Pace and Critical Gradient for Hill Runners: An Analysis of Race Records

Route choice through mountainous terrain requires a knowledge of how pace (the reciprocal of speed) varies with gradient of ascent or descent. To model this variation for runners, we analyse record times for 82 uphill and 14 downhill races or race stages. The pace is modelled as a polynomial function of gradient and a linear function of race duration, using ordinary least squares to obtain a best fit. For the gradient-dependence, a quartic function is derived which fits the data well; however, at steep gradients the quartic model is unrealistic and it may be argued that a linear model is more appropriate. Critical gradients, at which a runner’s vertical speed (uphill or downhill) is maximised, may be calculated from the quartic. However, there remains considerable uncertainty over the true value, and even the existence, of these critical gradients.