# Average acceleration

#### Prerequisites

The conceptual idea behind acceleration is the same kind of thing we did with velocity:

Average acceleration = (How much did your velocity change?) /

(How long did it take to make that change?)

As with velocity, we have to be careful to note that while this captures the conceptual content of average acceleration, it misses the vector character of acceleration. Since velocity is a **vector**, we really mean "how much did the vector velocity change" for the numerator — and this makes acceleration a vector too. So it's not only the change of speed that acceleration is about. If the velocity changes direction, that's an acceleration too.

Our equation for (vector) average acceleration uses the angle brackets "<..>" to indicate "average", just as we did for velocity.

$$\langle \overrightarrow{a} \rangle = \frac{\Delta \overrightarrow{v}}{\Delta t}$$

If we are more explicit about the initial and final positions and times, we might want to write this as

$$\langle \overrightarrow{a} \rangle = \frac{\overrightarrow{v_f} - \overrightarrow{v_i}}{ t_f - t_i}$$

where the "i" subscript means "initial" and the "f" subscript means "final"; so for example, $t_i$ means the starting (initial) time.

If we multiply both sides of our defining equation by the time interval, we can get a better sense of what the average acceleration means:

$$\Delta \overrightarrow{v} = \langle \overrightarrow{a} \rangle \Delta t$$

So if you changed your velocity by an amount $Δv$ in a time $Δt$, the average acceleration is "that constant acceleration that you would have to have to change your velocity that much in that time". Of course you might not have moved with a constant acceleration in that time interval.

## Dimensionality of acceleration

Since acceleration is a ratio of a velocity (dimensionality L/T) to a time (dimensionality T), it has dimensionality L/T^{2}:

[$a$] = [$v/t$] = (L/T)/T = L/T^{2}.

## Constant Acceleration

If you are speeding up at a constant rate, say your acceleration is the constant $a_0$, then you can easily figure out how much your velocity has changed in a given time interval. But you can also easily find how far you've gone in that time interval. Let's first see how to get the velocity change. As with average velocity, there are a lot of different ways to write this, depending on what you know and on what you want to find. Here are a few:

$$\langle \overrightarrow{a} \rangle = \frac{\Delta \overrightarrow{v}}{\Delta t} = \overrightarrow{a_0}$$

$$\Delta \overrightarrow{v} = \overrightarrow{a_0} \Delta t$$

$$ \overrightarrow{v_2} - \overrightarrow{v_1} = \overrightarrow{a_0} \Delta t$$

$$ \overrightarrow{v}_{final} = \overrightarrow{v}_{initial} + \overrightarrow{a_0} \Delta t$$

Note that as with velocity, if there is a vector on one side of the equation there has to be one on the other. Vectors and scalars (numbers that don't have a direction) are different kinds of physical objects and can't be equated (or added).

With constant acceleration we can also find how the position changes using the help of average velocity. See the follow-on page, calculating with constant acceleration.

Julia Gouvea and Joe Redish 8/20/13

#### Follow-ons

Last Modified: July 12, 2019