1 00:00:00,391 --> 00:00:01,671 - [Voiceover] There's a miniature rocketship 2 00:00:01,671 --> 00:00:04,195 and it's full of tiny aliens that just got done 3 00:00:04,195 --> 00:00:07,268 investigating a new moon with lunar pools 4 00:00:07,268 --> 00:00:10,314 and all kinds of organic new life forms. 5 00:00:10,314 --> 00:00:12,800 But they're done investigating, so they're gonna blast off 6 00:00:12,800 --> 00:00:15,732 and take their findings home to tell all their friends. 7 00:00:15,732 --> 00:00:18,039 Let's say at some moment during their ascent, 8 00:00:18,039 --> 00:00:20,227 they're moving at four meters per second. 9 00:00:20,227 --> 00:00:21,207 And they're tiny aliens, 10 00:00:21,207 --> 00:00:25,034 their spaceship is only 2.9 kilograms, 11 00:00:25,034 --> 00:00:26,826 but they need to know, are they gonna be able 12 00:00:26,826 --> 00:00:28,375 to get off this moon or not? 13 00:00:28,375 --> 00:00:29,957 So they gotta pay attention to their speed, 14 00:00:29,957 --> 00:00:32,142 but instead of using a speedometer, 15 00:00:32,142 --> 00:00:35,478 they clever aliens, they use a force versus time graph. 16 00:00:35,478 --> 00:00:37,952 So on their dashboard, they've got a force versus time graph 17 00:00:37,952 --> 00:00:40,958 and it tells them what the net force is on them, 18 00:00:40,958 --> 00:00:44,107 so let's say this is the net force, not just any force, 19 00:00:44,107 --> 00:00:47,104 but this is the total force on them from rocket boosters 20 00:00:47,104 --> 00:00:49,913 and the force of gravity and whatever other forces 21 00:00:49,913 --> 00:00:52,672 there might be, they've got advanced force sensors. 22 00:00:52,672 --> 00:00:54,646 I mean come on, they can determine their net force, 23 00:00:54,646 --> 00:00:56,339 let's say, and it gives them this force 24 00:00:56,339 --> 00:00:57,799 as a function of time. 25 00:00:57,799 --> 00:01:00,581 But they want to know what is their velocity gonna be 26 00:01:00,581 --> 00:01:02,372 after nine seconds? 27 00:01:02,372 --> 00:01:04,319 So they check their force versus time readout, 28 00:01:04,319 --> 00:01:05,615 and this is the graph they get. 29 00:01:05,615 --> 00:01:06,883 And now they can determine it. 30 00:01:06,883 --> 00:01:07,785 And here's how they do it. 31 00:01:07,785 --> 00:01:09,614 They say, "All right there's a force, a net force 32 00:01:09,614 --> 00:01:13,631 "of three newtons acting for the first four seconds." 33 00:01:13,631 --> 00:01:15,890 So during this entire first four seconds 34 00:01:15,890 --> 00:01:18,610 there's a constant force of three newtons. 35 00:01:18,610 --> 00:01:20,312 And every alien worth his weight knows 36 00:01:20,312 --> 00:01:24,479 that force, the net force multiplied by the time duration 37 00:01:25,633 --> 00:01:27,584 during which that force is applied, 38 00:01:27,584 --> 00:01:29,340 gives you the net impulse. 39 00:01:29,340 --> 00:01:30,544 So this gives us the net impulse. 40 00:01:30,544 --> 00:01:34,032 If we take this constant three newtons that acts, 41 00:01:34,032 --> 00:01:37,171 multiplied by four seconds during which it acts, 42 00:01:37,171 --> 00:01:40,053 we get that there's an impulse of 12 newton seconds. 43 00:01:40,053 --> 00:01:42,543 And you might be like, "Wait, who cares 44 00:01:42,543 --> 00:01:45,402 "about newton seconds here, I want the velocity. 45 00:01:45,402 --> 00:01:46,811 "I don't want the force and the time, 46 00:01:46,811 --> 00:01:50,040 "I want to know the velocity at nine seconds." 47 00:01:50,040 --> 00:01:52,794 But they teach you at the alien space academy, 48 00:01:52,794 --> 00:01:55,067 that the net impulse is not only equal 49 00:01:55,067 --> 00:01:57,302 to the net force times the time, 50 00:01:57,302 --> 00:02:00,863 it's also equal to the change in momentum of the object 51 00:02:00,863 --> 00:02:02,525 that the force was exerted on. 52 00:02:02,525 --> 00:02:03,422 And this is good. 53 00:02:03,422 --> 00:02:04,978 We know the mass of the object. 