2440.78 – Figure Eight Racing

In an animated action film, five cars are tearing along a figure-eight track that has a total length of 7 miles (see the diagram below).

The five cars, $A$, $B$, $C$, $D$, and $E$, travel at average speeds, respectively, of 60, 72, 75, 80, and 96 mph. Tall barriers to each side prevent any car from leaving the track. If two cars should meet in the center, they could collide unless they do some fancy steering and braking.

If all five start in the center of the track and all travel the same way, could two of them eventually collide? If so, which two, after how many miles and how many minutes?

Solution

This is a messy problem. One way (the best way?) to crack it is to act it out on a spreadsheet. If you do this, you will see that car $B$ goes 1.5 laps in 8.75 minutes and $E$ goes 2 laps in the same time. $B$ will have gone 10.5 miles and $E$ will have gone 14 miles.