The power of a force in motion: P = F × v

From W = F × d to P = F × v

Let us return to the work done by a constant force acting in the same direction as the displacement: W = F × d. As the average power is expressed as P = W/t, we obtain:

P = F × d / t = F × (d/t) = F × v

where v is the average velocity of the moving object. This relationship holds true at every instant if we consider the instantaneous velocity: P = F × v, provided that F and v are collinear (in the same direction).

General case: force and velocity not aligned

If the force makes an angle θ with the velocity, the power becomes:

P = F × v × cos(θ)

Thus, a force perpendicular to the direction of motion (theta = 90 deg, cos(theta) = 0) produces no power: this is the case with the weight of an object moving horizontally, or the centripetal force in uniform circular motion.

Example

A cyclist is travelling at a constant speed v = 5 m/s and exerts a driving force F = 150 N in the direction of motion.

P = F × v = 150 × 5 = 750 W

Common pitfall

Do not forget cos(theta) when the force and displacement are not parallel: applying P = F × v without considering the angle is a common mistake, particularly on an inclined plane where only the component of the force along the direction of displacement produces work (and therefore power).