Average velocity describes motion over a finite interval, but an object may speed up, slow down or reverse direction within that interval. Instantaneous velocity describes the rate of change of position at one particular time.

If position is $x(t)$, then

$$v(t)=\frac{dx}{dt}.$$

From average to instantaneous velocity

Over a short interval $\Delta t$,

$$\frac{x(t+\Delta t)-x(t)}{\Delta t}$$

is an average velocity. Letting the interval shrink to zero gives the derivative and therefore the instantaneous velocity.

Position-time graph

On a graph of position against time, average velocity is the slope of a secant line between two points. Instantaneous velocity is the slope of the tangent line at the chosen time.

A positive slope means motion in the positive coordinate direction, a negative slope motion in the negative direction, and zero slope means the object is instantaneously at rest.

Example

If

$$x(t)=t^2,$$

then

$$v(t)=2t.$$

At $t=3,\mathrm s$,

$$v=6,\mathrm{m/s}$$

when position is measured in metres.

Instantaneous velocity is therefore not an average over a hidden small interval; it is the derivative obtained in the limit as that interval vanishes.