• Ulf Holmlund
  • Raimo von Hertzen
  • Matti A. Ranta
Keywords: sprint models, physiological parameters, resistive force


INTRODUCTION: Hill’s model of sprinting, based on Newton’s second law of motion, uses two physiological parameters to characterize the sprinter, the maximum propulsive force per unit mass and the resistance-to-motion parameter related to the runner’s internal energy losses. Furusawa et al. (1927) suggested a resistive force law linear in the running speed. Later Keller (1973) and many others based their studies on Hill’s model. Senator (1982) added the effects of air resistance by a term quadratic in speed. Vaughan (1983) used a modification of these approaches by introducing a 0.7-power law. Recently, utilizing the rotational equation of motion for the leg and experimental data for stride frequency, we have shown that the internal and external resistive forces may well be approximated by a combination of linear and quadratic terms in running speed. We have also derived an expression for the internal resistive force in terms of physiological quantities. METHODS: The different models may be classified according to the resistive force law as linear (L), Vaughan (Va), quadratic (Q) and linearquadratic (LQ) models. We give analytical solutions for the distance-time relationships, except for the Va-model. We have used a numerical gradient method to fit the models with measured 100m data at ten equally spaced time stations of the 1988 Olympic Games in Seoul. Wind velocity and reaction times are also taken into account. RESULTS: By considering the residual errors we found that the best fit was given by the Va-model, followed by the LQ-, L-, and Q-models. The average residual error per time station for the Va- and LQ-models was about 0.01 s, which means a good fit throughout the run. We compared the values of the calculated physiological parameters with those in the existing literature. For the L-model the propulsive and resistive forces found by Vaughan and Matravers (1977) are in close agreement with ours, whereas those found by Woodside (1991) and Keller (1973) are too high, as can also be inferred from recent starting block data. For the parameters of the Va-model Vaughan (1983) obtained values somewhat lower than ours, using Ben Johnson, Carl Lewis and Linford Christie. This is to be expected, however, since Vaughan obtained his data for national university sprinters. As far as we know, the LQ-model has not been previously used. CONCLUSIONS: Our results show that the Va- and LQ-models best fit the Olympic 100 m data. There is a noteworthy difference between the interpretation of the linear resistive term in Hill’s theory and in the LQmodel presented in this paper: Hill and his colleagues invoked the concept of the viscosity of the muscles, while we arrived at the linear term by writing the rotational equation of the leg. It must be noted that already Fenn (1930) criticized the viscosity concept and attributed the resistive force to antagonistic muscles and other kinesiological factors. In view of our derivation of the LQ-model, the resistive force stems mainly from the rotational inertia of the leg, whereas the energy losses occur in the antagonistic muscles during the decelerating phases of the back and forth motion of the legs.