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Why distance is area under velocity-time line

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

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• 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).
• Calling it velocity is more accurate, because the positive and negative speeds can be considered directions. In this case right would be positive and left would be negative (even though Sal didn't include a negative speed in this example). Hope this helps!
• what is terminal velocity???
• 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.
• 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.
• just to enlarge slightly the answer of krytek
(m/s/s) = (m/s) / (s/1) = (m/s) * (1/s) = m/s^2
• Why does Sal use two lines on either side of something to show displacement?
• 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.
• 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.
• 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.
• 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?
• 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|.
• 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?
• 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.
• What is the difference between the velocity, magnitude of velocity, and the average velocity

Thanks to those of you who answer!
• 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.