1 00:00:00,744 --> 00:00:03,600 So I'm curious about how much acceleration 2 00:00:03,600 --> 00:00:05,600 does a pilot, or the pilot and the plane, 3 00:00:05,600 --> 00:00:08,336 experience when they need to take off 4 00:00:08,336 --> 00:00:09,854 from an aircraft carrier? 5 00:00:09,854 --> 00:00:11,400 So I looked up a few statistics 6 00:00:11,400 --> 00:00:13,166 on the Internet, this right here is 7 00:00:13,166 --> 00:00:16,417 a picture of an F/A-18 Hornet right over here. 8 00:00:16,417 --> 00:00:21,400 It has a take-off speed of 260 kilometers per hour. 9 00:00:21,400 --> 00:00:24,067 If we want that to be a velocity, 260 km/hour 10 00:00:24,067 --> 00:00:27,330 in this direction, if it's taking off from 11 00:00:27,330 --> 00:00:30,667 this Nimitz class carrier right over here. 12 00:00:30,667 --> 00:00:32,867 And I also looked it up, and I found 13 00:00:32,867 --> 00:00:34,867 the runway length, or I should say 14 00:00:34,867 --> 00:00:36,467 the catapult length, because these planes 15 00:00:36,467 --> 00:00:38,824 don't take off just with their own power. 16 00:00:38,824 --> 00:00:40,658 They have their own thrusters going, 17 00:00:40,658 --> 00:00:42,400 but they also are catapulted off, 18 00:00:42,400 --> 00:00:44,164 so they can be really accelerated quickly 19 00:00:44,164 --> 00:00:46,467 off of the flight deck of this carrier. 20 00:00:46,467 --> 00:00:49,000 And the runway length of a Nimitz class carrier 21 00:00:49,000 --> 00:00:51,084 is about 80 meters. So this is where 22 00:00:51,084 --> 00:00:52,933 they take off from. This right over here 23 00:00:52,933 --> 00:00:54,533 is where they take off from. 24 00:00:54,533 --> 00:00:56,587 And then they come in and they land over here. 25 00:00:56,587 --> 00:00:58,421 But I'm curious about the take-off. 26 00:00:58,421 --> 00:01:01,667 So to do this, let's figure out, well let's just 27 00:01:01,667 --> 00:01:03,333 figure out the acceleration, and from that 28 00:01:03,333 --> 00:01:04,933 we can also figure out how long it takes 29 00:01:04,933 --> 00:01:07,400 them to be catapulted off the flight deck. 30 00:01:07,400 --> 00:01:10,067 So, let me get the numbers in one place, 31 00:01:10,067 --> 00:01:11,933 so the take-off velocity, I could say, 32 00:01:11,933 --> 00:01:16,733 is 260 km/hour, so let me write this down. 33 00:01:16,733 --> 00:01:18,414 So that has to be your final velocity 34 00:01:18,414 --> 00:01:20,200 when you're getting off, of the plane, 35 00:01:20,200 --> 00:01:21,757 if you want to be flying. 36 00:01:21,757 --> 00:01:24,867 So your initial velocity is going to be 0, 37 00:01:24,867 --> 00:01:26,667 and once again I'm going to use the convention 38 00:01:26,667 --> 00:01:30,800 that the direction of the vector is implicit. 39 00:01:30,800 --> 00:01:33,200 Positive means going in the direction of take-off, 40 00:01:33,200 --> 00:01:35,749 negative would mean going the other way. 41 00:01:35,749 --> 00:01:39,933 My initial velocity is 0, I'll denote it as a vector 42 00:01:39,933 --> 00:01:43,886 right here. My final velocity over here 43 00:01:43,886 --> 00:01:48,739 has to be 260 km/hour. 44 00:01:48,739 --> 00:01:50,667 And I want to convert everything to meters 45 00:01:50,667 --> 00:01:53,000 and seconds, just so that I can get my, 46 00:01:53,000 --> 00:01:56,333 at least for meters, so that I can use my runway length 47 00:01:56,333 --> 00:01:58,200 in meters. So let's just do it in meters per second, 48 00:01:58,200 --> 00:02:00,000 I have a feeling it'll be a little bit easier 49 00:02:00,000 --> 00:02:02,137 to understand when we talk about acceleration 50 00:02:02,137 --> 00:02:04,067 in those units as well. 51 00:02:04,067 --> 00:02:06,090 So if we want to convert this into seconds, 52 00:02:06,090 --> 00:02:08,639 we have, we'll put hours in the numerator, 53 00:02:08,639 --> 00:02:10,728 1 hour, so it cancels out with this hour, 54 00:02:10,728 --> 00:02:14,857 is equal to 3600 seconds. 