Work, energy and power: the equations and the traps

Work, energy, and power sound like synonyms in everyday speech, and that's exactly why they get muddled in physics. They're three different, precisely defined quantities — and keeping them apart is the whole game.

Work done

Work is energy transferred when a force moves something through a distance:

That $\cos\theta$ is the trap. Only the part of the force along the direction of motion does work. The angle $\theta$ is between the force and the displacement:

• Force along the motion ($\theta = 0$): full work, $W = Fs$.
• Force perpendicular to the motion ($\theta = 90°$): zero work — $\cos 90° = 0$. This is why gravity does no work on a satellite in a circular orbit, and why carrying a bag horizontally does no work against gravity.

Work is measured in joules (J) — the same unit as energy, because work is a transfer of energy.

The two energy stores you'll use most

The work–energy principle ties them together: the work done on an object equals its change in kinetic energy. Push a trolley and the work you do shows up as its gain in $\tfrac{1}{2}mv^2$.

Power: the rate of doing work

Power is how fast energy is transferred — work per second, in watts (W):

Two engines can do the same total work; the more powerful one just does it sooner. The second form, $P = Fv$, is handy for vehicles moving at steady speed.

Conservation, with a caveat

In an ideal drop, $E_p$ converts fully to $E_k$: $mgh = \tfrac{1}{2}mv^2$. In reality some energy is transferred to heat by friction and air resistance — not lost, just moved to a store you can't get back.

Resolve the force along the motion first. If a force acts at an angle, the $\cos\theta$ in $W = Fs\cos\theta$ is non-negotiable. Forgetting it — using the whole force when only part of it pushes along the path — is the defining mistake of this topic.

Last revised 12 December 2025.