*Class content I* > *The Main Question: Motion > Kinematics > Kinematic Variables > Velocity*

3.2.1.3

**Prerequisites**

The conceptual idea behind average velocity is fairly simple. Suppose that you have moved a certain displacement in a certain time. Your motion might have been quite complicated -- starting, stopping, back and forth. If you then ask, "If I went at a constant velocity -- instead of the mess I actually did -- what velocity would I have to go at to do that displacement in the same time?" Since

average velocity = (distance traveled) / (time taken)

or

<v> = Δx/Δt

then we can see that if we know the average velocity, then given one of the other variables -- the displacement or the time -- we can calculate the missing one. So we have the equations

Δx = <v>Δt

Δt = Δx/<v>.

(Check the dimensions on these equations.)

## Special Case: Constant Acceleration

If the velocity is changing at a uniform rate (constant acceleration so <a> = a_{0}, some constant), then it's pretty obvious that over a time interval in which the velocity changes from *v*_{i} to *v*_{f}, the average velocity will be the average of the initial and final values:

<v> = (v_{i} + v_{f})/2

Since we know the rate of change equations

<v> = Δx/Δt = (x_{f} - x_{i})/Δt

<a> = a_{0} = Δv/Δt = (v_{f} - v_{i})/Δt

we can use these three equations to find lots of things. Since each of these equations make sense, and if you remember that the "delta means change", you **won't** have to memorize all the different possible equations that describe this situation.

**Follow-ons:**

Joe Redish 9/7/11

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