Following a 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:
Another type of race consists in running backwards (or retrorunning). The linear link is similar
with a slightly larger power
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.
Statisfaction 
Nice and impressionnant!
There’s a nice paper on power laws like this listed below. It appears that race time predictors like McMillan’s one use a power law similar to the one that you found.
“Approximate Law of Fatigue in the Speeds of Racing Animals” A. E. Kennelly
Proceedings of the American Academy of Arts and Sciences, Vol. 42, No. 15 (Dec., 1906), pp. 275-331.
Thanks for the reference. This is indeed a well-known relation, as studied in this paper:
“Power laws and athletic performance”, JS Katz and L Katz
Journal of sports sciences, 1999, 17, 467-476
The authors analyzed the evolution of c and n in T=cD^n along time (and show that the power n is decreasing with epoch)
Thanks for that reference. The paper looks very interesting.
http://cscs.umich.edu/~crshalizi/weblog/491.html
In particuar: “Lots of distributions give you straight-ish lines on a log-log plot. “
Thanks for your very interesting paper!
Savaglio, S. and V. Carbone (2000). “Human performance: Scaling in athletic world records.” Nature 404(6775): 244-244.