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  • So I'm curious about how much acceleration

  • does a pilot, or the pilot and the plane,

  • experience when they need to take off

  • from an aircraft carrier?

  • So I looked up a few statistics

  • on the Internet, this right here is

  • a picture of an F/A-18 Hornet right over here.

  • It has a take-off speed of 260 kilometers per hour.

  • If we want that to be a velocity, 260 km/hour

  • in this direction, if it's taking off from

  • this Nimitz class carrier right over here.

  • And I also looked it up, and I found

  • the runway length, or I should say

  • the catapult length, because these planes

  • don't take off just with their own power.

  • They have their own thrusters going,

  • but they also are catapulted off,

  • so they can be really accelerated quickly

  • off of the flight deck of this carrier.

  • And the runway length of a Nimitz class carrier

  • is about 80 meters. So this is where

  • they take off from. This right over here

  • is where they take off from.

  • And then they come in and they land over here.

  • But I'm curious about the take-off.

  • So to do this, let's figure out, well let's just

  • figure out the acceleration, and from that

  • we can also figure out how long it takes

  • them to be catapulted off the flight deck.

  • So, let me get the numbers in one place,

  • so the take-off velocity, I could say,

  • is 260 km/hour, so let me write this down.

  • So that has to be your final velocity

  • when you're getting off, of the plane,

  • if you want to be flying.

  • So your initial velocity is going to be 0,

  • and once again I'm going to use the convention

  • that the direction of the vector is implicit.

  • Positive means going in the direction of take-off,

  • negative would mean going the other way.

  • My initial velocity is 0, I'll denote it as a vector

  • right here. My final velocity over here

  • has to be 260 km/hour.

  • And I want to convert everything to meters

  • and seconds, just so that I can get my,

  • at least for meters, so that I can use my runway length

  • in meters. So let's just do it in meters per second,

  • I have a feeling it'll be a little bit easier

  • to understand when we talk about acceleration

  • in those units as well.

  • So if we want to convert this into seconds,

  • we have, we'll put hours in the numerator,

  • 1 hour, so it cancels out with this hour,

  • is equal to 3600 seconds.

  • I'll just write 3600 s. And then

  • if we want to convert it to meters,

  • we have 1000 meters is equal to 1 km,

  • and this 1 km will cancel out with those kms

  • right over there.

  • And whenever you're doing any type of

  • this dimensional analysis, you really should see

  • whether it makes sense.

  • If I'm going 260 km in an hour,

  • I should go much fewer km in a second

  • because a second is so much shorter

  • amount of time, and that's why we're dividing

  • by 3600. If I can go a certain

  • number of km in an hour a second,

  • I should be able to go a lot,

  • many many more meters in that same amount

  • of time, and that's why we're multiplying

  • by 1000. When you multiply these out,

  • the hours cancel out, you have km canceling out,

  • and you have 260 times 1000

  • divided by 3600 meters per second.

  • So let me get my trusty TI-85 out,

  • and actually calculate that.

  • So I have 260 times 1000 divided by 3600

  • gets me, I'll just round it to 72, because

  • that's about how many significant digits

  • I can assume here. 72 meters per second.

  • So all I did here is I converted the take-off velocity,

  • so this is 72 m/s, this has to be the final velocity

  • after accelerating. So let's think about

  • what that acceleration could be, given that we know

  • the length of the runway, and we're going to assume

  • constant acceleration here, just to simplify things

  • a little bit. But what does that

  • constant acceleration have to be?

  • So let's think a little bit about it.

  • The total displacement, I'll do that in purple,

  • the total displacement is going to be

  • equal to our average velocity while we're accelerating,

  • times the difference in time, or the amount of time

  • it takes us to accelerate.

  • Now, what is the average velocity here?

  • It's going to be our final velocity, plus our initial velocity,

  • over 2. It's just the average of the initial and final.

  • And we can only do that because we are dealing

  • with a constant acceleration.

  • And what is our change in time over here?

  • What is our change in time?

  • Well our change in time is how long does it take

  • us to get to that velocity? Or another way to think about it is:

  • it is our change in velocity divided by our acceleration.

