Power: how fast work is done

A 60.0 kg woman running up 3.00 m of stairs in 3.50 s puts out 538 W, a little under one horsepower.

Same work, different power

Walk up a flight of stairs or run up it: you end at the same height, so you do the same work against gravity. Running still feels harder, because you do that work in less time. Power measures how fast work is done or energy is transferred: P = W ÷ t. The unit is the watt (W), one joule per second. A 60 W bulb uses 60 J every second.

An older unit is still common for engines: the horsepower. OpenStax gives 1 hp = 746 W. A horse can keep up about one horsepower for hours. People can go above one horsepower with their legs for short bursts, but not for long.

Worked example: running up the stairs

OpenStax works out the stairs. A 60.0 kg woman runs up a 3.00 m flight in 3.50 s, starting from rest and finishing at 2.00 m/s. She gains potential energy m g h = 60.0 × 9.80 × 3.00 = 1,764 J and kinetic energy ½ × 60.0 × 2.00² = 120 J. Her power is (1,764 + 120) ÷ 3.50 = 538 W, which is 0.72 hp. Notice that almost all the energy went into height, not speed. Climb the same stairs more slowly with the same start and finish speeds and the work is the same, but the average power is smaller.

You can measure your own power this way. Measure the height of a flight of stairs, time yourself running up, and work out m g h ÷ t. OpenStax suggests you should not expect much more than about half a horsepower.

A ladder of powers

Power covers a huge range. OpenStax Table 7.3 lists a person at rest at about 100 W, almost all of it heat. A resting heart works at about 8 W, an electric clock at 3 W, a clothes dryer at about 4,000 W, a car at about 80,000 W counting its heat, and the Sun at about 4 × 10²⁶ W. Light bulbs are a good reminder that power in is not the same as useful power out: a 60 W incandescent bulb gives only about 5 W of light, and the other 55 W is heat.

From power back to energy: E = P t

Turn P = W ÷ t around and you get the energy used: E = P t. Electricity companies bill in kilowatt-hours, the energy used by one kilowatt for one hour, 3.6 × 10⁶ J. OpenStax's example is a 0.200 kW computer used 6.00 hours a day for 30.0 days: 0.200 × 6.00 × 30.0 = 36.0 kWh. At $0.120 per kWh that costs $4.32 a month. Power and time both count. A 2,000 W kettle running three minutes uses 2,000 × 180 = 360,000 J, only 0.1 kWh, while a low-power router left on all year adds up.

Horsepower makes these numbers easier to picture. OpenStax lists a car at about 8 × 10⁴ W, counting the heat as well as the useful work. Divide by 746 and that is about 107 hp. A person at rest puts out about 100 W, almost all as body heat, which is why a room full of people warms up. OpenStax puts a resting heart at about 8 W, counting the heat it gives off as well as the work of pumping blood, less than a tenth of the body's total.