Optimal Velocity in a Race


We have used the biomechanics of muscle contraction and oxygen consumption described above, together with the laws of mechanics and thermodynamics, to determine optimal performance strategies for participants in athletic competitions.  Among these determinations, for each participating athlete, are the optimal choice of a baseball bat weight, the optimal choice of a gear on a bicycle, the optimal choice of weight and loft angle of a golf club, and the optimal choice of an ice skating blade.  This unique procedure will be illustrated here by the determination of the optimal strategy for a runner to use in a racing competition.

Let v(t) be the (positive) speed of the runner at time t during a race of distance D, let f(t) be the forward force exerted on the ground by the runner at time t, and let E(t) be the internal energy available for running at time t.  The problem to be solved is to determine how the runner should choose f(t) in order to minimize the time T required to run the distance D, subject to the constraints

                                                                                        f(t) ≤ F,   E(t) ≥0,

for all times 0 ≤ t ≤T, where F is the runner’s maximum possible exerted force.  The functions f, v, and E are related by Newton’s second law of motion and the first law of thermodynamics:

                                                                                   f = mv’ + cv,    E’ =q - fv ,

where cv is the resistive force (from breaking), fv is the exerted power, and q is the rate of energy consumption (Fig. 5.3).  The initial conditions are

                                                                                 v(0) = 0,   E(0) = S,  f(0) = F,

where S is the runner’s initial anaerobic energy store available for running.  Since v(0) = 0, the optimal initial force is F since the exerted power fv is 0 at t = 0 for any f, so it costs nothing to exert the maximum force F.  Let t1 be the (possibly zero) time interval for which f remains equal to F.

The problem is to find v(t), f(t), and E(t) satisfying the above equations such that T is minimized, given the four parameters F, c, q, and S, and the distance D.  The solution to this problem, obtained by applying optimal control theory, is the following:  f(t) = F  from t = 0 until a later time t = t1, v(t) is constant between t = t1 and a later time t = t2, and E(t) = 0 from t = t2 until t = T.  The values of t1 and t2 depend on the values of the four parameters and D.  If D is small enough, t1 = t2 = T, and the solution is f(t) = F throughout the race.  In that case 


These races are called sprints.  (The longest sprint distance is


where s is the solution of E(s) = 0 with E(t) evaluated using f = F and v(t) = u(t).)  For longer races, the values of t1 and t2 are less than T, and are determined by the optimization equations.  For these races, the fact that the theory predicts that v(t) is constant throughout most of the race is in agreement with what is observed in long-distance races.  However, the theory states that the runner has to slow down during the last few seconds of the race, and long-distance runners do not do that.  On the contrary, runners usually end a race with a final burst of speed, but the theory states that these runners would finish the race in less time if they did not have energy left near the end of the race to maintain, or briefly increase, their speed.

The values of the four parameters for elite runners, and the values of the times t1, t2, and T, for each D, can be determined by comparing the optimization solution with the world records.  Using these results, the average speed D/T for any race distance D can be calculated.  A plot of these calculated speeds is given in Fig. 5.4 for D ≤ 2000 m.  The record speeds for races between 50 yards and 2000 m are shown as points on the plot.  For D less then about 300 m, the races are sprints, and the average speed increases with increasing D because the speed u(t) in a sprint steadily increases with increasing t.  For the long-distance races, the average speed decreases with increasing D because slower speeds require smaller energy consumptions.

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  Fig. 5.4. Average speed D/T vs. race distance D theory (solid curve) and world records (points)

Fig. 5.4. Average speed D/T vs. race distance D theory (solid curve) and world records (points)