  • If we're trying to get to 10 m/s, or we're trying to get

  • 10 m/s faster, and we're accelerating at 2 m/s squared,

  • it'll take us 5 seconds.

  • Or if you want to see that explicitly written in a formula,

  • we know that acceleration is equal to

  • change in velocity over change in time.

  • You multiply both sides by change in time,

  • and you divide both sides by acceleration,

  • so let's do that, multiply both sides by change in time

  • and divide by acceleration.

  • Multiply by change in time and divide by acceleration.

  • And you get, that cancels out, and then you have

  • that cancels out, and you have change in time

  • is equal to change in velocity divided by acceleration.

  • Change in velocity divided by acceleration.

  • So what's the change in velocity?

  • Change in velocity, so this is going to be

  • change in velocity divided by acceleration.

  • Change in velocity is the same thing as your

  • final velocity minus your initial velocity,

  • all of that divided by acceleration.

  • So this delta t part we can re-write as

  • our final velocity minus our initial velocity, over acceleration.

  • And just doing this simple little derivation here

  • actually gives us a pretty cool result!

  • If we just work through this math, and I'll try to

  • write a little bigger, I see my writing is getting smaller,

  • our displacement can be expressed as

  • the product of these two things.

  • And what's cool about this, well let me just

  • write it this way: so this is our final velocity

  • plus our initial velocity, times our final velocity

  • minus our initial velocity,

  • all of that over 2 times our acceleration.

  • Our assumed constant acceleration.

  • And you probably remember from algebra class

  • this takes the form: a plus b times a minus b.

  • And so this equal to -- and you can multiply it out

  • and you can review in our algebra playlist

  • how to multiply out two binomials like this,

  • but this numerator right over here,

  • I'll write it in blue, is going to be equal to

  • our final velocity squared minus our initial velocity squared.

  • This is a difference of squares, you can factor it out

  • into the sum of the two terms times the difference

  • of the two terms, so that when you multiply these two out

  • you just get that over there, over 2 times the acceleration.

  • Now what's really cool here is we were able to derive

  • a formula that just deals with the displacement,

  • our final velocity, our initial velocity, and the acceleration.

  • And we know all of those things except for the acceleration.

  • We know that our displacement is 80 meters.

  • We know that this is 80 meters.

  • We know that our final velocity, just before

  • we square it, we know that our final velocity is

  • 72 meters per second. And we know that

  • our initial velocity is 0 meters per second.

  • And so we can use all of this information

  • to solve for our acceleration.

  • And you might see this formula, displacement,

  • sometimes called distance, if you're just using

  • the scalar version, and really we are thinking only

  • in the scalar, we're thinking about the magnitudes

  • of all of these things for the sake of this video.

  • We're only dealing in one dimension.

  • But sometimes you'll see it written like this,

  • sometimes you'll multiply both sides times the 2 a,

  • and you'll get something like this, where you have

  • 2 times, really the magnitude of the acceleration,

  • times the magnitude of the displacement,

  • which is the same thing as the distance,

  • is equal to the final velocity, the magnitude of

  • the final velocity, squared, minus the initial velocity squared.

  • Or sometimes, in some books, it'll be written as 2 a d

  • is equal to v f squared minus v i squared.

  • And it seems like a super mysterious thing,

  • but it's not that mysterious. We just very simply

  • derived it from displacement, or if you want to say distance,

  • if you're just thinking about the scalar quantity,

  • is equal to average velocity times the change in time.

  • So, so far we've just derived ourselves a kind of a

  • neat formula that is often not derived in

  • physics class, but let's use it to actually

  • figure out the acceleration that a pilot experiences

  • when they're taking off of a Nimitz class carrier.

  • So we have 2 times the acceleration

  • times the distance, that's 80 meters,

  • times 80 meters, is going to be equal to

  • our final velocity squared.

  • What's our final velocity? 72 meters per second.

  • So 72 meters per second, squared,

  • minus our initial velocity.

  • So our initial velocity in this situation is just 0.

