Solution 1

A distance-time graph with 2 different drivers' lines drawn on. Both lines are straight segments, driver 1 driving 4 miles over a long time, and driver 2 driving 3 miles over a much shorter time. They both start at the same time.

The graph above shows the information we were given in the table, with the speeds of the drivers represented by the gradients of the lines. How can we use it to help us solve the problem?

It is always a good idea to sketch a graph of a situation. It can help you visualise situations, highlight differences and similarities, and offer you a geometric way to solve a problem.

  1. How far apart will the cars be after the first driver has gone \(\quantity{1}{mile}\)?
The distance time plot according with to the table with the positions when passing the second camera highlighted.

After driving \(\quantity{1}{mile}\) at \(\quantity{50}{mph}\) the first driver has been travelling for \(\dfrac{1}{50}\)th of an hour or \(1.2\) minutes. Therefore the second driver has travelled a distance of \(75\times\dfrac{1}{50} = \quantity{1.5}{miles}\) so the cars are \(\quantity{0.5}{miles}\) apart.

We could also have thought about the ratio of the speeds. The second driver is going \(1\frac{1}{2}\) times as fast as the first driver, so he will travel \(1\frac{1}{2}\) times as far in the same time.

  1. What will be the greatest distance between the two drivers and when will it occur?

Both drivers arrive at the second speed camera together so we could add in the second driver’s journey to the graph.

The second drivers journey after the first 3 miles.

The cars will be the greatest distance away from each other when the second driver has travelled \(\quantity{3}{miles}\). This occurs at \(\dfrac{3}{75}\)ths of an hour or \(2.4\) minutes. In this time, the first driver will have travelled \(50\times\dfrac{3}{75}=\quantity{2}{miles}\). The gap between them will be \(\quantity{1}{mile}\).

  1. What is the average speed of the second driver?

Both cars travel the same distance in the same amount of time, so the second driver must also be averaging \(50\) mph in the situation above.

  1. What was the second driver’s speed for the last mile?

The line we have drawn in represents the second part of their journey. The speed of the car is equal to the distance divided by the time taken.

The gradient of the journey from 3 miles which is point A to 4 miles which is point B in the distance time plot.

We know the coordinates of A and B so we can calculate the speed as, \[\dfrac{4-3}{\frac{4}{50}-\frac{3}{75}}=\quantity{25}{mph}.\]

If we hadn’t drawn the graph we could have taken an algebraic approach to finding the solution. To work out the speed of the second driver once he has slowed down we need to consider his overall journey in two sections.

In the first section we know the speed and the distance so we have \[t_1=\dfrac{3}{75}.\]

In the second section we only know the distance so we can write \[t_2=\dfrac{1}{v},\] where \(v\) is the speed.

Subscripts can be very useful in questions like this, so as not to get confused. The times for each section will be different, which could be easily forgotten if we just used \(t\) for both equations.

As \(\text{average speed} = \dfrac{\text{total distance}}{\text{total time}}\), for the overall journey, with an average speed of \(50\) mph and a distance of \(\quantity{4}{miles}\), we have

\[ 50 = \dfrac{4}{t_1+t_2}.\]

Rearranging this equation gives us the total time,

\[t_1 +t_2=\dfrac{4}{50}.\]

You might have also found the total time by working out the time that the first driver took.

Substituting in the equations for \(t_1\) and \(t_2\) from above we get

\[\dfrac{3}{75}+\dfrac{1}{v}= \dfrac{4}{50}.\]

Solving for \(v\) gives us a speed of \(\quantity{25}{mph}\). So our driver must travel at \(\quantity{25}{mph}\) to make their average speed \(\quantity{50}{mph}\) over the \(\quantity{4}{miles}\).