Power

Power ($P$) is the rate at which Energy is transferred or Work is done.

Average power:

$$P_{avg}=\frac{\Delta E}{\Delta t}=\frac{W}{\Delta t}$$

Instantaneous power:

$$P=\frac{dE}{dt}=\frac{dW}{dt}$$

When a constant force acts on an object moving with velocity $\vec{v}$,

$$P=\vec{F}\cdot \vec{v}$$

Again, the dot product matters: only force along the direction of motion transfers mechanical energy.

Intuition

Power is not how much energy you used; it is how fast you used it. Lifting a box to a shelf requires roughly the same work whether you do it slowly or quickly, but doing it quickly requires greater power.

Common checks

• Energy is measured in Joule; power is measured in Watt.
• A machine with high power can transfer the same energy in less time.
• If power is constant, $E=Pt$.

Example scale

If two motors both lift the same mass through the same height, they do the same work: $W=mgh$. The motor that finishes in half the time has twice the average power. Power is therefore the right quantity when comparing engines, athletes, batteries, or any system where delivery rate matters.