55 00:02:14,857 --> 00:02:17,323 I'll just write 3600 s. And then 56 00:02:17,323 --> 00:02:20,200 if we want to convert it to meters, 57 00:02:20,200 --> 00:02:27,400 we have 1000 meters is equal to 1 km, 58 00:02:27,400 --> 00:02:29,133 and this 1 km will cancel out with those kms 59 00:02:29,133 --> 00:02:30,667 right over there. 60 00:02:30,667 --> 00:02:32,000 And whenever you're doing any type of 61 00:02:32,000 --> 00:02:33,667 this dimensional analysis, you really should see 62 00:02:33,667 --> 00:02:35,318 whether it makes sense. 63 00:02:35,318 --> 00:02:37,067 If I'm going 260 km in an hour, 64 00:02:37,067 --> 00:02:40,078 I should go much fewer km in a second 65 00:02:40,078 --> 00:02:41,402 because a second is so much shorter 66 00:02:41,402 --> 00:02:43,074 amount of time, and that's why we're dividing 67 00:02:43,074 --> 00:02:45,233 by 3600. If I can go a certain 68 00:02:45,233 --> 00:02:47,400 number of km in an hour a second, 69 00:02:47,400 --> 00:02:48,333 I should be able to go a lot, 70 00:02:48,333 --> 00:02:50,867 many many more meters in that same amount 71 00:02:50,867 --> 00:02:52,547 of time, and that's why we're multiplying 72 00:02:52,547 --> 00:02:54,600 by 1000. When you multiply these out, 73 00:02:54,600 --> 00:02:59,400 the hours cancel out, you have km canceling out, 74 00:02:59,400 --> 00:03:01,800 and you have 260 times 1000 75 00:03:01,800 --> 00:03:05,295 divided by 3600 meters per second. 76 00:03:05,295 --> 00:03:07,733 So let me get my trusty TI-85 out, 77 00:03:07,733 --> 00:03:12,052 and actually calculate that. 78 00:03:12,052 --> 00:03:26,754 So I have 260 times 1000 divided by 3600 79 00:03:26,754 --> 00:03:29,467 gets me, I'll just round it to 72, because 80 00:03:29,467 --> 00:03:31,133 that's about how many significant digits 81 00:03:31,133 --> 00:03:34,645 I can assume here. 72 meters per second. 82 00:03:34,645 --> 00:03:37,641 So all I did here is I converted the take-off velocity, 83 00:03:37,641 --> 00:03:41,667 so this is 72 m/s, this has to be the final velocity 84 00:03:41,667 --> 00:03:44,560 after accelerating. So let's think about 85 00:03:44,560 --> 00:03:46,867 what that acceleration could be, given that we know 86 00:03:46,867 --> 00:03:49,267 the length of the runway, and we're going to assume 87 00:03:49,267 --> 00:03:51,667 constant acceleration here, just to simplify things 88 00:03:51,667 --> 00:03:52,933 a little bit. But what does that 89 00:03:52,933 --> 00:03:54,667 constant acceleration have to be? 90 00:03:54,667 --> 00:03:56,890 So let's think a little bit about it. 91 00:03:56,890 --> 00:03:59,630 The total displacement, I'll do that in purple, 92 00:03:59,630 --> 00:04:02,467 the total displacement is going to be 93 00:04:02,467 --> 00:04:07,966 equal to our average velocity while we're accelerating, 94 00:04:07,966 --> 00:04:13,800 times the difference in time, or the amount of time 95 00:04:13,800 --> 00:04:16,046 it takes us to accelerate. 96 00:04:16,046 --> 00:04:19,466 Now, what is the average velocity here? 97 00:04:19,466 --> 00:04:23,600 It's going to be our final velocity, plus our initial velocity, 98 00:04:23,600 --> 00:04:26,144 over 2. It's just the average of the initial and final. 99 00:04:26,144 --> 00:04:28,608 And we can only do that because we are dealing 100 00:04:28,608 --> 00:04:30,733 with a constant acceleration. 