Slot car racing is an application that requires no advanced mathematics -- just the formula for the circumference of a circle.
Youngsters are familiar with the small electrically powered cars that race along parallel slots in track sections that can be
assembled in a variety of ways. The figures below show some possible layouts. Notice that the curved track sections can be
joined to form quarter-circles and semicircles of different radii and that overpasses/underpasses are possible.
Question #1: In the simple track layout shown below (figure 1), what head start should the car in the dotted slot be allowed?
The slots are 4 centimeters apart.
Solution: Straightaway sections can be ignored. We can imagine a solid circle of radius R centimeters and a concentric dotted one
of radius (R + 4) centimeters. The difference in their circumference is as follows:
2 π (R + 4) - 2πR = 8π cm ~ 25 cm.
The car in the dotted slot should be given a 25-centimeter head start. Or putting it differently, if equally fast cars begin at the
same starting line, then the inside car should win a one lap race by 8π centimeters. Emphasize that this winning margin is
independent of how sharp or gradual the semicircular ends of the track happen to be.
Question #2: Suppose the track is laid out as in figure 2, and suppose two equally fast cars start off together from the starting line. Which
should win a one-lap race -- the dotted or the solid -- and by how much?
Solution: We have seen that for equally fast cars on a circular track, the inside car wins by 8π centimeters; so on a semicircular
curve the inside car gains 4π centimeters and on a quarter-circular curve it picks up 2π centimeters. We can chart the
progress of the race as follows:
| After completing turn
|
Dotted car ahead by
|
| I
|
4π
|
| II
|
6π
|
| III
|
2π
|
| IV
|
0
|
The race should end in a tie. In other words, this is a fair track. No handicap needs to be given to either car.
The solid path and the dotted path have the same length. Note: for this question, the distance between slots, 4 centimeters, turned
out to be irrelevant, but it made the argument a bit more concrete.
Question #3: Determine by how much the dotted car should be expected to win a one-lap race on each of the layouts in figure 3.
