On minimizing cyclists' ascent times

TL;DR

This paper proves that maintaining a constant ground speed during uphill cycling minimizes ascent time for a given average power, providing practical strategies for cyclists based on their power constraints.

Contribution

It introduces a theoretical proof that constant ground speed minimizes ascent time under average power constraints, with implications for training strategies.

Findings

Constant speed minimizes ascent time at fixed average power.

Maintaining constant power results in longer ascent times unless slope is constant.

Modified constant-speed strategies are optimal under maximum power constraints.

Abstract

We prove that, given an average power, the ascent time is minimized if a cyclist maintains a constant ground speed regardless of the slope. Herein, minimizing the time is equivalent to maximizing -- for a given uphill -- the corresponding mean ascent velocity (VAM: velocit\`a ascensionale media), which is a common training metric. We illustrate the proof with numerical examples, and show that, in general, maintaining a constant instantaneous power results in longer ascent times; both strategies result in the same time if the slope is constant. To remain within the athlete's capacity, we examine the effect of complementing the average-power constraint with a maximum-power constraint. Even with this additional constraint, the ascent time is the shortest with a modified constant-speed -- not constant-power -- strategy; as expected, both strategies result in the same time if the maximum and average powers are equal to one another. Given standard available information -- including level of fitness, quantified by the power output, and ascent profile -- our results allow to formulate reliable and convenient strategies of uphill timetrials.