Definition

Velocity

v = \frac{d r}{d t}

Statement

The instantaneous rate of change of position with time.

Symbols and units

$v$	instantaneous velocity	m/s
$r$	position	m

When it applies

Any motion; v is the derivative of position with respect to time.

Derivation

Average velocity over an interval is displacement divided by time; shrinking the interval to zero gives the instantaneous value:

v = Δ t \to 0 lim \frac{Δ r}{Δ t} = \frac{d r}{d t}

Worked examples

Average vs instantaneous velocity

x(t) = t² (metres). Average velocity from t=1 s to t=3 s:

v ˉ = \frac{x ( 3 ) - x ( 1 )}{3 - 1} = \frac{9 - 1}{2} = 4 m/s

Instantaneous velocity at t = 2 s (the midpoint):

v (2) = \frac{d x}{d t}_{t = 2} = 2 t_{t = 2} = 4 m/s

They agree here only because x(t) is quadratic and 2 s is the midpoint — not true in general.

Watch out for

Average velocity (total displacement over total time) is not the average of the instantaneous speeds unless the motion is at constant velocity.

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