Power Formula — P = W/t

The power formula P = W/t explained: rate of doing work, watts vs joules, average vs instantaneous power, and a worked example.

P = \frac{W}{t}

Power equals work done divided by the time taken.

What it means

Power is how fast energy changes hands. A joule is an amount of energy; a watt is a joule per second — the speed of delivery. Two machines can do the same work, but the more powerful one finishes sooner. That is why engines are rated in watts or horsepower rather than joules: buyers care how quickly the work gets done. Because P = W/t is an average over the interval, a sprinter's instantaneous power at push-off can be several times the average over a whole race.

Variables

Symbol	Name	Unit	Meaning
$P$	Power	W	Rate of energy transfer
$W$	Work	J	Work done (energy transferred)
$t$	Time	s	Time taken to do the work

Assumptions

W is the useful work output (or total energy transferred)
t is the time over which the work is done
For varying power, this gives the average — instantaneous power is dW/dt

Derivation

Power is defined as the rate of energy transfer: $P = dW/dt$ . For a constant rate over time $t$ , this becomes $P = W/t$ . Combining with $W = Fd$ gives the handy form $P = Fv$ for a force acting at speed $v$ .

Worked example

Problem

A 70 kg person climbs a 3 m flight of stairs in 5 s. What average power does she develop?

Given:

m = 70 kg
h = 3 m
t = 5 s
g = 9.8 m/s²

First find the work against gravity: $W = mgh = 70 \times 9.8 \times 3$
Calculate: $W = 686 \times 3 = 2058\ \mathrm{J}$
Apply the power formula: $P = W/t = 2058 / 5$
Result: $P = 411.6\ \mathrm{W}$

≈ 412 W (about 0.55 horsepower)

The climber's muscles delivered energy at 412 watts — roughly seven times the power of sustained cycling for a fit amateur, which is why stairs leave you breathless so fast. Power, not just energy, is what your body feels.