Modelling the effect of curves on distance running performance

Background Although straight ahead running appears to be faster, distance running races are predominately contested on tracks or roads that involve curves. How much faster could world records be run on straight courses? Methods Here,we propose a model to explain the slower times observed for races involving curves compared to straight running. For a given running velocity, on a curve, the average axial leg force (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\overline{F}}_{a}$\end{document}F¯a) of a runner is increased due to the need to exert centripetal force. The increased \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\overline{F}}_{a}$\end{document}F¯a presumably requires a greater rate of metabolic energy expenditure than straight running at the same velocity. We assumed that distance runners maintain a constant metabolic rate and thus slow down on curves accordingly. We combined published equations to estimate the change in the rate of gross metabolic energy expenditure as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\overline{F}}_{a}$\end{document}F¯a, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\overline{F}}_{a}$\end{document}F¯a depends on curve radius and velocity, with an equation for the gross rate of oxygen uptake as a function of velocity. We compared performances between straight courses and courses with different curve radii and geometries. Results The differences between our model predictions and the actual indoor world records, are between 0.45% in 3,000 m and 1.78% in the 1,500 m for males, and 0.59% in the 5,000 m and 1.76% in the 3,000 m for females. We estimate that a 2:01:39 marathon on a 400 m track, corresponds to 2:01:32 on a straight path and to 2:02:00 on a 200 m track. Conclusion Our model predicts that compared to straight racecourses, the increased time due to curves, is notable for smaller curve radii and for faster velocities. But, for larger radii and slower speeds, the time increase is negligible and the general perception of the magnitude of the effects of curves on road racing performance is not supported by our calculations.