  • So it's just going to be minus 0 squared,

  • which is just going to be 0, so we don't even have to

  • write it down. And so to solve for acceleration,

  • to solve for acceleration, you just divide,

  • so this is the same thing as 160 meters,

  • well, let's just divide both sides by 2 times 80,

  • so we get acceleration is equal to 72 m/s squared

  • over 2 times 80 meters.

  • And what we're gonna get is,

  • I'll just write this in one color,

  • it's going to be 72 divided by 160,

  • times, we have in the numerator,

  • meters squared over seconds squared,

  • we're squaring the units, and then we're

  • going to be dividing by meters.

  • So times, I'll do this in blue,

  • times one over meters. Right?

  • Because we have a meters in the denominator.

  • And so what we're going to get is this

  • meters squared divided by meters,

  • that's going to cancel out, we're going to get

  • meters per second squared. Which is cool

  • because that's what acceleration should be in.

  • And so let's just get the calculator out,

  • to calculate this exact acceleration.

  • So we have to take, oh sorry, this is 72 squared,

  • let me write that down. So this is, this is going to be

  • 72 squared, don't want to forget about this part

  • right over here. 72 squared divided by 160.

  • So we have, and we can just use the original number

  • right over here that we calculated,

  • so let's just square that, and then divide that by 160,

  • divided by 160. And if we go to 2 significant digits,

  • we get 33, we get our acceleration is, our acceleration

  • is equal to 33 meters per second squared.

  • And just to give you an idea of how much acceleration

  • that is, is if you are in free fall over Earth,

  • the force of gravity will be accelerating you,

  • so g is going to be equal to 9.8 meters per second squared.

  • So this is accelerating you 3 times more than what

  • Earth is making you accelerate if you were to

  • jump off of a cliff or something.

  • So another way to think about this is that the force,

  • and we haven't done a lot on force yet,

  • we'll talk about this in more depth,

  • is that this pilot would be experiencing

  • more than 3 times the force of gravity,

  • more than 3 g's. 3 g's would be about

  • 30 meters per second squared, this is more than that.

  • So an analogy for how the pilot would feel

  • is when he's, you know, if this is the chair right here,

  • his pilot's chair, that he's in, so this is the chair,

  • and he's sitting on the chair, let me do my best

  • to draw him sitting on the chair, so this is him

  • sitting on the chair, flying the plane, and this is the pilot,

  • the force he would feel, or while this thing is accelerating

  • him forward at 33 meters per second squared,

  • it would feel very much to him like if he was lying down

  • on the surface of the planet, but he was 3 times heavier,

  • or more than 3 times heavier. Or if he was lying down,

  • or if you were lying down, like this, let's say this is you,

  • this is your feet, and this is your face, this is your hands,

  • let me draw your hands right here, and if you had

  • essentially two more people stacked above you,

  • roughly, I'm just giving you the general sense of it,

  • that's how it would feel, a little bit more than two people,

  • that squeezing sensation. So his entire body

  • is going to feel 3 times heavier than it would

  • if he was just laying down on the beach or something

  • like that. So it's very very very interesting, I guess,

  • idea, at least to me. Now the other question

  • that we can ask ourselves is how long will it take

  • to get catapulted off of this carrier? And if he's

  • accelerating at 33 meters per second squared,

  • how long would it take him to get from 0

  • to 72 meters per second?

  • So after 1 second, he'll be going 33 meters per second,

  • after 2 seconds, he'll be going 66 meters per second,

  • so it's going to take, and so it's a little bit more

  • than 2 seconds. So it's going to take him

  • a little bit more than 2 seconds.

  • And we can calculate it exactly if you take

  • 72 meters per second, and you divide it by 33,

  • it'll take him 2.18 seconds, roughly, to be catapulted

  • off of that carrier.

So I'm curious about how much acceleration

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航母起飛的加速度|一維運動|物理學|可汗學院 (Acceleration of aircraft carrier take-off | One-dimensional motion | Physics | Khan Academy)

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    楊凱翔 發佈於 2021 年 01 月 14 日
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