 599 00:32:29 --> 00:32:35 and I will go through that in general terms... 600 00:32:33 --> 00:32:39 I put that on the Web for you. 601 00:32:34 --> 00:32:40 It should be there tonight, 602 00:32:36 --> 00:32:42 so you may want to take very few notes. 603 00:32:38 --> 00:32:44 The entire derivation, 604 00:32:40 --> 00:32:46 of which I will just highlight the important issues, 605 00:32:45 --> 00:32:51 is on the Web. 606 00:32:46 --> 00:32:52 607 00:32:48 --> 00:32:54 And the way that Tipler is doing it 608 00:32:51 --> 00:32:57 is exclusively from your frame of reference. 609 00:32:55 --> 00:33:01 You're sitting in 26.100, 610 00:32:58 --> 00:33:04 and you are seeing this rocket go up. 611 00:33:02 --> 00:33:08 612 00:33:04 --> 00:33:10 Let's keep that equation, which is the thrust of the rocket, 613 00:33:11 --> 00:33:17 except that in the case of a rocket we would call this u, 614 00:33:16 --> 00:33:22 which is the speed of the exhaust relative to the rocket, 615 00:33:20 --> 00:33:26 and this is what we call the thrust. 616 00:33:22 --> 00:33:28 So let's now take a rocket, which at time t 617 00:33:27 --> 00:33:33 as seen from your frame of reference-- 618 00:33:31 --> 00:33:37 where you are sitting. 619 00:33:33 --> 00:33:39 I use v-- it's your frame of reference. 620 00:33:36 --> 00:33:42 It's going up with a velocity v. 621 00:33:38 --> 00:33:44 622 00:33:40 --> 00:33:46 The mass of the rocket is m. 623 00:33:43 --> 00:33:49 And now we're going to look at time t plus delta t. 624 00:33:47 --> 00:33:53 The rocket has increased its speed, v plus delta v. 625 00:33:56 --> 00:34:02 626 00:33:58 --> 00:34:04 The mass is m-- let me do that in... not in color. 627 00:34:03 --> 00:34:09 The mass is now m minus delta m, 628 00:34:06 --> 00:34:12 and here is a little bit of exhaust 629 00:34:09 --> 00:34:15 that was spewed out with velocity u 630 00:34:13 --> 00:34:19 relative to the rocket-- u relative to the rocket. 631 00:34:18 --> 00:34:24 So you in 26.100 will see the velocity 632 00:34:22 --> 00:34:28 of this little piece delta m-- 633 00:34:25 --> 00:34:31 you will see this to be v minus u. 634 00:34:29 --> 00:34:35 635 00:34:31 --> 00:34:37 If the velocity of the rocket is larger than u-- could be-- 636 00:34:34 --> 00:34:40 you will see the exhaust going up from your frame of reference. 637 00:34:38 --> 00:34:44 If the velocity of the exhaust 638 00:34:40 --> 00:34:46 is larger than the velocity of the rocket, 639 00:34:42 --> 00:34:48 from you frame of reference, you will see it go down. 640 00:34:46 --> 00:34:52 That's taken care of in the signs. 641 00:34:48 --> 00:34:54 642 00:34:51 --> 00:34:57 Now I'm going to compare the momentum here 643 00:34:56 --> 00:35:02 with the momentum there, 644 00:34:59 --> 00:35:05 and we take the situation 645 00:35:00 --> 00:35:06 that there are no external forces acting upon it-- 646 00:35:02 --> 00:35:08 somewhere in outer space, this rocket burns. 647 00:35:05 --> 00:35:11 Momentum must be conserved. 648 00:35:09 --> 00:35:15 The momentum at time t equals m times v. 649 00:35:15 --> 00:35:21 That is very simple. 650 00:35:17 --> 00:35:23 That is time t. 651 00:35:18 --> 00:35:24 The momentum at time t plus delta t of the entire system 652 00:35:23 --> 00:35:29 including the exhaust-- 653 00:35:25 --> 00:35:31 if I'm telling you that momentum is conserved, 654 00:35:29 --> 00:35:35 you can't ignore the exhaust. 655 00:35:31 --> 00:35:37 It's momentum of the system that is conserved. 656 00:35:33 --> 00:35:39 The momentum of the rocket is going to change 657 00:35:36 --> 00:35:42 and the momentum of the exhaust is going to change, 658 00:35:39 --> 00:35:45 but not of the system. 