The aim of this paper is to bring a mathematical justification to the optimal way of organizing ones effort when running. It is well known from physiologists that all running exercises of duration less than 3mn are run with a strong initial acceleration and a decelerating end; on the contrary, long races are run with a final sprint. This can be explained using a mathematical model describing the evolution of the velocity, the anaerobic energy, and the propulsive force: a system of ordinary differential equations, based on Newtons second law and energy conservation, is coupled to the condition of optimizing the time to run a fixed distance. We show that the monotony of the velocity curve vs time is the opposite of that of the oxygen uptake ($dot{VO2}$) vs time. Since the oxygen uptake is monotone increasing for a short run, we prove that the velocity is exponentially increasing to its maximum and then decreasing. For longer races, the oxygen uptake has an increasing start and a decreasing end and this accounts for the change of velocity profiles. Numerical simulations are compared to timesplits from real races in world championships for 100m, 400m and 800m and the curves match quite well.
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