### Calculus meets Functions

Problem requiring decisions

## 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?

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 $d_t=d_s$?