659 00:35:40 --> 00:35:46 660 00:35:41 --> 00:35:47 So we're going to get mass times velocity-- 661 00:35:48 --> 00:35:54 (m minus delta m) times (v plus delta v) plus delta m, 662 00:35:57 --> 00:36:03 which has a velocity v minus u. 663 00:36:02 --> 00:36:08 664 00:36:03 --> 00:36:09 And how large is this? 665 00:36:06 --> 00:36:12 Well, there is mv plus m delta v. 666 00:36:09 --> 00:36:15 667 00:36:14 --> 00:36:20 Here you see the m delta v, here you see the mv, 668 00:36:18 --> 00:36:24 and then you get minus u delta m, 669 00:36:22 --> 00:36:28 and the delta mv here cancels minus delta mv here, 670 00:36:26 --> 00:36:32 and this term, delta m delta v, 671 00:36:28 --> 00:36:34 is the product of two incredibly small numbers-- I ignore that. 672 00:36:33 --> 00:36:39 673 00:36:35 --> 00:36:41 So this is the momentum at t plus delta t 674 00:36:38 --> 00:36:44 and this is the momentum at time t, 675 00:36:41 --> 00:36:47 and so the change in momentum, delta p-- 676 00:36:44 --> 00:36:50 which must be zero because momentum is conserved-- 677 00:36:48 --> 00:36:54 is m delta v minus u delta m. 678 00:36:52 --> 00:36:58 m delta v minus u delta m. 679 00:36:57 --> 00:37:03 680 00:36:59 --> 00:37:05 I can take the derivative of this equation, 681 00:37:04 --> 00:37:10 so I get on the left side dp/dt. 682 00:37:09 --> 00:37:15 dp/dt is going to be zero, 683 00:37:13 --> 00:37:19 so I get zero equals m times dv/dt, 684 00:37:18 --> 00:37:24 but dv/dt is the acceleration of the rocket minus u dm/dt. 685 00:37:27 --> 00:37:33 686 00:37:29 --> 00:37:35 And that is the thrust on the rocket. 687 00:37:34 --> 00:37:40 So what you see here 688 00:37:35 --> 00:37:41 is something that is very easy to digest-- 689 00:37:38 --> 00:37:44 that ma, which is the... this is the acceleration of the rocket, 690 00:37:42 --> 00:37:48 this is the mass of the rocket at time t-- 691 00:37:44 --> 00:37:50 equals the thrust of the rocket, and that equals u dm/dt. 692 00:37:51 --> 00:37:57 And some people call this the... the "rocket equation." 693 00:37:59 --> 00:38:05 Now, this is true if there is no external force on the system. 694 00:38:06 --> 00:38:12 It is interesting to include a real launch from Earth, 695 00:38:11 --> 00:38:17 and if you have a real launch from Earth, 696 00:38:15 --> 00:38:21 then the rocket is going up in this direction, 697 00:38:19 --> 00:38:25 but then gravity is exactly in the opposite direction. 698 00:38:24 --> 00:38:30 In other words, only when you launch vertically from Earth 699 00:38:28 --> 00:38:34 would you have a thrust like this, 700 00:38:31 --> 00:38:37 and you would have mg like this. 701 00:38:34 --> 00:38:40 In that case, this equation has to be adjusted, 702 00:38:40 --> 00:38:46 and then you get ma equals m thrust minus mg. 703 00:38:46 --> 00:38:52 Only if you have a launch from Earth vertically up. 704 00:38:51 --> 00:38:57 Now you have to do a little bit of massaging, 705 00:38:55 --> 00:39:01 and I will leave you with that massaging. 706 00:38:58 --> 00:39:04 A couple of integrals are necessary to convert this 707 00:39:02 --> 00:39:08 into the final velocity of the rocket after the burn 708 00:39:06 --> 00:39:12 compared to the initial velocity. 709 00:39:10 --> 00:39:16 And that part I will leave you with, 710 00:39:12 --> 00:39:18 but you will see that worked out in all detail 711 00:39:14 --> 00:39:20 on the notes that I left on the Web. 712 00:39:16 --> 00:39:22 And then you come up with a very famous equation 713 00:39:20 --> 00:39:26 that the final velocity of the rocket 714 00:39:22 --> 00:39:28 minus the initial velocity of the rocket 715 00:39:25 --> 00:39:31 equals minus u times the logarithm 716 00:39:27 --> 00:39:33 of the final mass of the rocket 717 00:39:30 --> 00:39:36 divided by the initial mass of the rocket. 