IMA Journal of Management Mathematics Advance Access originally published online on August 29, 2008
IMA Journal of Management Mathematics 2009 20(2):109-120; doi:10.1093/imaman/dpn022
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This article appears in the following IMA Journal of Management Mathematics issue: Special Issue Mathematics in Sport [View the issue table of contents]
A new modelling approach demonstrating the inability to make up for lost time in endurance running events
Institute of Food, Nutrition and Human Health, Massey University, Palmerston North 4410, New Zealand
Email: h.morton{at}massey.ac.nz
Received on 27 July 2007. Accepted on 30 July 2008.
The tolerable duration of high-intensity exercise can be described by a simple hyperbolic function of power or velocity, with an asymptote referred to as the ‘critical power/velocity’ and a curvature constant referred to as the ‘anaerobic work/distance capacity’. More recently, this hyperbola has been generalized by permitting a non-zero temporal asymptote. Using this three-parameter model, we consider the consequences of running the initial part of a race at a speed different from the constant rate proscribed by the hyperbola. We show that for any distance split, an improved time is achievable and that the least time occurs when both parts of the race are run at speeds determined by applying the hyperbola to each part. Further improvement is possible by an appropriate selection of initial distance, with the first part being run at a higher speed than the second. Still further improvement is possible if the athlete follows an all-out running strategy, and we prove that for this model an all-out strategy is uniquely optimal. Significant performance gains appear possible for events of less than 10-min duration. Thus, under this model, an athlete, who at any time during a race drops below their all-out pace, can never make up for lost time. This result is contrary to conventional wisdom. Accordingly, we examine some recent empirical evidence which confirms the predicted nature of all-out power development over short time periods and suggests that pace variation, at least to some degree, may not be as suboptimal as previously assumed.
Keywords: all-out effort; critical power; hyperbolic model; optimal strategy; power–duration curve