The red line represents the distance of the car from the starting-point, the blue lines, that of the walkers. The graph shows one complete cycle of the scheme, whereby the three friends are all together again, ready to begin another cycle.
We need to find the overall speed in mph from the beginning to the end of this cycle; that is, the fraction:Any further cycles will have exactly the same average speed (as we shall see) and therefore this fraction will represent how far they went in one hour.
Suppose that a time hours elapses before Bob drops Chris, and returns to pick up Peter, taking a further time to meet him. These times are represented by the distances and on the graph. Then, using , we have:Now:
The second part of this first cycle is where Bob and Peter overtake Chris, represented by the line-segment . Now since the car travels at a constant speed, and the two walkers walk at the same speed, and . is therefore a parallelogram, and hence:
, from (1)
So, from (2) and (3):
, from (1)
which is independent of , and hence represents the average speed over any complete cycle. Hence it is the number of miles travelled in one hour (assuming that at the end of the hour the three friends are at the same point).