December 15, 2010

Running track lengths

Yesterday, Rhett at Dot Physics had a comment about last week's puzzler on Car Talk.  He had an alternate solution to the question about how a pair of people could walk side by side for an hour and cover different distances.  His solution was that they were walking on a circular track. (Read his solution for a full explanation.)

At the end of the post he said:

It doesn’t even have to be a circle – it can be just curved at the ends and straight in the middle. Of course in this case, the husband would have to slow down on the straight parts (or the wife would have to speed up) in order for them to stay side by side. But it could be done.

There was something about this that didn't seem right.  I remember the track I used to jog on had one lane that was 8 laps for a mile.  Let's use that as the lane for the husband in our example.

Here's the layout of the track:
Let's use Rhett's values for the inner and outer radii: 4 m and 5 m, respectively. If the outer track is an 1/8th of a mile, then how long are the straight sections?

1/8th of a mile is about 200 m. The circumference of the curved part of the outer track is d = 2×π×5 m = 31.4 m. So the length of each straight section for the outer track is 84.9 m.

The circumference of the curved part for the inner track is d = 2×π×4 m = 25.1 m. But the length of the straight sections is the same.  So the total length of the inner track is 25.1 m + 2 × 84.9 m = 194.9 m.

The wife has to walk 33 laps to go 4 miles. If the husband and wife are walking side-by-side the whole way, he also walks 33 laps.  His lane was 1/8th of a mile, so he has gone 4 miles, plus an extra 1/8th.

I still like Dot Physics.