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A dropped tool

Energy and power · level 5, lesson 5 of 6

A question to start

An astronaut on a 10 m scaffold drops a 0.5 kg spanner. How fast is it going when it hits the ground on Mars? On Earth? And how much energy does it hit with?

The idea

As the spanner falls, its potential energy turns into kinetic energy. With almost no air (as on Mars), almost none is lost:

\[mgh = \tfrac12 mv^2 \qquad\Rightarrow\qquad v = \sqrt{2gh}.\]

The mass cancels, so all dropped things land at the same speed from the same height. But the energy they hit with, \(mgh\), does depend on mass.

Dropped from 10 m Mars Earth (no air)
Speed at the ground, \(\sqrt{2gh}\) 8.6 m/s, 31 km/h 14.0 m/s, 50 km/h
Energy of a 0.5 kg spanner 18.6 J 49.1 J
Time to fall, \(\sqrt{2h/g}\) 2.3 s 1.4 s
Height on Earth that gives the same landing speed 3.8 m 10 m
EarthMarstopground potential 49 Jkinetic 49 Jpotential 18.6 Jkinetic 18.6 J

So a fall on Mars from 10 m is like a fall on Earth from 3.8 m. Still dangerous, but much less so. Helmets, toe caps and tool tethers would still be needed on any building site on Mars.

Worked examples

1. Speed on Mars. \(v = \sqrt{2 \times 3.71 \times 10} = \sqrt{74.2} = 8.6\) m/s.

2. The same speed on Earth. \(h = v^2/(2g) = 74.2/19.62 = 3.8\) m.

3. A person jumping down. From 1 m: on Mars \(\sqrt{7.42} = 2.7\) m/s, on Earth \(\sqrt{19.6} = 4.4\) m/s. Landing from a jump of 2.6 m on Mars feels like landing from 1 m on Earth.

Try it

  1. A rock falls 5 m on Mars. Find its landing speed.
  2. From what height on Earth does a rock land at the same speed?
  3. A 2 kg tool falls 6 m on Mars. With what kinetic energy does it land?
  4. Why does the landing speed not depend on the mass?
Answers
  1. \(\sqrt{2 \times 3.71 \times 5} = \sqrt{37.1} = 6.1\) m/s.
  2. \(37.1 / 19.62 = 1.9\) m.
  3. \(2 \times 3.71 \times 6 = 44.5\) J.
  4. Both \(mgh\) and \(\tfrac12 mv^2\) are proportional to \(m\), so it cancels.

Common mistake

"Weaker gravity means the speed is 0.38 times as much." The landing speed is \(\sqrt{2gh}\), so it goes as \(\sqrt{g}\): \(\sqrt{0.38} = 0.62\) of Earth's. The energy is 0.38 of Earth's.

Where it leads