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
Current time:0:00Total duration:9:26

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)

Video transcript

let's save 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 this is my velocity so I'm going to actually going to only plot the magnitude of the velocity and you could specify that like this so that's the this is the magnitude of the velocity and then on this axis I'm going to plot I am going to plot time so we have a constant velocity of five meters per second so it's magnitude is five meters per second 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 it's just 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 second four second 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 we know that we know that velocity is equal to displacement is equal to displacement over change in time and displacement is just change in position over change so this is 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 is equal to displacement so what was the displacement over here well I know what the velocity is it's five meters per second I know 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 here it is five seconds it is five seconds and so you get the seconds cancel out the seconds you get five times five twenty five meters is equal to twenty five meters and that's pretty straightforward but the slightly more interesting thing is is that's exactly the area under this rectangle right over here that is exactly this area right over here now 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 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 it's just if you just take out a rectangle right over there so let me draw a slightly different one where the velocity is changing and 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 all that's going to look a little different now so this is my velocity axis give myself a little bit more space so this is my velocity axis so I'm just going to draw out the magnitude of the velocity and this right over here is my time axis so this is time 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 time is going to be measured in seconds so what's going to happen here assuming that we start with so my initial velocity my initial velocity or I could say the magnitude of my initial velocity the magnitude of my initial velocity so just this 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 what's happened well now I'm going another meter per second faster than that after another second if I if I go forward in time if change in time is 1 second then I'm going a second faster than that and if you remember the idea of slope from your Algebra one class that's exactly what the acceleration is in this diagram right over here the acceleration we know we know that acceleration is equal to change in velocity 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 so in this when we when we plot velocity or the magnitude of velocity relative to time the slope of that line is the acceleration and since the 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 accelerated 1 meter per second squared and we do it for we do it for so the time so the change in time is going to be is going to be 5 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 0 and then for 5 seconds we accelerate at 1 meter per second squared so 1 2 3 4 5 so this is where we go this is where we are so after 5 seconds we know our velocity our velocity is now 5 meters per second 5 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 we were at maybe right over here we had a velocity of 1 meter per second so if I say 1 meter per second times a second that'll give me that'll give me a little bit of distance and then the next one I have a little bit more of distance calculate it the same way I could keep drawing these rectangles here but then you're like wait those rectangles are missing there you know because I wasn't for the whole second I wasn't only going 1 meter per second I kept accelerating so 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 at that velocity for a half second velocity times the time will give me the displacement and then I do it the next half second same exactly idea here give me the displacement so on and so forth well I think what you see is you're getting is the more accurate your the smaller the rectangles you try to make here the closer you're going to get to the area under this under this curve the area under this curve and just like the situation here this area under the curve is going to be the distance is going to be the distance traveled 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 a triangle so the area of a triangle is so area of a triangle is equal to 1/2 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 the distance traveled 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 1/2 times the base which is 5 seconds which is 5 seconds 5 seconds times the height which is 5 meters per second times 5 meters let me do that in that other color 5 meters per second 5 meters per second the seconds cancel out with the seconds and we're left with 1/2 times 5 times 5 meters so it's 1/2 times 25 which is equal to 12 point 5 meters and so there's an interesting thing here well one there's a couple of interesting things hopefully you 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 is the slope of the curve tells you your acceleration what's the slope over here well where it's completely flat and that's because the velocity isn't changing so in this situation we have a constant acceleration we have a constant acceleration the magnitude of that acceleration is exactly zero our velocity is not changing here we have a 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 you want to even if you have constant acceleration you can still figure out the distance by just taking the area under the curve like this we were able to figure out that we were able to get twelve point five meters the last thing I want to introduce you to actually let me just do it into the 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 this the velocity versus time curve