Average Acceleration

What this formula says

Over a time interval from $t_1$ to $t_2$, a body's velocity changes from $v_1$ to $v_2$. The average acceleration over this interval is:

$\bar{a} = \frac{v_2 - v_1}{t_2 - t_1} = \frac{\Delta v}{\Delta t}$

It measures how much the velocity changed per unit time, on average, over that interval.

Acceleration is about velocity change — not speed change

Acceleration is the rate of change of velocity, not speed. Since velocity is a vector (it has direction), acceleration occurs whenever:

• The speed changes (body speeds up or slows down), or
• The direction changes (body turns), or
• Both change simultaneously

A car turning a corner at constant speed is still accelerating — its direction is changing so its velocity vector is changing.

Derivation

This is a definition. Average acceleration is defined as:

$\bar{a} = \frac{\Delta v}{\Delta t} = \frac{v_2 - v_1}{t_2 - t_1}$

where $\Delta v = v_2 - v_1$ is the change in velocity and $\Delta t = t_2 - t_1$ is the time elapsed.

Sign of average acceleration

• $\bar{a} > 0$: velocity increased (or changed in the positive direction)
• $\bar{a} < 0$: velocity decreased (deceleration, or retardation)
• $\bar{a} = 0$: velocity did not change — uniform motion

Important: Negative acceleration does not always mean the body is slowing down. If a body moves in the negative direction and $\bar{a}$ is also negative, the body is speeding up (in the negative direction).

The body slows down when $v$ and $a$ have opposite signs. It speeds up when they have the same sign.

Units

$[a] = \frac{[v]}{[t]} = \frac{\text{m/s}}{\text{s}} = \text{m/s}^2$

Relation to instantaneous acceleration

Average acceleration gives the overall change. As the time interval $\Delta t$ shrinks to zero, average acceleration approaches the instantaneous acceleration at that moment:

$a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

For uniform acceleration, average acceleration equals instantaneous acceleration at every point — they are the same constant value throughout.