101 00:04:30,733 --> 00:04:33,133 And what is our change in time over here? 102 00:04:33,133 --> 00:04:35,067 What is our change in time? 103 00:04:35,067 --> 00:04:37,800 Well our change in time is how long does it take 104 00:04:37,800 --> 00:04:42,133 us to get to that velocity? Or another way to think about it is: 105 00:04:42,133 --> 00:04:45,884 it is our change in velocity divided by our acceleration. 106 00:04:45,884 --> 00:04:50,505 If we're trying to get to 10 m/s, or we're trying to get 107 00:04:50,505 --> 00:04:54,267 10 m/s faster, and we're accelerating at 2 m/s squared, 108 00:04:54,267 --> 00:04:55,467 it'll take us 5 seconds. 109 00:04:55,467 --> 00:04:57,610 Or if you want to see that explicitly written in a formula, 110 00:04:57,610 --> 00:05:01,467 we know that acceleration is equal to 111 00:05:01,467 --> 00:05:05,760 change in velocity over change in time. 112 00:05:05,760 --> 00:05:08,267 You multiply both sides by change in time, 113 00:05:08,267 --> 00:05:10,000 and you divide both sides by acceleration, 114 00:05:10,000 --> 00:05:12,400 so let's do that, multiply both sides by change in time 115 00:05:12,400 --> 00:05:14,259 and divide by acceleration. 116 00:05:14,259 --> 00:05:18,200 Multiply by change in time and divide by acceleration. 117 00:05:18,200 --> 00:05:22,267 And you get, that cancels out, and then you have 118 00:05:22,267 --> 00:05:24,499 that cancels out, and you have change in time 119 00:05:24,499 --> 00:05:29,667 is equal to change in velocity divided by acceleration. 120 00:05:29,667 --> 00:05:32,800 Change in velocity divided by acceleration. 121 00:05:32,800 --> 00:05:34,507 So what's the change in velocity? 122 00:05:34,507 --> 00:05:36,733 Change in velocity, so this is going to be 123 00:05:36,733 --> 00:05:41,357 change in velocity divided by acceleration. 124 00:05:41,357 --> 00:05:42,867 Change in velocity is the same thing as your 125 00:05:42,867 --> 00:05:46,133 final velocity minus your initial velocity, 126 00:05:46,133 --> 00:05:49,467 all of that divided by acceleration. 127 00:05:49,467 --> 00:05:52,267 So this delta t part we can re-write as 128 00:05:52,267 --> 00:06:00,867 our final velocity minus our initial velocity, over acceleration. 129 00:06:00,867 --> 00:06:03,400 And just doing this simple little derivation here 130 00:06:03,400 --> 00:06:05,459 actually gives us a pretty cool result! 131 00:06:05,459 --> 00:06:08,872 If we just work through this math, and I'll try to 132 00:06:08,872 --> 00:06:11,667 write a little bigger, I see my writing is getting smaller, 133 00:06:11,667 --> 00:06:13,733 our displacement can be expressed as 134 00:06:13,733 --> 00:06:15,954 the product of these two things. 135 00:06:15,954 --> 00:06:17,626 And what's cool about this, well let me just 136 00:06:17,626 --> 00:06:21,620 write it this way: so this is our final velocity 137 00:06:21,620 --> 00:06:29,027 plus our initial velocity, times our final velocity 138 00:06:29,027 --> 00:06:32,067 minus our initial velocity, 139 00:06:32,067 --> 00:06:36,783 all of that over 2 times our acceleration. 140 00:06:36,783 --> 00:06:40,733 Our assumed constant acceleration. 141 00:06:40,733 --> 00:06:43,133 And you probably remember from algebra class 142 00:06:43,133 --> 00:06:47,200 this takes the form: a plus b times a minus b. 143 00:06:47,200 --> 00:06:50,800 And so this equal to -- and you can multiply it out 144 00:06:50,800 --> 00:06:52,667 and you can review in our algebra playlist 145 00:06:52,667 --> 00:06:54,801 how to multiply out two binomials like this, 146 00:06:54,801 --> 00:06:56,933 but this numerator right over here, 147 00:06:56,933 --> 00:06:59,027 I'll write it in blue, is going to be equal to 148 00:06:59,027 --> 00:07:03,867 our final velocity squared minus our initial velocity squared. 