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The power of a person

Energy and power · level 4, lesson 5 of 6

A question to start

Run up a flight of stairs as fast as you can, and time yourself. With your mass and the height of the stairs, you can work out your power in watts. What would you get on Mars?

The idea

Climbing stairs lifts your whole body. The energy needed is your weight times the height:

\[E = mgh,\]

with \(m\) your mass in kg, \(g\) the pull of gravity (9.81 on Earth, 3.71 on Mars), and \(h\) the height in metres. Divide by the time it took, and you have your power:

\[P = \frac{mgh}{t}.\]

(Your muscles waste a lot as heat, so your body really uses about four times as much. This is the useful power.)

h = 3 m
A 50 kg person running up 3 m in 4 s Mars Earth
Energy to lift their body, \(mgh\) 557 J 1,472 J
Power, \(mgh/t\) 139 W 368 W
Time for the same climb at 368 W 1.5 s 4 s

On Mars, the same climb takes only 0.38 of the energy. With the same legs and the same power, you would go up the stairs much faster, or climb much higher before tiring.

Worked examples

1. A trained cyclist. A fit amateur can keep up about 200 W for an hour; top racing cyclists, around 400 W. A sprinter can give over 1,000 W for a few seconds.

2. Climbing a crater rim on Mars. A 120 kg astronaut in a suit climbs 100 m. \(E = 120 \times 3.71 \times 100 = 44{,}500\) J. At a steady 100 W of useful power, that takes 445 s, about 7.5 minutes. On Earth, with the same suit, it would take nearly 20 minutes.

Try it

  1. A 40 kg child climbs 4 m of stairs in 8 s on Earth. Find the power.
  2. Find the power for the same climb, in the same time, on Mars.
  3. You can give 300 W. How long would it take you (60 kg) to climb 6 m on Mars?
  4. Why do your muscles use more energy than \(mgh\)?
Answers
  1. \(40 \times 9.81 \times 4 / 8 = 196\) W.
  2. \(40 \times 3.71 \times 4 / 8 = 74\) W.
  3. \(E = 60 \times 3.71 \times 6 = 1{,}336\) J; \(t = 1336 / 300 = 4.5\) s.
  4. Muscles are not perfectly efficient: most of the food energy they use becomes heat.

Common mistake

Using mass instead of weight. The energy to lift is weight times height, \(mgh\), not \(m\) times \(h\). Leave out \(g\) and you get the wrong number on every planet.

Where it leads