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Followinga few entries on sports here and there, I was wondering what kind of law follow the running records with respect to the distance. The data are available on Wikipedia, or here for a tidied version. It collects 18 distances, from 100 meters to 100 kilometers. A log-log scale is in order:

It is nice to find a clear power law: the relation between the logarithms of time T and of distance D is linear. Its slope (given by the lm function) defines the power in the following relation:

$T\propto D^{1.11}$

Another type of race consists in running backwards (or retrorunning). The linear link is similar

with a slightly larger power

$T\propto D^{1.13}$

So it gets harder to run longer distances backwards than forwards…

It would be interesting to compare the powers for other sports like swimming and cycling.

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