On the brachistochrone problem for cycling ascents

VAM ( velocità ascensionale media ) is a measurement that quantifies a cyclist’s climbing ability. We show that to minimize the time to attain a given height gain—which is tantamount to maximizing VAM—a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. This article is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that—for given start and end points, and for any fixed average-power constraint—the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.