MATHEMATICAL-MODELS OF RUNNING

A number of mathematical theories to describe the momentum and energy balances associated with running have been advocated since the pioneering work of the British biologist A.V. Hill in the 1920s. Since the various models lead to quite different conclusions and predictions about the balances associated with running, it is instructive to examine critically the assumptions underlying them. So, for example, depending on the model employed, it has been predicted that, for a sprinter, the relative expenditure of energy in overcoming air resistance lies somewhere in the range of 2 to 16%. This wide range of values derives mainly from the various estimates of the propulsive force exerted during sprinting, indicating just how disparate the predictions among the models are. In the early 1970s Keller built on the ideas of Hill to develop a complete theoretical model for competitive running in which the goal of the athlete is to cover the race distance in a minimum amount of time (subject to certain constraints on the runner.) While the Hill-Keller model provides only a first approximation, in that no attempt is made to model separately the anaerobic and the aerobic pathways for energy conversion and only the simplest model for the resistance to motion is considered, it nevertheless gives a good overall representation of the world running records from 60 yds. to 10 km. Based on a careful appraisal of the various models, it is suggested here that approximately 3% of a sprinter's effort is spent on overcoming air resistance, a figure that is supported by an independent estimate derived from the long jump. An interesting consequence of the analysis is the feature that the benefits of sprinting at altitude cannot be accounted for wholly by the reduced air density, suggesting that there may be some physiological benefit to sprinting al altitude. As a special application of the estimate for the cost of overcoming air resistance, an appraisal has been made of the series of 100 m races run at the U.S. Olympic Trials in 1988 in which Florence Griffith-Joyner set a new world record. This record has been treated with great suspicion in the athletics community because of a questionable wind-gauge reading. It is shown here that the record was almost certainly wind aided, with a trailing wind of 4 ms-1 or more.