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?

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 \(d_t=d_s\)?