718 00:39:33 --> 00:39:39 This is if there were no gravity at all. 719 00:39:36 --> 00:39:42 Just in case, and only in case of a vertical launch from Earth, 720 00:39:42 --> 00:39:48 there is also here a term, minus gt-- 721 00:39:45 --> 00:39:51 only if you have a vertical launch from Earth. 722 00:39:50 --> 00:39:56 723 00:39:52 --> 00:39:58 When I watched my lecture, I noticed that in my enthusiasm 724 00:39:55 --> 00:40:01 I put brackets around the minus gt. 725 00:39:57 --> 00:40:03 This is quite misleading, 726 00:39:59 --> 00:40:05 because it may give you the wrong impression 727 00:40:01 --> 00:40:07 that there's a product at stake here, which is not so. 728 00:40:06 --> 00:40:12 So the brackets really should not be around the minus gt. 729 00:40:10 --> 00:40:16 It is this term, 730 00:40:12 --> 00:40:18 minus u logarithm of mf divided by m of i minus gt. 731 00:40:16 --> 00:40:22 732 00:40:17 --> 00:40:23 Let's now look at this equation in a little bit more detail, 733 00:40:21 --> 00:40:27 so that we get a little bit of feeling for it. 734 00:40:23 --> 00:40:29 Suppose we had a vertical launch from Earth, 735 00:40:26 --> 00:40:32 but we had no rocket. 736 00:40:27 --> 00:40:33 It's possible. 737 00:40:30 --> 00:40:36 So this term does not exist. 738 00:40:32 --> 00:40:38 What do you see? 739 00:40:33 --> 00:40:39 That the velocity equals v initial-- 740 00:40:36 --> 00:40:42 we have called that before in 801 v zero-- 741 00:40:40 --> 00:40:46 that's the initial speed-- minus gt. 742 00:40:44 --> 00:40:50 Ha! 743 00:40:45 --> 00:40:51 We had that result during our first lecture-- 744 00:40:47 --> 00:40:53 completely consistent with this equation. 745 00:40:49 --> 00:40:55 If you have no rocket, you get that v equals v zero minus gt, 746 00:40:53 --> 00:40:59 if you throw the object vertically up 747 00:40:55 --> 00:41:01 and we had a vertical launch. 748 00:40:56 --> 00:41:02 So that looks good. 749 00:40:59 --> 00:41:05 We launch from the Earth, 750 00:41:02 --> 00:41:08 and we now have an initial speed which is zero. 751 00:41:07 --> 00:41:13 The rocket is standing there, and we fire the rocket. 752 00:41:11 --> 00:41:17 t, by the way, is the burn time of the rocket. 753 00:41:14 --> 00:41:20 Let me write that down. 754 00:41:17 --> 00:41:23 t is the burn time. 755 00:41:19 --> 00:41:25 756 00:41:21 --> 00:41:27 So initial speed is zero. 757 00:41:25 --> 00:41:31 So now, this final velocity... if we want this final velocity 758 00:41:30 --> 00:41:36 to be anything physically meaningful-- a positive number-- 759 00:41:34 --> 00:41:40 this thing has to come out positive. 760 00:41:36 --> 00:41:42 And you will say, "But how can that be?" 761 00:41:39 --> 00:41:45 Because we have a minus sign here 762 00:41:41 --> 00:41:47 and we have a minus sign there. 763 00:41:43 --> 00:41:49 How can that ever become positive? 764 00:41:45 --> 00:41:51 Well, don't forget that the final mass 765 00:41:48 --> 00:41:54 is smaller always than the initial mass, 766 00:41:51 --> 00:41:57 because you burn fuel, 767 00:41:53 --> 00:41:59 and so this logarithm is always going to be negative, 768 00:41:57 --> 00:42:03 and so this term is always going to be positive, 769 00:42:00 --> 00:42:06 if you burn any fuel. 770 00:42:01 --> 00:42:07 Now, of course for this velocity 771 00:42:03 --> 00:42:09 to be a real value-- physically meaningful-- 772 00:42:06 --> 00:42:12 this term has to kill this one, has to be larger than this one, 773 00:42:11 --> 00:42:17 otherwise the rocket won't even go up. 774 00:42:13 --> 00:42:19 You could be spewing out fuel all the time, 775 00:42:15 --> 00:42:21 and the rocket would just be sitting there, 776 00:42:18 --> 00:42:24 because the thrust up effectively 777 00:42:20 --> 00:42:26 is not enough to lift it off. 778 00:42:21 --> 00:42:27 So the first term has to win from the second term 779 00:42:26 --> 00:42:32 in case of a vertical launch. 