1 00:00:03,382 --> 00:00:05,090 Now that we have all the pieces in place, 2 00:00:05,090 --> 00:00:07,460 we'd like to apply the momentum principle to the rocket 3 00:00:07,460 --> 00:00:08,310 problem. 4 00:00:08,310 --> 00:00:10,490 Recall that the momentum principle 5 00:00:10,490 --> 00:00:14,360 is that the external force on the rocket 6 00:00:14,360 --> 00:00:17,510 causes-- on the system-- causes the momentum of the system 7 00:00:17,510 --> 00:00:18,690 to change. 8 00:00:18,690 --> 00:00:22,460 And the fundamental definition of a derivative 9 00:00:22,460 --> 00:00:26,720 is to look at the change in momentum between sometimes t 10 00:00:26,720 --> 00:00:32,030 plus delta t-- our two states that we've identified-- 11 00:00:32,030 --> 00:00:35,210 divided by delta t. 12 00:00:35,210 --> 00:00:40,390 Now recall that we had the momentum of the system 13 00:00:40,390 --> 00:00:44,270 at time t was the mass of the rocket times 14 00:00:44,270 --> 00:00:46,730 d-bar of the rocket a t. 15 00:00:46,730 --> 00:00:54,530 I'll just quickly show mass of the rocket V-rt. 16 00:00:54,530 --> 00:01:02,750 And we had the system at time t plus delta t 17 00:01:02,750 --> 00:01:08,660 where we had delta M-fuel equals minus delta M-rocket Here we 18 00:01:08,660 --> 00:01:15,680 had M-r plus delta M. And we had the velocity in the ground 19 00:01:15,680 --> 00:01:21,350 frame, which we saw was equal to-- so let's 20 00:01:21,350 --> 00:01:26,150 write out the momentum of the system at time t plus delta t. 21 00:01:26,150 --> 00:01:28,420 That was a little bit longer. 22 00:01:28,420 --> 00:01:34,070 p system at t plus delta t had two pieces. 23 00:01:34,070 --> 00:01:42,050 It had M-r plus delta M-r times V of r of t plus delta t. 24 00:01:42,050 --> 00:01:46,670 And we were subtracting-- now, because we 25 00:01:46,670 --> 00:01:51,270 made this change, that's minus then delta M-r u 26 00:01:51,270 --> 00:01:55,580 plus V-bar of t plus delta t. 27 00:01:55,580 --> 00:02:00,590 And this, recall, was the velocity of the fuel. 28 00:02:00,590 --> 00:02:04,880 Now we're in position to apply our momentum principle, 29 00:02:04,880 --> 00:02:06,950 because we have expressions for the momentum 30 00:02:06,950 --> 00:02:11,150 of the system at time t and at time t 31 00:02:11,150 --> 00:02:13,160 and at time t plus delta t. 32 00:02:13,160 --> 00:02:15,980 So now this will be a big expression. 33 00:02:15,980 --> 00:02:23,120 So we'll write external force is the limit as delta t goes to 0. 34 00:02:23,120 --> 00:02:30,230 Now here, our first term, is of M of r plus delta M-r times 35 00:02:30,230 --> 00:02:35,960 the velocity of the rocket at time t plus delta t. 36 00:02:35,960 --> 00:02:42,440 And now we have the fuel term, minus delta M-r times u 37 00:02:42,440 --> 00:02:48,020 plus V of r of t plus delta t. 38 00:02:48,020 --> 00:02:49,400 And we have to subtract from that 39 00:02:49,400 --> 00:02:53,480 and I'll indicate that with a slightly different color. 40 00:02:53,480 --> 00:02:57,590 M-r V of r of t. 41 00:02:57,590 --> 00:03:03,860 And the whole thing, we're dividing by delta t. 42 00:03:03,860 --> 00:03:05,810 Now let's look at this expression 43 00:03:05,810 --> 00:03:08,510 first because there is some very nice simplifications. 44 00:03:08,510 --> 00:03:11,260 The first thing we can see, let's look at this term delta 45 00:03:11,260 --> 00:03:13,010 M-r times V of r. 46 00:03:13,010 --> 00:03:17,520 Notice we have minus delta M-r V of r here. 47 00:03:17,520 --> 00:03:19,790 So those two terms cancel. 48 00:03:19,790 --> 00:03:22,430 And we're just left with three other terms 49 00:03:22,430 --> 00:03:24,360 and let's write them out. 50 00:03:24,360 --> 00:03:29,210 So now we have the limit as delta t goes to 0. 51 00:03:29,210 --> 00:03:31,920 And I'm going to combine terms in the following way. 52 00:03:31,920 --> 00:03:38,750 M-r times V of r of t plus delta t. 53 00:03:38,750 --> 00:03:41,630 And over here, I have M-r minus V of r. 54 00:03:41,630 --> 00:03:50,090 So I have a minus V of r of t divided by delta t. 55 00:03:50,090 --> 00:03:52,520 And now I have one more term here. 56 00:03:52,520 --> 00:03:58,160 And I'm going to write this as minus the limit as delta t goes 57 00:03:58,160 --> 00:04:02,120 to 0 of delta M-r over delta t. 58 00:04:02,120 --> 00:04:03,440 Remember, this term cancelled. 59 00:04:03,440 --> 00:04:09,890 We only have u, the speed of the fuel relative to the rocket. 60 00:04:09,890 --> 00:04:14,030 And in both cases-- this is our first term 61 00:04:14,030 --> 00:04:15,450 and here's our second term. 62 00:04:15,450 --> 00:04:17,260 Let's look at these limits. 63 00:04:17,260 --> 00:04:18,950 Notice in here, we're just taking 64 00:04:18,950 --> 00:04:23,930 V of r of t plus delta t minus V of r-t divided by delta t. 65 00:04:23,930 --> 00:04:27,650 And that's precisely the definition of the derivative 66 00:04:27,650 --> 00:04:29,540 of the velocity of the rocket. 67 00:04:29,540 --> 00:04:32,030 So this first term, our expression 68 00:04:32,030 --> 00:04:36,620 becomes the external force is equal to the mass 69 00:04:36,620 --> 00:04:40,820 of our system times V of r, the derivative 70 00:04:40,820 --> 00:04:43,100 of the velocity of the rocket. 71 00:04:43,100 --> 00:04:45,800 And the second term, this is the rate 72 00:04:45,800 --> 00:04:49,180 that the fuel is changing in the rocket. 73 00:04:49,180 --> 00:04:53,010 This is rather the rate that the mass of the rocket is changing. 74 00:04:53,010 --> 00:05:00,350 So that's the derivative dM-r dt times the relative velocity 75 00:05:00,350 --> 00:05:02,540 of the fuel with respect to the rocket. 76 00:05:02,540 --> 00:05:08,710 So this equation here-- I'll box it off-- 77 00:05:08,710 --> 00:05:14,810 is called the rocket equation, which 78 00:05:14,810 --> 00:05:17,100 we've derived from the momentum principle 79 00:05:17,100 --> 00:05:19,580 using our momentum diagrams. 80 00:05:19,580 --> 00:05:23,030 And you can see that this is what people refer to 81 00:05:23,030 --> 00:05:25,660 as rocket science.