For a given braking force the greater the speed of the vehicle, the…

The idea

The two parts of the stopping distance grow differently with speed. Thinking distance is speed × reaction time, so it is directly proportional to speed: at a reaction time of 0.6 s, doubling the speed from 10 m/s to 20 m/s doubles the thinking distance from 6 m to 12 m. Braking distance comes from energy. The kinetic energy to be removed is ½ m v², and the work done by the brakes is braking force × braking distance, so with a fixed braking force the braking distance goes with the square of the speed. Doubling the speed from 10 m/s to 20 m/s takes the braking distance from 6 m to 24 m, four times as far. The stopping distance therefore rises far more steeply than the speed.

What earns the marks

• Stating that thinking distance is directly proportional to speed.
• Supporting that with figures, such as 6 m at 10 m/s rising to 12 m at 20 m/s.
• Stating that braking distance is not proportional to speed but increases more steeply.
• The supporting braking figures, such as 6 m at 10 m/s rising to 24 m at 20 m/s.

Common mistakes and how to put them right

• Saying both distances double when the speed doubles.

• Calling the braking distance graph proportional simply because it rises as speed rises.

• Assuming the driver brakes harder at higher speeds, when the question fixes the braking force.

• Confusing the braking distance with the time taken to stop.

Exam tips

• Quote a pair of numbers from the table when you judge a proportionality claim; the figures carry a mark of their own.
• Test proportionality by dividing: if distance ÷ speed is the same for every row, it is proportional.
• Use the words 'increases by a factor of four' rather than 'increases a lot'.