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Why distance is area under velocity-time line

Explore the relationship between velocity, time, and displacement. Discover how the area under a velocity-time graph represents distance traveled. Understand how the slope of the graph indicates acceleration. Delve into scenarios of constant velocity and constant acceleration, and learn to calculate distance in each case. Created by Sal Khan.

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Video transcript

Let's say I have something moving with a constant velocity of five meters per second. And we're just assuming it's moving to the right, just to give us a direction, because this is a vector quantity, so it's moving in that direction right over there. And let me plot its velocity against time. So this is my velocity. So I'm actually going to only plot the magnitude of the velocity, and you can specify that like this. So this is the magnitude of the velocity. And then on this axis I'm going to plot time. So we have a constant velocity of five meters per second. So its magnitude is five meters per second. And it's constant. It's not changing. As the seconds tick away the velocity does not change. So it's just moving five meters per second. Now, my question to you is how far does this thing travel after five seconds? So after five seconds-- so this is one second, two second, three seconds, four seconds, five seconds, right over here. So how far did this thing travel after five seconds? Well, we could think about it two ways. One, we know that velocity is equal to displacement over change in time. And displacement is just change in position over change in time. Or another way to think about it-- If you multiply both sides by change in time-- you get velocity times change in time, is equal to displacement. So what was of the displacement over here? Well, I know what the velocity is-- it's five meters per second. That's the velocity, let me color-code this. That is the velocity. And we know what the change in time is, it is five seconds. And so you get the seconds cancel out the seconds, you get five times five-- 25 meters-- is equal to 25 meters. And that's pretty straightforward. But the slightly more interesting thing is that's exactly the area under this rectangle right over here. What I'm going to show you in this video, that is in general, if you plot velocity, the magnitude of velocity. So you could say speed to versus time. Or let me just stay with the magnitude of the velocity versus time. The area under that curve is going to be the distance traveled, because, or the displacement. Because displacement is just the velocity times the change in time. So if you just take out a rectangle right over there. So let me draw a slightly different one where the velocity is changing. So let me draw a situation where you have a constant acceleration . The acceleration over here is going to be one meter per second, per second. So one meter per second, squared. And let me draw the same type of graph, although this is going to look a little different now. So this is my velocity axis. I'll give myself a little bit more space. So this is my velocity axis. I'm just going to draw the magnitude of the velocity, and this right over here is my time axis. So this is time. And let me mark some stuff off here. So one, two, three, four, five, six, seven, eight, nine, ten. And one, two, three, four, five, six, seven, eight, nine, ten. And the magnitude of velocity is going to be measured in meters per second. And the time is going to be measured in seconds. So my initial velocity, or I could say the magnitude of my initial velocity-- so just my initial speed, you could say, this is just a fancy way of saying my initial speed is zero. So my initial speed is zero. So after one second what's going to happen? After one second I'm going one meter per second faster. So now I'm going one meter per second. After two seconds, whats happened? Well now I'm going another meter per second faster than that. After another second-- if I go forward in time, if change in time is one second, then I'm going a second faster than that. And if you remember the idea of the slope from your algebra one class, that's exactly what the acceleration is in this diagram right over here. The acceleration, we know that acceleration is equal to change in velocity over change in time. Over here change in time is along the x-axis. So this right over here is a change in time. And this right over here is a change in velocity. When we plot velocity or the magnitude of velocity relative to time, the slope of that line is the acceleration. And since we're assuming the acceleration is constant, we have a constant slope. So we have just a line here. We don't have a curve. Now what I want to do is think about a situation. Let's say that we accelerate it one meter per second squared. And we do it for-- so the change in time is going to be five seconds. And my question to you is how far have we traveled? Which is a slightly more interesting question than what we've been asking so far. So we start off with an initial velocity of zero. And then for five seconds we accelerate it one meter per second squared. So one, two, three, four, five. So this is where we go. This is where we are. So after five seconds, we know our velocity. Our velocity is now five meters per second. But how far have we traveled? So we could think about it a little bit visually. We could say, look, we could try to draw rectangles over here. Maybe right over here, we have the velocity of one meter per second. So if I say one meter per second times the second, that'll give me a little bit of distance. And then the next one I have a little bit more of distance, calculated the same way. I could keep drawing these rectangles here, but then you're like, wait, those rectangles are missing, because I wasn't for the whole second, I wasn't only going one meter per second. I kept accelerating. So I actually, I should maybe split up the rectangles. I could split up the rectangles even more. So maybe I go every half second. So on this half-second I was going at this velocity. And I go that velocity for a half-second. Velocity times the time would give me the displacement. And I do it for the next half second. Same exact idea here. Gives me the displacement. So on and so forth. But I think what you see as you're getting-- is the more accurate-- the smaller the rectangles, you try to make here, the closer you're going to get to the area under this curve. And just like the situation here. This area under the curve is going to be the distance traveled. And lucky for us, this is just going to be a triangle, and we know how to figure out the area for triangle. So the area of a triangle is equal to one half times base times height. Which hopefully makes sense to you, because if you just multiply base times height, you get the area for the entire rectangle, and the triangle is exactly half of that. So the distance traveled in this situation, or I should say the displacement, just because we want to make sure we're focused on vectors. The displacement here is going to be-- or I should say the magnitude of the displacement, maybe, which is the same thing as the distance, is going to be one half times the base, which is five seconds, times the height, which is five meters per second. Times five meters. Let me do that in another color. Five meters per second. The seconds cancel out with the seconds. And we're left with one half times five times five meters. So it's one half times 25, which is equal to 12.5 meters. And so there's an interesting thing here, well one, there's a couple of interesting things. Hopefully you'll realize that if you're plotting velocity versus time, the area under the curve, given a certain amount of time, tells you how far you have traveled. The other interesting thing is that the slope of the curve tells you your acceleration. What's the slope over here? Well, It's completely flat. And that's because the velocity isn't changing. So in this situation, we have a constant acceleration. The magnitude of that acceleration is exactly zero. Our velocity is not changing. Here we have an acceleration of one meter per second squared, and that's why the slope of this line right over here is one. The other interesting thing, is, if even if you have constant acceleration, you could still figure out the distance by just taking the area under the curve like this. We were able to figure out there we were able to get 12.5 meters. The last thing I want to introduce you to-- actually, let me just do it until next video, and I'll introduce you to the idea of average velocity. Now that we feel comfortable with the idea, that the distance you traveled is the area under the velocity versus time curve.