54 00:02:04,978 --> 00:02:06,733 We wanna know something about velocity. 55 00:02:06,733 --> 00:02:08,880 So we know momentum is M times V, 56 00:02:08,880 --> 00:02:11,541 this net impulse is gonna help us get there. 57 00:02:11,541 --> 00:02:13,288 But his 12 newton seconds was only 58 00:02:13,288 --> 00:02:14,832 for the first four seconds. 59 00:02:14,832 --> 00:02:17,199 How do we figure it out for the next three seconds? 60 00:02:17,199 --> 00:02:18,032 Look at this. 61 00:02:18,032 --> 00:02:20,781 During the next three seconds, there's not a constant force, 62 00:02:20,781 --> 00:02:23,640 this force is varying, the force is getting smaller. 63 00:02:23,640 --> 00:02:24,486 So how do I do this? 64 00:02:24,486 --> 00:02:26,080 The force isn't a constant value, 65 00:02:26,080 --> 00:02:29,019 so I can't just simply take force times time, 66 00:02:29,019 --> 00:02:31,617 because, I mean, what force do I pick? 67 00:02:31,617 --> 00:02:33,356 So we're gonna use a trick. 68 00:02:33,356 --> 00:02:35,370 We're gonna use a trick, because if you notice 69 00:02:35,370 --> 00:02:38,028 for this first section, for the first four seconds, 70 00:02:38,028 --> 00:02:40,441 we took the force and multiplied 71 00:02:40,441 --> 00:02:42,909 by the time interval, four seconds. 72 00:02:42,909 --> 00:02:44,771 So what we did really is we just took 73 00:02:44,771 --> 00:02:47,235 the height of this rectangle times 74 00:02:47,235 --> 00:02:48,791 the width of this rectangle 75 00:02:48,791 --> 00:02:50,871 and that gives us the area of this rectangle. 76 00:02:50,871 --> 00:02:52,933 So what we really did is we found the area 77 00:02:52,933 --> 00:02:55,313 under the force versus time graph. 78 00:02:55,313 --> 00:02:58,648 That gave us the impulse and that's not a coincidence. 79 00:02:58,648 --> 00:03:02,815 The impulse equals the area under a force versus time graph, 80 00:03:04,058 --> 00:03:05,666 and this is extremely useful to know, 81 00:03:05,666 --> 00:03:08,949 because now in this section, where the force was varying, 82 00:03:08,949 --> 00:03:11,529 we can still use this, we can just find the impulse 83 00:03:11,529 --> 00:03:14,916 by determining the area under that curve. 84 00:03:14,916 --> 00:03:18,130 And by area under the curve, we mean from the line curve, 85 00:03:18,130 --> 00:03:21,486 in general, to the x-axis, which, in this case, 86 00:03:21,486 --> 00:03:24,846 the x-axis is of the time axis. 87 00:03:24,846 --> 00:03:25,679 So let's do this. 88 00:03:25,679 --> 00:03:27,925 We found the impulse for this first section. 89 00:03:27,925 --> 00:03:29,919 That was 12 newton seconds. 90 00:03:29,919 --> 00:03:32,620 Now we can find the impulse for this next section 91 00:03:32,620 --> 00:03:34,378 by just determining the area. 92 00:03:34,378 --> 00:03:35,376 So this is a triangle. 93 00:03:35,376 --> 00:03:40,221 We'll do 1/2 base, the base is one, two, three seconds, 94 00:03:40,221 --> 00:03:43,058 and the height is still three newtons. 95 00:03:43,058 --> 00:03:46,891 So we get a net impulse of 4.5 newton seconds. 96 00:03:48,044 --> 00:03:49,572 So we've got one more section to go, 97 00:03:49,572 --> 00:03:51,262 but this one's a little weird, this one's located, 98 00:03:51,262 --> 00:03:54,744 the area is located below the time axis, 99 00:03:54,744 --> 00:03:56,333 so this is still a triangle, 100 00:03:56,333 --> 00:03:58,350 but since the forces are negative, 101 00:03:58,350 --> 00:04:01,082 this is gonna count as a negative net impulse. 