Safe distances

The Official Highway Code describes typical stopping distances for cars (section 126 of the 2015 edition). They are given as a table showing distances for different speeds of travel. Each stopping distance is made up of two parts – a thinking distance and a braking distance. They are summarised below.

Speed, \(v\) Thinking distance Braking distance Stopping distance, \(d_s\)
\(\quantity{20}{mph}\) \(\quantity{6}{m}\) \(\quantity{6}{m}\) \(\quantity{12}{m}\)
\(\quantity{30}{mph}\) \(\quantity{9}{m}\) \(\quantity{14}{m}\) \(\quantity{23}{m}\)
\(\quantity{40}{mph}\) \(\quantity{12}{m}\) \(\quantity{24}{m}\) \(\quantity{36}{m}\)
\(\quantity{50}{mph}\) \(\quantity{15}{m}\) \(\quantity{38}{m}\) \(\quantity{53}{m}\)
\(\quantity{60}{mph}\) \(\quantity{18}{m}\) \(\quantity{55}{m}\) \(\quantity{73}{m}\)
\(\quantity{70}{mph}\) \(\quantity{21}{m}\) \(\quantity{75}{m}\) \(\quantity{96}{m}\)

Look at the data in the table. What relationships do you see between the distances and how do they vary with speed?

  • How is the stopping distance related to the thinking and braking distances?
  • Starting at \(\quantity{20}{mph}\) what happens to the distances when you double (or treble) the speed?
  • Can you suggest reasons why the distances are related to the speed like this?

Write an equation expressing the stopping distance, \(d_s\), in terms of speed, \(v\). Plot a graph of \(d_s\) against \(v\), for speeds between zero and \(\quantity{70}{mph}\).

An alternative model

The Highway Code goes on to say that when driving you should leave a gap of at least the stopping distance between you and the vehicle in front.

It also says that in faster-moving traffic, you should instead leave a “two-second gap”. In other words the front of your car should not reach a fixed point on the road until at least two seconds after the rear of the previous vehicle passed the same point.

Write down an equation for this two-second distance, \(d_t\), in terms of \(v\) and add it to your graph of the stopping distance.

At \(\quantity{60}{mph}\) which of the two distances is bigger? Why might the Highway Code make the two-second suggestion?

At what speeds is \(d_t=d_s\)?