149 00:07:03,867 --> 00:07:06,272 This is a difference of squares, you can factor it out 150 00:07:06,272 --> 00:07:10,267 into the sum of the two terms times the difference 151 00:07:10,267 --> 00:07:13,067 of the two terms, so that when you multiply these two out 152 00:07:13,067 --> 00:07:21,452 you just get that over there, over 2 times the acceleration. 153 00:07:21,452 --> 00:07:24,667 Now what's really cool here is we were able to derive 154 00:07:24,667 --> 00:07:27,267 a formula that just deals with the displacement, 155 00:07:27,267 --> 00:07:31,600 our final velocity, our initial velocity, and the acceleration. 156 00:07:31,600 --> 00:07:35,067 And we know all of those things except for the acceleration. 157 00:07:35,067 --> 00:07:38,176 We know that our displacement is 80 meters. 158 00:07:38,176 --> 00:07:40,521 We know that this is 80 meters. 159 00:07:40,521 --> 00:07:42,774 We know that our final velocity, just before 160 00:07:42,774 --> 00:07:45,267 we square it, we know that our final velocity is 161 00:07:45,267 --> 00:07:50,133 72 meters per second. And we know that 162 00:07:50,133 --> 00:07:53,362 our initial velocity is 0 meters per second. 163 00:07:53,362 --> 00:07:55,133 And so we can use all of this information 164 00:07:55,133 --> 00:07:59,267 to solve for our acceleration. 165 00:07:59,267 --> 00:08:02,267 And you might see this formula, displacement, 166 00:08:02,267 --> 00:08:03,933 sometimes called distance, if you're just using 167 00:08:03,933 --> 00:08:07,271 the scalar version, and really we are thinking only 168 00:08:07,271 --> 00:08:08,757 in the scalar, we're thinking about the magnitudes 169 00:08:08,757 --> 00:08:10,684 of all of these things for the sake of this video. 170 00:08:10,684 --> 00:08:12,200 We're only dealing in one dimension. 171 00:08:12,200 --> 00:08:14,200 But sometimes you'll see it written like this, 172 00:08:14,200 --> 00:08:18,022 sometimes you'll multiply both sides times the 2 a, 173 00:08:18,022 --> 00:08:19,800 and you'll get something like this, where you have 174 00:08:19,800 --> 00:08:23,400 2 times, really the magnitude of the acceleration, 175 00:08:23,400 --> 00:08:26,067 times the magnitude of the displacement, 176 00:08:26,067 --> 00:08:27,133 which is the same thing as the distance, 177 00:08:27,133 --> 00:08:31,600 is equal to the final velocity, the magnitude of 178 00:08:31,600 --> 00:08:41,845 the final velocity, squared, minus the initial velocity squared. 179 00:08:41,845 --> 00:08:44,733 Or sometimes, in some books, it'll be written as 2 a d 180 00:08:44,733 --> 00:08:46,933 is equal to v f squared minus v i squared. 181 00:08:46,933 --> 00:08:48,602 And it seems like a super mysterious thing, 182 00:08:48,602 --> 00:08:50,600 but it's not that mysterious. We just very simply 183 00:08:50,600 --> 00:08:54,268 derived it from displacement, or if you want to say distance, 184 00:08:54,268 --> 00:08:55,847 if you're just thinking about the scalar quantity, 185 00:08:55,847 --> 00:08:59,533 is equal to average velocity times the change in time. 186 00:08:59,533 --> 00:09:02,023 So, so far we've just derived ourselves a kind of a 187 00:09:02,023 --> 00:09:05,274 neat formula that is often not derived in 188 00:09:05,274 --> 00:09:07,267 physics class, but let's use it to actually 189 00:09:07,267 --> 00:09:10,127 figure out the acceleration that a pilot experiences 190 00:09:10,127 --> 00:09:14,267 when they're taking off of a Nimitz class carrier. 