# 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

#### Follow-on

#### Associated Problem

Last Modified: February 18, 2021