780 00:42:28 --> 00:42:34 781 00:42:30 --> 00:42:36 Let's take a example with some numbers-- 782 00:42:33 --> 00:42:39 it always gives a little bit of insight. 783 00:42:36 --> 00:42:42 We have a burn time of about 100 seconds, 784 00:42:40 --> 00:42:46 and the initial speed is zero, 785 00:42:43 --> 00:42:49 and let us take a case whereby u is 1,000 meters per second. 786 00:42:49 --> 00:42:55 That is less than the Saturn rocket, 787 00:42:51 --> 00:42:57 but it's still a sizable... it's a kilometer per second. 788 00:42:54 --> 00:43:00 So the exhaust is coming out relative to the rocket 789 00:42:57 --> 00:43:03 with one kilometer per second. 790 00:42:59 --> 00:43:05 And the final mass divided by the initial mass of the rocket 791 00:43:04 --> 00:43:10 is 0.1, so 90% was burned away, the fuel that is gone. 792 00:43:09 --> 00:43:15 You have only ten percent left when the burn is over. 793 00:43:13 --> 00:43:19 794 00:43:15 --> 00:43:21 So now we can calculate-- 795 00:43:17 --> 00:43:23 if this was a launch from Earth, a vertical launch up-- 796 00:43:20 --> 00:43:26 we can calculate how large this term is, 797 00:43:23 --> 00:43:29 take the logarithm of this value, 798 00:43:25 --> 00:43:31 multiply this by minus u, 799 00:43:28 --> 00:43:34 and then we find that the final velocity... 800 00:43:32 --> 00:43:38 The first term is 2,300, 801 00:43:35 --> 00:43:41 and the second term, 100 seconds, 802 00:43:38 --> 00:43:44 and this is ten, is minus 1000. 803 00:43:41 --> 00:43:47 So you pay a price for the gravitational field. 804 00:43:46 --> 00:43:52 And so you have about 1.3 kilometers per second 805 00:43:50 --> 00:43:56 is the final speed of that rocket. 806 00:43:53 --> 00:43:59 These are meters per second, 807 00:43:54 --> 00:44:00 and that is in kilometers per second. 808 00:43:58 --> 00:44:04 Now, if there were no gravity, 809 00:44:00 --> 00:44:06 then of course the gain in speed-- this difference-- 810 00:44:04 --> 00:44:10 would be 2,300 meters per second. 811 00:44:06 --> 00:44:12 Also, if there is no vertical launch 812 00:44:10 --> 00:44:16 but if, for instance, the rocket were in orbit around the Earth-- 813 00:44:16 --> 00:44:22 here we have the rocket 814 00:44:18 --> 00:44:24 and going in orbit around the Earth-- 815 00:44:21 --> 00:44:27 then, of course, even though there is gravity, 816 00:44:26 --> 00:44:32 gravity is now not doing any work. 817 00:44:28 --> 00:44:34 And therefore if you fire this rocket for 100 seconds, 818 00:44:33 --> 00:44:39 the change in velocity, the tangential change, 819 00:44:36 --> 00:44:42 will also be 2,300 meters per second. 820 00:44:39 --> 00:44:45 You cannot say now there is gravity 821 00:44:41 --> 00:44:47 and therefore we have to take the gt term into account. 822 00:44:45 --> 00:44:51 It will be the same for this equation. 823 00:44:47 --> 00:44:53 If you had an object going in this direction, 824 00:44:50 --> 00:44:56 this minus gt term wouldn't be there either. 825 00:44:53 --> 00:44:59 It's only, of course, when you deal with a vertical motion. 826 00:44:57 --> 00:45:03 827 00:44:58 --> 00:45:04 It is very important to realize, and very non-intuitive, 828 00:45:03 --> 00:45:09 that when you burn a certain amount of fuel 829 00:45:06 --> 00:45:12 for a given amount of time 830 00:45:08 --> 00:45:14 you obtain a fixed change in the velocity. 831 00:45:11 --> 00:45:17 This is fixed for a given amount of fuel. 832 00:45:15 --> 00:45:21 The change in kinetic energy is not fixed, 833 00:45:17 --> 00:45:23 and I'll give you some numbers 834 00:45:19 --> 00:45:25 so that you can immediately check that for yourself. 