191 00:09:14,267 --> 00:09:18,867 So we have 2 times the acceleration 192 00:09:18,867 --> 00:09:21,467 times the distance, that's 80 meters, 193 00:09:21,467 --> 00:09:25,267 times 80 meters, is going to be equal to 194 00:09:25,267 --> 00:09:27,400 our final velocity squared. 195 00:09:27,400 --> 00:09:30,421 What's our final velocity? 72 meters per second. 196 00:09:30,421 --> 00:09:35,667 So 72 meters per second, squared, 197 00:09:35,667 --> 00:09:37,867 minus our initial velocity. 198 00:09:37,867 --> 00:09:40,467 So our initial velocity in this situation is just 0. 199 00:09:40,467 --> 00:09:42,200 So it's just going to be minus 0 squared, 200 00:09:42,200 --> 00:09:43,933 which is just going to be 0, so we don't even have to 201 00:09:43,933 --> 00:09:46,838 write it down. And so to solve for acceleration, 202 00:09:46,838 --> 00:09:49,578 to solve for acceleration, you just divide, 203 00:09:49,578 --> 00:09:52,086 so this is the same thing as 160 meters, 204 00:09:52,086 --> 00:09:55,667 well, let's just divide both sides by 2 times 80, 205 00:09:55,667 --> 00:10:03,708 so we get acceleration is equal to 72 m/s squared 206 00:10:03,708 --> 00:10:11,361 over 2 times 80 meters. 207 00:10:11,361 --> 00:10:12,267 And what we're gonna get is, 208 00:10:12,267 --> 00:10:13,733 I'll just write this in one color, 209 00:10:13,733 --> 00:10:16,867 it's going to be 72 divided by 160, 210 00:10:16,867 --> 00:10:20,832 times, we have in the numerator, 211 00:10:20,832 --> 00:10:22,400 meters squared over seconds squared, 212 00:10:22,400 --> 00:10:26,173 we're squaring the units, and then we're 213 00:10:26,173 --> 00:10:28,680 going to be dividing by meters. 214 00:10:28,680 --> 00:10:30,533 So times, I'll do this in blue, 215 00:10:30,533 --> 00:10:33,733 times one over meters. Right? 216 00:10:33,733 --> 00:10:35,000 Because we have a meters in the denominator. 217 00:10:35,000 --> 00:10:37,341 And so what we're going to get is this 218 00:10:37,341 --> 00:10:39,431 meters squared divided by meters, 219 00:10:39,431 --> 00:10:40,940 that's going to cancel out, we're going to get 220 00:10:40,940 --> 00:10:42,267 meters per second squared. Which is cool 221 00:10:42,267 --> 00:10:44,841 because that's what acceleration should be in. 222 00:10:44,841 --> 00:10:46,933 And so let's just get the calculator out, 223 00:10:46,933 --> 00:10:48,766 to calculate this exact acceleration. 224 00:10:48,766 --> 00:10:53,267 So we have to take, oh sorry, this is 72 squared, 225 00:10:53,267 --> 00:10:57,600 let me write that down. So this is, this is going to be 226 00:10:57,600 --> 00:10:59,200 72 squared, don't want to forget about this part 227 00:10:59,200 --> 00:11:02,001 right over here. 72 squared divided by 160. 228 00:11:02,001 --> 00:11:06,267 So we have, and we can just use the original number 229 00:11:06,267 --> 00:11:07,867 right over here that we calculated, 230 00:11:07,867 --> 00:11:12,845 so let's just square that, and then divide that by 160, 231 00:11:12,845 --> 00:11:17,140 divided by 160. And if we go to 2 significant digits, 232 00:11:17,140 --> 00:11:22,551 we get 33, we get our acceleration is, our acceleration 233 00:11:22,551 --> 00:11:28,286 is equal to 33 meters per second squared. 234 00:11:28,286 --> 00:11:31,800 And just to give you an idea of how much acceleration 235 00:11:31,800 --> 00:11:35,467 that is, is if you are in free fall over Earth, 236 00:11:35,467 --> 00:11:40,709 the force of gravity will be accelerating you, 237 00:11:40,709 --> 00:11:46,630 so g is going to be equal to 9.8 meters per second squared. 