字幕表 動画を再生する 英語字幕をプリント 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,