If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Why distance is area under velocity-time line

AP.PHYS:
CHA‑4.A (EU)
,
CHA‑4.A.1 (EK)
,
CHA‑4.A.1.1 (LO)
,
CHA‑4.A.2 (EK)
,
CHA‑4.A.2.1 (LO)
,
CHA‑4.A.2.3 (LO)
Understanding why distance is area under velocity-time line. Created by Sal Khan.

Want to join the conversation?

  • orange juice squid orange style avatar for user Chris Pilcher
    Since you are only looking at the magnitude of the velocity for the y-axis, couldn't you just call it the speed, since you only care (for the purpose of this example) about the scalar quantities that make up part of the velocity?

    I do understand that usually velocity and speed are technically two different things, and I guess maybe you're just trying to introduce/reinforce the symbols used in physics (with the symbols you used to indicate the magnitude of the velocity).
    (129 votes)
    Default Khan Academy avatar avatar for user
  • spunky sam blue style avatar for user christi
    what is terminal velocity???
    (18 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user RedBengalTiger
      Building on the above, you also have to remember that she is still falling, because her velocity towards the ground is positive. The idea is that gravity isn't pulling her down anymore, at least in a sense, because the air resistance counters that. But for the counterbalance to work, her velocity must remain where it is and cannot change.
      (5 votes)
  • male robot johnny style avatar for user Andrew C.
    If it is 5 meters per second per second, then why is it referred to as 5 meters per second squared? When you say "per [...]", you are implying that you are dividing, while an exponent would imply multiplication.
    (6 votes)
    Default Khan Academy avatar avatar for user
  • leaf orange style avatar for user jandcnix
    Why does Sal use two lines on either side of something to show displacement?
    (7 votes)
    Default Khan Academy avatar avatar for user
    • leafers seed style avatar for user P
      Those lines mean magnitude. Like at "", when he says, according to the transcript,"I'm actually going to only plot the magnitude of velocity and you can specify that like this:||v||" See? The same thing with displacement.
      (12 votes)
  • leaf green style avatar for user utkarsh rai
    at ,why did sal took area under the graph as distance whereas in V-T graph area under the graph is displacement?please simplify
    thanks.
    (7 votes)
    Default Khan Academy avatar avatar for user
    • leaf red style avatar for user nbk186
      You can't fully represent displacement by finding the area, since as a vector quantity, displacement also requires a direction. Finding the area only gives an amount, no direction.


      The area under the curve is the magnitude of the displacement, which is equal to the distance traveled (only for constant acceleration). So in this case, they are interchangeable although it was probably a mistake by Sal to use both.
      (7 votes)
  • blobby green style avatar for user adityaysv05
    Hi, so this is not much related to the supposed content of the video, but rather related to the notation used.

    At the beginning of the video, a v/t graph is sketched and the narrator picks up modulus of velocity |v| which is to show the magnitude, I get that part, however-

    He uses double bars- ||v|| so I just want to ask does that give a special/different meaning to the magnitude of velocity?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Charles LaCour
      The notation of |x| and ||x|| both indicate the magnitude of a vector.

      The notation |x| is also used for scalar values to indicate the absolute value of x where as ||x|| is us usually only used for vectors.

      There is a more generic usage of ||x|| which is called the norm of a vector which the euclidean norm of a vector is what we would consider the length of the vector x. There are other types of norms that are not the same a the vector's length. In the more generic version of ||x|| ≥ |x|.
      (9 votes)
  • blobby green style avatar for user Jazmine Crutchfield
    I understand most of whats happening here but I do not understand where the half came from. I know that displacement is velocity times time but where does that half come in?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Mitchell Devisser
      That is a great question. The 1/2 comes from the fact that for the area of a triangle: Area = bh/2. Since we know that the area under the curve of a Velocity vs. Time graph represents the total displacement (on that time interval) it is just a matter of calculating the area under the given triangle.
      (6 votes)
  • blobby green style avatar for user nishal.f.dsouza
    What is the difference between the velocity, magnitude of velocity, and the average velocity

    Thanks to those of you who answer!
    (4 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Jesse William Olson
      Velocity is a speed in a direction at one specific time.Magnitude of velocity is speed (you get rid of the direction part) at one specific time.Average (magnitude of) velocity is average speed over a period of time.

      I think the direction doesn't matter at the moment, because we're assuming forward movement in these examples so far.
      (4 votes)
  • purple pi purple style avatar for user antru338
    does initial velocity have a different alphabet representing it than final velocity?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user utkarsh rai
    what is escape velocity ? please answer in simple terms...
    thanks
    (1 vote)
    Default Khan Academy avatar avatar for user

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.