Instantaneous power

Intermediate

Force dotted with velocity.

Key relationP = F·v

The power at an instant is P = F·v = Fv cosθ, useful for engines and moving loads.

Worked example

A car resisting a 600 N drag at 30 m/s needs 18 kW.

Common mistake

If a force is perpendicular to velocity it does no work and supplies no power.

In more detail

Instantaneous power is P = F·v, the dot product of force and velocity. It shows how the power changes as speed changes, even with constant force.

Car engine power changes with speed, and is highest at a particular engine speed.
Cycling power meters measure the force and speed to find instantaneous power.
A car's engine power at a given speed equals the driving force times the speed, which is why hills need more power.

Why is power F times v?

Because power is the rate of work, and work rate is the component of force along motion times speed.

Can power be negative?

Yes, when the force opposes the motion, such as braking.

Why does power depend on the speed?

Work per second equals force times speed, so for a fixed force, faster motion needs more power.