238 00:11:46,630 --> 00:11:52,400 So this is accelerating you 3 times more than what 239 00:11:52,400 --> 00:11:54,667 Earth is making you accelerate if you were to 240 00:11:54,667 --> 00:11:56,034 jump off of a cliff or something. 241 00:11:56,034 --> 00:11:59,470 So another way to think about this is that the force, 242 00:11:59,470 --> 00:12:00,800 and we haven't done a lot on force yet, 243 00:12:00,800 --> 00:12:02,048 we'll talk about this in more depth, 244 00:12:02,048 --> 00:12:05,533 is that this pilot would be experiencing 245 00:12:05,533 --> 00:12:08,400 more than 3 times the force of gravity, 246 00:12:08,400 --> 00:12:11,800 more than 3 g's. 3 g's would be about 247 00:12:11,800 --> 00:12:15,533 30 meters per second squared, this is more than that. 248 00:12:15,533 --> 00:12:19,625 So an analogy for how the pilot would feel 249 00:12:19,625 --> 00:12:22,867 is when he's, you know, if this is the chair right here, 250 00:12:22,867 --> 00:12:26,707 his pilot's chair, that he's in, so this is the chair, 251 00:12:26,707 --> 00:12:29,133 and he's sitting on the chair, let me do my best 252 00:12:29,133 --> 00:12:32,733 to draw him sitting on the chair, so this is him 253 00:12:32,733 --> 00:12:37,133 sitting on the chair, flying the plane, and this is the pilot, 254 00:12:37,133 --> 00:12:41,000 the force he would feel, or while this thing is accelerating 255 00:12:41,000 --> 00:12:44,401 him forward at 33 meters per second squared, 256 00:12:44,401 --> 00:12:48,067 it would feel very much to him like if he was lying down 257 00:12:48,067 --> 00:12:52,467 on the surface of the planet, but he was 3 times heavier, 258 00:12:52,467 --> 00:12:56,475 or more than 3 times heavier. Or if he was lying down, 259 00:12:56,475 --> 00:12:59,958 or if you were lying down, like this, let's say this is you, 260 00:12:59,958 --> 00:13:04,667 this is your feet, and this is your face, this is your hands, 261 00:13:04,667 --> 00:13:07,040 let me draw your hands right here, and if you had 262 00:13:07,040 --> 00:13:11,475 essentially two more people stacked above you, 263 00:13:11,475 --> 00:13:14,067 roughly, I'm just giving you the general sense of it, 264 00:13:14,067 --> 00:13:16,467 that's how it would feel, a little bit more than two people, 265 00:13:16,467 --> 00:13:19,733 that squeezing sensation. So his entire body 266 00:13:19,733 --> 00:13:22,876 is going to feel 3 times heavier than it would 267 00:13:22,876 --> 00:13:25,600 if he was just laying down on the beach or something 268 00:13:25,600 --> 00:13:29,518 like that. So it's very very very interesting, I guess, 269 00:13:29,518 --> 00:13:32,733 idea, at least to me. Now the other question 270 00:13:32,733 --> 00:13:34,486 that we can ask ourselves is how long will it take 271 00:13:34,486 --> 00:13:39,067 to get catapulted off of this carrier? And if he's 272 00:13:39,067 --> 00:13:45,930 accelerating at 33 meters per second squared, 273 00:13:45,930 --> 00:13:47,675 how long would it take him to get from 0 274 00:13:47,675 --> 00:13:50,133 to 72 meters per second? 275 00:13:50,133 --> 00:13:53,067 So after 1 second, he'll be going 33 meters per second, 276 00:13:53,067 --> 00:13:56,533 after 2 seconds, he'll be going 66 meters per second, 277 00:13:56,533 --> 00:13:58,800 so it's going to take, and so it's a little bit more 278 00:13:58,800 --> 00:14:00,067 than 2 seconds. So it's going to take him 279 00:14:00,067 --> 00:14:01,533 a little bit more than 2 seconds. 280 00:14:01,533 --> 00:14:04,161 And we can calculate it exactly if you take 281 00:14:04,161 --> 00:14:07,733 72 meters per second, and you divide it by 33, 282 00:14:07,733 --> 00:14:13,600 it'll take him 2.18 seconds, roughly, to be catapulted 283 00:14:13,600 --> 00:00:00,000 off of that carrier.