Definition

Power

P = \frac{d W}{d t} = F \cdot v

Statement

The instantaneous rate of doing work; equivalently, force dotted with velocity.

Symbols and units

$P$	power	W
$F$	force	N
$v$	velocity	m/s

When it applies

Any force and velocity, instantaneous or averaged over an interval.

Derivation

Differentiate work with respect to time, and use ds/dt = v:

P = \frac{d W}{d t} = F \cdot \frac{d s}{d t} = F \cdot v

Worked examples

Power to climb stairs

A 70 kg person climbs 3.0 m in 5.0 s:

P = \frac{m g h}{t} = \frac{( 70 ) ( 9.8 ) ( 3.0 )}{5.0} \approx 412 W

Watch out for

A force delivers zero power the instant velocity is zero, however large the force — power depends on how fast the object is already moving, not on the force alone.