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Timing a fall

Gravity and orbits · level 3, lesson 1 of 6

A question to start

Drop a ball from your hand. At first it is slow; by the time it hits the floor it is moving quite fast. How much faster does it get, and how quickly?

The idea

Scientists have timed falling things very carefully. Without air in the way, every falling thing near the Earth's surface gets faster by the same amount every second:

About 10 metres per second faster, every second. (More exactly 9.8.)

That steady speeding up is called acceleration. Here it is, second by second, for a stone dropped from a very high cliff (ignoring the air):

Time after letting go Speed on Earth Speed on Mars
0 s 0 m/s 0 m/s
1 s 10 m/s 3.7 m/s
2 s 20 m/s 7.4 m/s
3 s 30 m/s 11.1 m/s
4 s 40 m/s 14.8 m/s

On Mars a falling thing gains only 3.7 m/s every second. It still speeds up steadily, but more gently.

How far does it go? In the first second on Earth it falls about 5 m. By the end of 2 seconds, about 20 m; by 3 seconds, 45 m. Divide by 5 and you get 1, 4, 9: the square numbers. The distance grows faster and faster because the speed keeps growing. On Mars it falls about 1.9 m in the first second, 7.4 m in two, and 17 m in three.

1 s, 5 m2 s, 20 m3 s, 45 m2 s, 7.4 m3 s, 17 m EarthMars

Worked examples

1. The well. A stone dropped into a well hits the water after 2 seconds. About how fast is it going? \(2 \times 10 = 20\) m/s. How deep is the well? About 20 m.

2. A rock on Mars. A rock falls for 3 seconds on Mars. Its speed is \(3 \times 3.7 = 11.1\) m/s. On Earth, after 3 seconds, it would be going 30 m/s, nearly three times as fast.

3. The ruler test. Hold a ruler at the top, with a friend's open fingers at the bottom. Let go without warning. The farther it falls before they catch it, the slower their reactions. A catch at 20 cm means about 0.2 s.

Try it

  1. A ball is dropped on Earth. How fast is it going after 5 seconds (ignoring air)?
  2. The same ball on Mars: how fast after 5 seconds?
  3. How far does a stone fall on Earth in 4 seconds? (Use the square numbers.)
  4. A ball is going 30 m/s when it hits the ground on Earth. How long was it falling?
Answers
  1. \(5 \times 10 = 50\) m/s.
  2. \(5 \times 3.7 = 18.5\) m/s.
  3. \(5 \times 16 = 80\) m (more exactly 78 m).
  4. 3 seconds.

Common mistake

"It falls at 10 m/s." It does not fall at one speed. It gains 10 m/s every second: 10 after one second, 20 after two, 30 after three. The 10 is how fast the speed grows, not the speed itself.

Where it leads