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Speeding up and slowing down

Forces and motion · level 4, lesson 4 of 6

A question to start

Drop a stone. After one second it is falling at about 10 m/s on Earth. After two seconds, 20 m/s. On Mars, after one second only about 3.7 m/s. How do we describe how quickly something gains speed?

The idea

Acceleration is how fast the velocity changes:

\[\text{acceleration} = \frac{\text{change in velocity}}{\text{time taken}}.\]

Velocity in m/s, divided by time in s, gives acceleration in metres per second per second, written m/s².

  • A car going from 0 to 20 m/s in 10 s accelerates at \(20 \div 10 = 2\) m/s²: it gains 2 m/s every second.
  • Slowing down is acceleration too, with a minus sign, sometimes called deceleration.

A falling thing, with no air in the way, accelerates at a steady rate called g:

Free fall Mars Earth
Acceleration, g 3.71 m/s² 9.81 m/s²
Speed after 1 s 3.7 m/s 9.8 m/s
Speed after 3 s 11.1 m/s 29.4 m/s

A speed-time graph shows this. A straight sloping line means a steady acceleration, and its slope is the acceleration. A flat line means a steady speed, no acceleration.

3 stimespeed Earth: 29.4 m/sMars: 11.1 m/s

Worked examples

1. A train. From 30 m/s to 0 in 60 s: \((0 - 30) \div 60 = -0.5\) m/s². It loses 0.5 m/s every second.

2. A Mars lander's last seconds. Falling at 30 m/s, its rockets slow it to 2 m/s in 14 s: \((2 - 30) \div 14 = -2\) m/s².

3. How long to reach a speed. A stone dropped on Mars: how long to reach 15 m/s? \(15 \div 3.71 = 4.0\) s.

Try it

  1. A bike goes from 2 m/s to 8 m/s in 3 s. What is its acceleration?
  2. A sprinter reaches 10 m/s in 2.5 s from a standing start. What is her acceleration?
  3. How fast is a stone falling on Mars after 5 s, with no air?
  4. On a speed-time graph, what does a flat line mean?
Answers
  1. \((8 - 2) \div 3 = 2\) m/s².
  2. \(10 \div 2.5 = 4\) m/s².
  3. \(5 \times 3.71 = 18.6\) m/s.
  4. A steady speed: no acceleration.

Common mistake

"Acceleration means going fast." Something can go very fast with no acceleration at all, like a spacecraft coasting at 30 km/s. Acceleration means the velocity is changing.

Where it leads