102 00:04:01,082 --> 00:04:03,062 So when the area lies above the time axis, 103 00:04:03,062 --> 00:04:04,797 it counts as a positive impulse, 104 00:04:04,797 --> 00:04:06,966 and when the area lies below the time axis, 105 00:04:06,966 --> 00:04:08,858 it counts as a negative net impulse. 106 00:04:08,858 --> 00:04:10,362 So how much negative net impulse? 107 00:04:10,362 --> 00:04:11,787 We still find the area. 108 00:04:11,787 --> 00:04:15,456 So the area of a triangle, again, is gonna be 1/2, 109 00:04:15,456 --> 00:04:17,682 the base this time is two seconds, 110 00:04:17,682 --> 00:04:20,422 and the height is negative two. 111 00:04:20,422 --> 00:04:23,780 So negative two newtons, which gives us a net impulse 112 00:04:23,780 --> 00:04:26,116 of negative two newton seconds. 113 00:04:26,116 --> 00:04:27,585 And now we can figure out the velocity 114 00:04:27,585 --> 00:04:30,733 of the spaceship at nine seconds. 115 00:04:30,733 --> 00:04:34,381 So, assuming that this force readout started at this moment 116 00:04:34,381 --> 00:04:38,018 right over here at t equals zero seconds 117 00:04:38,018 --> 00:04:40,665 was the moment when it was going four meters per second, 118 00:04:40,665 --> 00:04:44,441 then we could say the total net impulse should equal 119 00:04:44,441 --> 00:04:47,374 the total change in momentum of this spaceship. 120 00:04:47,374 --> 00:04:49,328 And we can find the total net impulse 121 00:04:49,328 --> 00:04:52,428 by just adding up all the individual impulses. 122 00:04:52,428 --> 00:04:53,988 So during the first four seconds, 123 00:04:53,988 --> 00:04:56,405 there was 12 newton seconds of impulse. 124 00:04:56,405 --> 00:04:58,044 During the next three seconds 125 00:04:58,044 --> 00:05:00,972 there was 4.5 newton seconds of impulse. 126 00:05:00,972 --> 00:05:02,257 And during this last portion 127 00:05:02,257 --> 00:05:05,461 there was negative two newton seconds of impulse, 128 00:05:05,461 --> 00:05:06,854 which if you add all of those up, 129 00:05:06,854 --> 00:05:09,697 12 plus 4.5 plus negative two, you're gonna get 130 00:05:09,697 --> 00:05:13,030 positive 14.5 newton seconds of impulse. 131 00:05:14,502 --> 00:05:16,899 That's good news for our alien buddies over here, 132 00:05:16,899 --> 00:05:18,552 they need to get off this moon, 133 00:05:18,552 --> 00:05:21,426 which means they need positive impulse, upward impulse. 134 00:05:21,426 --> 00:05:22,648 They got some positive impulse. 135 00:05:22,648 --> 00:05:24,284 Let's see what their final velocity was. 136 00:05:24,284 --> 00:05:27,033 We know that delta P is the change in momentum, 137 00:05:27,033 --> 00:05:31,145 so this is final momentum minus initial momentum, 138 00:05:31,145 --> 00:05:32,249 which we could write as 139 00:05:32,249 --> 00:05:36,774 mass times v final minus mass times v initial. 140 00:05:36,774 --> 00:05:38,834 Now, if this were an earth rocket, this would be hard, 141 00:05:38,834 --> 00:05:41,234 'cause earth rockets using earth technology, 142 00:05:41,234 --> 00:05:44,735 eject fuel at a huge rate out the backend, 143 00:05:44,735 --> 00:05:46,171 and that loses mass. 144 00:05:46,171 --> 00:05:48,451 That means this mass isn't gonna stay constant. 145 00:05:48,451 --> 00:05:51,526 So earth rockets essentially push fuel down 146 00:05:51,526 --> 00:05:53,192 which causes and equal and opposite force 147 00:05:53,192 --> 00:05:55,112 back on the rocket upward. 148 00:05:55,112 --> 00:05:57,833 But if you're losing mass, this mass doesn't stay constant, 149 00:05:57,833 --> 00:05:59,624 and this whole process is a lot harder, 150 00:05:59,624 --> 00:06:02,940 because m final and m initial aren't gonna be the same. 151 00:06:02,940 --> 00:06:05,251 Maybe this is where the phrase, "It's not rocket science" 152 00:06:05,251 --> 00:06:07,534 comes from, 'cause rocket science is a little harder 153 00:06:07,534 --> 00:06:08,833 when that mass changes. 