Minimum-time running and swimming: an optimal control approach

During analysis of the competitor's velocity in a run, strong assumptions are imposed upon the runner's tactic. It is assumed that the competitor uses his/her maximal propulsive force in short-distance events. The runner's velocity is assumed constant in long-distance races. None of these assumptions is satisfied during middle-distance races. In this study, the competitor's velocity, minimizing the time taken to cover the distance, is determined by means of extremization of linear integrals using Green's theorem (Miele's method). The model of the competitor's motion is based on two differential equations: the first one derives from Newton's second law, the second one is the equation for power balance. The theory is illustrated with two examples referring to competitive running and swimming. The minimum-time competitive run can be broken into three phases: -the acceleration, -the cruise with the constant velocity, and -the negative kick at the end of the race. The problem has a similar solution in competitive swimming, however, the acceleration is replaced by the gliding phase.