5

I went for a run with a friend last night. As we set off, we started time keeping apps on our phones at the same time. Every kilometre, Endomondo speaks out pace, time etc.

I noticed after 3km (roughly one lap), my phone would trigger the 3km alert about 10 metres before my friend's. My feeling for this is I was running on the outside.

Lets assume the track is perfectly circular and I am running 2 metres to the outside, the extra distance I will run will be

$2\pi r_{outside} - 2\pi r_{inside} = 2\pi (r_{inside}+2) - 2\pi r_{inside}$

$= 4\pi \approx 12m$

My first surprise is that regardless of the length of the lap, I will only ever do 12m more

Would this formula hold for a non-circular track, e.g. oval or something even more complicated like the Suzuka Formula 1 circuit

enter image description here

Note: I am not a mathematician and I apologise for the way I've asked the question.

Gerry Myerson
  • 179,216
John Oxley
  • 153
  • 4

1 Answers1

3

Your formula holds for any track that doesn't cross itself. For a figure-of-eight track like Suzuka, the difference is 0, because you are on the outside in one part and on the inside in the other part. The relative lengths of the two parts are not important, because one part contributes $+4\pi m$, and the other part contributes $-4\pi m$.

In general, the difference is $4\pi T$, where $T$ is the turning number of the track (see this Geometry Center article for an informal explanation with pictures).

TonyK
  • 64,559