835 00:45:23 --> 00:45:29 And it is very non-intuitive. 836 00:45:25 --> 00:45:31 It is, in fact, a mistake that is made by many physicists 837 00:45:28 --> 00:45:34 who think that if you burn the same amount of fuel 838 00:45:31 --> 00:45:37 of the same rocket for the same amount of time 839 00:45:34 --> 00:45:40 that the increase in kinetic energy is a given. 840 00:45:37 --> 00:45:43 That is not true. 841 00:45:38 --> 00:45:44 It's the change in velocity that is a given. 842 00:45:42 --> 00:45:48 But suppose that v final minus v initial 843 00:45:44 --> 00:45:50 is 100 meters per second. 844 00:45:46 --> 00:45:52 That's just a given. 845 00:45:48 --> 00:45:54 I have a rocket, I have a certain amount of fuel, 846 00:45:50 --> 00:45:56 I burn it, and that is my change in velocity. 847 00:45:54 --> 00:46:00 I start off with v initial zero, 848 00:45:57 --> 00:46:03 so my kinetic energy increase is one-half m times 100 squared, 849 00:46:04 --> 00:46:10 which is ten to the fourth. 850 00:46:07 --> 00:46:13 That's what I got out of this burn. 851 00:46:10 --> 00:46:16 Now I use the very same rocket, 852 00:46:12 --> 00:46:18 the same amount of fuel, the same burn time, 853 00:46:15 --> 00:46:21 so I get again that the final velocity 854 00:46:17 --> 00:46:23 minus the initial velocity is still 100. 855 00:46:20 --> 00:46:26 That's exactly the same. 856 00:46:22 --> 00:46:28 But before the burn, this rocket had an initial velocity 857 00:46:27 --> 00:46:33 of 1,000 meters per second. 858 00:46:29 --> 00:46:35 What is now the gain of kinetic energy? 859 00:46:33 --> 00:46:39 The final velocity is now 1,100. 860 00:46:35 --> 00:46:41 That's non-negotiable, because the rocket changes the momentum 861 00:46:39 --> 00:46:45 changes the velocity by a fixed amount. 862 00:46:42 --> 00:46:48 So now the gain of kinetic energy, 863 00:46:45 --> 00:46:51 the increase of kinetic energy 864 00:46:48 --> 00:46:54 equals one-half m times 1,100 squared minus 1,000 squared. 865 00:46:54 --> 00:47:00 This is the new velocity and this was the old one. 866 00:47:00 --> 00:47:06 And this number is one-half m times 200,000. 867 00:47:05 --> 00:47:11 This could be in joules, and this is also in joules. 868 00:47:10 --> 00:47:16 This number is 20 times higher. 869 00:47:12 --> 00:47:18 So you see that the change in velocity was exactly the same. 870 00:47:17 --> 00:47:23 The rocket burned the same amount of time, 871 00:47:20 --> 00:47:26 the same amount of fuel, 872 00:47:21 --> 00:47:27 but the kinetic energy increase is way more 873 00:47:25 --> 00:47:31 if the rocket has a higher speed to start with. 874 00:47:28 --> 00:47:34 875 00:47:30 --> 00:47:36 We here in 26.100 made our own rocket. 876 00:47:34 --> 00:47:40 It's a very down-to-earth model, no pun implied. 877 00:47:40 --> 00:47:46 But it is quite powerful, and I would like to show it to you, 878 00:47:46 --> 00:47:52 because we are quite proud of it. 879 00:47:49 --> 00:47:55 As a rocket, we use a fire extinguisher-- carbon dioxide-- 880 00:47:54 --> 00:48:00 and this fantastic invention I have here. 881 00:47:58 --> 00:48:04 882 00:48:05 --> 00:48:11 And this powerful rocket 883 00:48:07 --> 00:48:13 is enough to reach the escape velocity of... 884 00:48:11 --> 00:48:17 (loud whooshing ) 885 00:48:12 --> 00:48:18 ...of 26.100. 886 00:48:14 --> 00:48:20 (students laughing ) 887 00:48:17 --> 00:48:23 I almost reached the escape velocity, but I crashed. 888 00:48:21 --> 00:48:27 See you next Friday. 889 00:48:24 --> 00:48:30 (laughter and scattered applause ) 890 00:48:27 --> 00:48:33 891 00:48:29 --> 00:48:35 (more whooshing ) 892 00:48:31 --> 00:48:37 893 00:48:38 --> 00:48:44.000