Marathon (and Zeno) Debrief

The news you're all waiting to hear is that I completed the Brighton Marathon in 4 hours 12 minutes 43 seconds.

But I know you're not really interested in how quickly I ran the marathon but, as I asked in an earlier post, what was the distance?

Despite not being able to post diagrams, I suspect Katie had the answer to the question posed (as I guessed she would before I even posted).

The answer is that Mr Tortoise, the slower runner, records the further distance on his GPS.

Suppose the following is the actual route that both runners take on a small fragment of the course. It's a sharp bend, to make things clearer, but the same principals would apply to any bend in the route. (Click the diagrams to enlarge.)

Now Mr Tortoise will complete this section in twice the time of Mr Hare. Given that their GPSs both sample their position once a second, that means Mr Tortoise will have twice as many samples (shown as red crosses) as Mr Hare (green circles).

Given these samples locations, calculation of the actual route taken is done by joining up these locations as shown.

(Note that it's an assumption that the GPS software joins the dot's as straight lines. Conceivably, a much more complex algorithm could be used.)

Clearly the red (Mr Tortoise's) route is going to be longer than Mr Hare's, regardless of the nature of the bend. The only circumstances when this is not the case will be on a straight line, where the intermediate red cross will lie on the green line and the distances will be the same.

So on a perfectly straight route, the total distance will be the same, but if there are any bends at all, the slower runner will always record the further distance.

And of course, both recorded distances will be less than the true route, represented by the black curve.

Which leads me to be puzzled as to why, at the Brighton Marathon, my GPS recorded a distance of 26.41 miles, when an official marathon is only 26.22 miles. Worth further sponsorship, surely?

Zeno’s Paradox Revisited

Two runners, Mr Hare and Mr Tortoise, run a marathon. Mr Hare is faster and completes the course in, let's say, three hours, whereas Mr Tortoise takes six hours.

Each of them wears an identical GPS running watch, which samples the runner's location once every second and uses the information to calculate speed and distance travelled.

The marathon follows a typical street circuit and each of the two runners follows exactly the same path, without cutting any corners or taking any shortcuts.

The question is, at the end of the race, which of Mr Hare and Mr Tortoise is likely to have travelled the furthest distance according to his GPS watch?

If you have a solution, post it as a comment, showing your working. (Note there are no tricks in this, it's purely an issue of mathematics, or possibly physics.)

I'll be posting my solution after the Brighton Marathon on Sunday April 18^th, for which you can still sponsor me at http://www.runningsponsorme.org/jasperkent. I hope that my time will lie somewhere between those of Messrs Tortoise and Hare.