154 00:06:08,833 --> 00:06:12,080 So let's just say, these clever aliens can eject 155 00:06:12,080 --> 00:06:14,928 only a little bit of fuel, they do so. 156 00:06:14,928 --> 00:06:16,054 You might say, "How?" 157 00:06:16,054 --> 00:06:18,319 Well, there's gotta be a certain amount of momentum, right, 158 00:06:18,319 --> 00:06:21,083 that they eject to give themselves momentum up. 159 00:06:21,083 --> 00:06:23,823 But let's say they can eject only a small mass 160 00:06:23,823 --> 00:06:26,577 at a huge speed, so there's not much fuel 161 00:06:26,577 --> 00:06:27,786 that they're losing. 162 00:06:27,786 --> 00:06:30,829 The fuel that they eject is ejected at enormous speed, 163 00:06:30,829 --> 00:06:32,842 so that they get their momentum upward, 164 00:06:32,842 --> 00:06:35,405 but they lose almost no mass. 165 00:06:35,405 --> 00:06:36,869 And that let's us solve this problem 166 00:06:36,869 --> 00:06:39,221 assuming that the mass is constant. 167 00:06:39,221 --> 00:06:41,390 So if we do that, if we assume the mass is constant, 168 00:06:41,390 --> 00:06:45,458 we get positive 14.5 newton seconds equals, 169 00:06:45,458 --> 00:06:47,562 we can pull the mass out, the mass is a constant, 170 00:06:47,562 --> 00:06:49,141 so we could just write it as 171 00:06:49,141 --> 00:06:51,808 m times v final minus v initial, 172 00:06:53,369 --> 00:06:55,626 since I can pull out a common factor of m, 173 00:06:55,626 --> 00:06:58,649 which means I can write this as 2.9 kilograms 174 00:06:58,649 --> 00:07:01,759 multiplied by the final velocity after nine seconds 175 00:07:01,759 --> 00:07:03,692 and I know it's after nine seconds, 'cause I added up 176 00:07:03,692 --> 00:07:06,136 all the impulse during the nine seconds, 177 00:07:06,136 --> 00:07:09,688 minus the initial velocity was four meters per second. 178 00:07:09,688 --> 00:07:13,004 So if I divide both sides by 2.9 kilograms, 179 00:07:13,004 --> 00:07:17,389 14.5 over 2.9 is 5 and that will be newton seconds 180 00:07:17,389 --> 00:07:21,144 over kilograms which has units of meters per second, 181 00:07:21,144 --> 00:07:22,155 and it's positive. 182 00:07:22,155 --> 00:07:24,633 And that's gonna equal the leftover over here 183 00:07:24,633 --> 00:07:28,622 is gonna be v final minus four meters per second. 184 00:07:28,622 --> 00:07:31,506 And now finally if I add four meters per second 185 00:07:31,506 --> 00:07:35,623 to both sides, I get the v final somewhere up here, 186 00:07:35,623 --> 00:07:40,319 v final of this rocket is gonna be nine meters per second. 187 00:07:40,319 --> 00:07:41,692 So after nine seconds 188 00:07:41,692 --> 00:07:44,123 it ended up going nine meters per second. 189 00:07:44,123 --> 00:07:46,268 That's just a numerical coincidence. 190 00:07:46,268 --> 00:07:49,103 The way you find it, so recapping the way we did this, 191 00:07:49,103 --> 00:07:51,335 we found the area under the curve, 192 00:07:51,335 --> 00:07:52,696 'cause the area under a curve 193 00:07:52,696 --> 00:07:55,160 under our force versus time graph represents 194 00:07:55,160 --> 00:07:57,276 the impulse on the object. 195 00:07:57,276 --> 00:07:58,664 We found it for the entire trip, 196 00:07:58,664 --> 00:08:00,642 noting that underneath the time axis, 197 00:08:00,642 --> 00:08:02,825 when this curve goes under the time axis, 198 00:08:02,825 --> 00:08:04,821 the net impulse is gonna be negative. 199 00:08:04,821 --> 00:08:06,275 We added up all the net impulse, 200 00:08:06,275 --> 00:08:08,861 we set it equal to the change in momentum, 201 00:08:08,861 --> 00:08:10,608 we plugged in our values and we solved 202 00:08:10,608 --> 00:00:00,000 for our velocity after nine seconds.