This right here is a picture of an Airbus A380 aircraft. And I was curious how long would it take this plane to take off? And I looked up its takeoff velocity. And the specs I got were 280 kilometers per hour. And to make this a velocity we have to specify a direction as well, not just a magnitude. So the course is in the direction of the runway. So that would be the specific direction right over there. So when we’re talking about acceleration or velocity in this, we’re going to assume it’s in this course, the direction of going down the runway. And I also looked up its specs, and this, I’m simplifying a little bit, because it’s not going to have a purely constant acceleration. But let’s just say from the moment that the pilot means we’re taking off to when it actually takes off it has a standard acceleration. Its engines are able to provide a constant acceleration of meters per second per second. So after every second, it can go one meter per second faster than it was going at the beginning of that second.

Or another way to write this is 1.0– let me write it this way– meters per second per second can also be written as meters per second squared. I find this a little bit more intuitive. This is a little bit neater to write. So let’s figure this out. So the first thing we’re trying to answer is, how long does take off last? That is the question we will try to explain. And to solve this, at least my brain wants to at least get the units right. So over here we have our acceleration regarding meters and seconds, or seconds squared. And over here we have our takeoff velocity regarding kilometers and hours. So let’s just convert this takeoff velocity into meters per second. And then it might simplify answering this question. So if we have 280 kilometers per hour, how do we convert that to meters per second? So let’s switch it to kilometers per second first. So we want to get rid of this hours. And the best way to do that, if we have an hour in the denominator, we want an hour in the numerator, and we want a second in the denominator.

And so what do we multiply this by? Or what do we put in front of the hours and seconds? So one hour, in one hour there are 3,600 seconds, 60 seconds in a minute, 60 minutes in an hour. And so you have one of the larger units is equal to 3,600 of the smaller group. And that we can multiply by that. And if we do that, the hours will cancel out. And we’ll get 280 divided by 3,600 kilometers per second.

But I want to do all my math at once. So let’s also do the conversion from kilometers to meters. So once again, we have kilometers in the numerator. So we want the kilometers in the denominator now. So it cancels out. And we want meters in the numerator. And what’s the smaller unit? It’s metered. And we have 1,000 meters for every 1 kilometer. And so when you multiply this out the kilometers are going to cancel out. And you are going to be left with 280 times 1, so we don’t have to write it down, times 1,000, all of that over 3,600, and the units we have left are meters per– and the only unit we have left here is second– meters per second. So let’s get my trusty TI-85 out and actually calculate this. So we have 280 times 1000, which is obviously 280,000, but let me just divide that by 3,600. And it gives me repeating indefinitely. And it looks like I had two significant digits in each of these original things. I had over here, not 100% clear how many significant digits over here. Was the spec rounded to the nearest 10 kilometers? Or is it exactly 280 kilometers per hour? Just to be safe I’ll assume that it’s rounded to the nearest 10 kilometers.

So we only have two significant digits here. So we should only have two significant digits in our answer. So we’re going to round this to 78 meters per second. So this is going to be 78 meters per second, which is pretty fast. For this thing to take off every second that goes by it has to travel 78 meters, roughly 3/4 of the length of a football field in every second. But that’s not what we’re trying to answer. We’re trying to say how long will take off last? Well we could just do this in our head if you think about it. The acceleration is 1 meter per second, per second. Which tells us after every second it’s going 1 meter per second faster.

So if you start at a velocity of 0 and then after 1 second it’ll be going 1 meter per second. After 2 seconds it will be going 2 meters per second. After 3 seconds it’ll be going 3 meters per second. So how long will it take to get to 78 meters per second? Well, it will take 78 seconds, or roughly a minute and 18 seconds. And just to verify this with our definition of our acceleration, so to speak, just remember acceleration, which is a vector quantity, and all the directions we’re talking about now are in the direction of this direction of the runway. The acceleration is equal to change in velocity over change in time. And we’re trying to solve for how much time does it take, or the change in time. So let’s do that. So let’s multiply both sides by change in time. You get change in time times acceleration is equal to change in velocity. And to solve for change in time, divide both sides by the acceleration. So divide both sides by the acceleration you get a change in time. I could go down here, but I just want to use all this real estate I have over here.

I have change in time is equal to change in velocity divided by acceleration. And in this situation, what is our change in velocity? Well, we’re starting off with the velocity, or we’re assuming we’re starting off with a velocity of 0 meters per second. And we’re getting up to 78 meters per second. So our change in velocity is the 78 meters per second.

So this is equal, in our situation, 78 meters per second is our change in velocity. I’m taking the final velocity, 78 meters per second, and subtract from that the initial velocity, which is 0 meters per second. And you just get this. Divided by the acceleration, divided by 1 meter per second per second, or 1 meter per second squared. So the numbers part are pretty easy. You have 78 divided by 1, which is just 78. And then the units you have meters per second. And then if you divide by meters per second squared, that’s the same thing as multiplying by seconds squared per meter. Right? Dividing by something the same thing as multiplying by its reciprocal. And you can do the same thing with units.

And then we see the meters cancel out. And then seconds squared divided by seconds, you’re just left with seconds. So once again, we get 78 seconds, a little over a minute for this thing to take off..

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