1 00:00:03,600 --> 00:00:06,780 Let's consider a gravitational slingshot. 2 00:00:06,780 --> 00:00:08,010 What is that? 3 00:00:08,010 --> 00:00:12,300 Well, once in a while, people like to send spacecrafts out 4 00:00:12,300 --> 00:00:14,370 into the solar system, particularly 5 00:00:14,370 --> 00:00:18,620 the outer solar system, to explore what's going on there. 6 00:00:18,620 --> 00:00:22,050 And because we can't on Earth give enough speed 7 00:00:22,050 --> 00:00:25,770 to these little spacecrafts, we need the big planets around us 8 00:00:25,770 --> 00:00:27,880 to help us a little. 9 00:00:27,880 --> 00:00:33,030 And so we can consider a large planet like Jupiter or Saturn. 10 00:00:33,030 --> 00:00:34,860 Here we have Saturn. 11 00:00:34,860 --> 00:00:37,290 And if we have a little spacecraft 12 00:00:37,290 --> 00:00:43,110 and we make it fly by close, then what actually happens 13 00:00:43,110 --> 00:00:46,080 is it will, due to the gravitational attraction 14 00:00:46,080 --> 00:00:49,420 of Saturn, acquire a kick in velocity. 15 00:00:49,420 --> 00:00:51,300 And that is a gravitational slingshot. 16 00:00:51,300 --> 00:00:53,180 So let's look at that. 17 00:00:53,180 --> 00:00:58,080 So our little spacecraft comes in with an initial velocity. 18 00:00:58,080 --> 00:01:03,180 And Saturn, of course, also has a velocity. 19 00:01:03,180 --> 00:01:07,620 And once it has passed, our little spacecraft 20 00:01:07,620 --> 00:01:11,430 will have a final velocity. 21 00:01:11,430 --> 00:01:15,030 And in order to calculate what this final velocity is going 22 00:01:15,030 --> 00:01:17,850 to be, what the increase in speed is going to be, 23 00:01:17,850 --> 00:01:20,160 we need the concept of relative velocity. 24 00:01:25,240 --> 00:01:30,789 And for that, we need to first consider some initial state. 25 00:01:30,789 --> 00:01:35,690 So we have the relative velocity initially. 26 00:01:35,690 --> 00:01:39,400 And that is the difference between those two velocities, 27 00:01:39,400 --> 00:01:41,811 between the spacecraft and Saturn. 28 00:01:41,811 --> 00:01:43,310 And what becomes very important here 29 00:01:43,310 --> 00:01:46,420 is that we look at the coordinate system. 30 00:01:46,420 --> 00:01:47,650 And keep that in mind. 31 00:01:47,650 --> 00:01:50,650 Otherwise, we're going to get a few sign errors. 32 00:01:50,650 --> 00:01:55,090 So the initial velocity of the spacecraft 33 00:01:55,090 --> 00:01:57,830 goes in the i hat direction. 34 00:01:57,830 --> 00:01:59,926 And the velocity-- the relative velocity 35 00:01:59,926 --> 00:02:01,300 is, of course, the difference, so 36 00:02:01,300 --> 00:02:04,180 minus the velocity of Saturn. 37 00:02:04,180 --> 00:02:08,240 But that one goes in the minus i hat direction. 38 00:02:08,240 --> 00:02:11,710 And then we have the final state. 39 00:02:11,710 --> 00:02:14,940 So V final relative. 40 00:02:14,940 --> 00:02:17,590 And here we have the final velocity 41 00:02:17,590 --> 00:02:20,170 of the spacecraft now going in the minus i hat 42 00:02:20,170 --> 00:02:24,040 direction minus the velocity of Saturn 43 00:02:24,040 --> 00:02:28,520 that also goes in the minus i hat direction. 44 00:02:28,520 --> 00:02:30,910 Now, there is one thing that we need to consider, 45 00:02:30,910 --> 00:02:33,370 which is this velocity of Saturn. 46 00:02:33,370 --> 00:02:35,200 This one actually is, of course, here, 47 00:02:35,200 --> 00:02:39,220 the initial velocity of Saturn, and this one is the final one. 48 00:02:39,220 --> 00:02:42,190 But because the mass of Saturn is 49 00:02:42,190 --> 00:02:45,730 much larger than the mass of the spacecraft, 50 00:02:45,730 --> 00:02:49,750 we can actually set the initial velocity of Saturn 51 00:02:49,750 --> 00:02:52,460 to the final velocity of Saturn. 52 00:02:52,460 --> 00:02:57,180 So we can turn this-- we can take this away here again, 53 00:02:57,180 --> 00:03:03,040 and just consider one velocity of Saturn. 54 00:03:03,040 --> 00:03:03,830 OK, good. 55 00:03:03,830 --> 00:03:06,820 There's one more thing that we need in order to solve this, 56 00:03:06,820 --> 00:03:10,330 because we need to know how the relative velocities are 57 00:03:10,330 --> 00:03:11,860 actually related. 58 00:03:11,860 --> 00:03:16,780 And the energy momentum law helps us there, 59 00:03:16,780 --> 00:03:20,050 because that gives us that the initial relative velocity 60 00:03:20,050 --> 00:03:25,000 equals minus the final relative velocity. 61 00:03:25,000 --> 00:03:27,840 OK, so we can plug that in now. 62 00:03:27,840 --> 00:03:31,360 What do we have here for the initial velocity? 63 00:03:31,360 --> 00:03:39,329 We have Vi minus two i's gives us a plus the Saturn velocity. 64 00:03:39,329 --> 00:03:43,390 And that equals-- minus minus gives us a 65 00:03:43,390 --> 00:03:47,230 plus-- the final velocity of the spacecraft. 66 00:03:47,230 --> 00:03:49,180 And then we have three minuses here, 67 00:03:49,180 --> 00:03:54,760 so that gives us a minus the Saturn velocity. 68 00:03:54,760 --> 00:03:59,890 And we know from the problem that the initial velocity here 69 00:03:59,890 --> 00:04:01,750 of the spacecraft was actually given 70 00:04:01,750 --> 00:04:04,520 at three Saturn velocities. 71 00:04:04,520 --> 00:04:05,890 So we can tally this up now. 72 00:04:05,890 --> 00:04:09,820 We have three-- we have three plus one is four. 73 00:04:09,820 --> 00:04:11,940 And we'll put this over on the other side. 74 00:04:11,940 --> 00:04:14,310 That gives us five. 75 00:04:14,310 --> 00:04:20,170 Five Saturn velocities equals our final velocity. 76 00:04:20,170 --> 00:04:24,370 So that's quite a good gain, I would say. 77 00:04:24,370 --> 00:04:28,630 And it nicely illustrates why big planets like Jupiter 78 00:04:28,630 --> 00:04:31,210 and Saturn are really, really helpful for the exploration 79 00:04:31,210 --> 00:04:32,980 of the solar system. 80 00:04:32,980 --> 00:04:37,480 And that is actually how the New Horizons mission made it out 81 00:04:37,480 --> 00:04:40,590 to Pluto, all the way out there.