3.4 Power

Power is the rate of doing work

Power measures how quickly energy is transferred or work is done: P = dW/dt. Average power is P_avg = W/t. The SI unit is the watt (W), equal to 1 joule per second. Power tells you how fast, not how much, energy is transferred.

Power from force and velocity

Since W = F·d, power can be written as P = F·v (dot product of force and velocity), or P = Fv cos θ. Instantaneous power uses instantaneous velocity; if force and velocity are constant, average power equals instantaneous power.

Applying calculus to power

Because P = dW/dt, integrating power over time gives total work: W = ∫P dt. This lets you find work done when power varies with time, such as an engine with changing output, by computing the area under a P vs. t graph.

Key terms

Power (definition)

The rate at which work is done or energy is transferred: P = dW/dt.

Average power formula

P_avg = W/t, total work divided by total time.

Power in terms of force and velocity

P = F·v = Fv cos θ, where θ is the angle between force and velocity.

Unit of power

Watt (W), equal to 1 joule per second.

Finding work from variable power

W = ∫P dt; integrate power over time to get total work done.

Try an AP-style question

Power P(t) = 4t (W) from t = 0 to 3 s. How much work is done?

1. 12 J
2. 4 J
3. 36 J
4. 18 J