# Calculating with average velocity

#### 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

$$\langle v \rangle = Δx/Δt$$

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

$$Δx = \langle v\rangle Δt$$

$$Δt = Δx/\langle v \rangle$$

(Check the dimensions on these equations.)

## Special Case: Constant Acceleration

If the velocity is changing at a uniform rate (constant acceleration so $\langle a \rangle = a_0$,  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:

$$\langle v \rangle = \frac{v_i + v_f}{2}$$

Since we know the rate of change equations

$$\langle v \rangle = \frac{Δx}{Δt} =\frac{(x_f - x_i)}{\Delta t}$$

$$\langle a \rangle =\frac{ Δv}{Δt} =\frac{(v_f - v_i)}{\Delta t}$$

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 along with the conditions when each one applies!

Joe Redish 9/7/11

Article 326