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On the brachistochrone problem for cycling ascents

Len Bos, Michael A. Slawinski, Raphaël A. Slawinski, Theodore Stanoev

TL;DR

This work addresses the brachistochrone problem for cycling ascents under a fixed average power constraint. It derives that the minimum-time ascent between two points is achieved along the straight-line segment at a constant ground speed, i.e., with constant power, contrasting the classical descent brachistochrone which follows a cycloid with nonuniform speed. The authors formulate a power-speed-elevation model and prove, via implicit differentiation, that the ascent time strictly increases with path length, thereby favoring the shortest path. Numerical analyses with representative parameters reveal that the fastest ascent corresponds to the steepest feasible grade, subject to practical limits on cadence and balance, and quantify the trade-offs between slope, speed, and ascent time. The findings have implications for climbing strategy and performance metrics like VAM, highlighting that, under power constraints, steeper constant-grade ascents can maximize ascent rate within physical and ergonomic limits.

Abstract

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. The limit of steepness is imposed, among others, by such factors as the efficiency of pedalling, which is related -- among others -- to feasible cadence, maintaining balance and preventing skidding of the rear wheel. This article, however, is focused on the primary issue of 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.

On the brachistochrone problem for cycling ascents

TL;DR

This work addresses the brachistochrone problem for cycling ascents under a fixed average power constraint. It derives that the minimum-time ascent between two points is achieved along the straight-line segment at a constant ground speed, i.e., with constant power, contrasting the classical descent brachistochrone which follows a cycloid with nonuniform speed. The authors formulate a power-speed-elevation model and prove, via implicit differentiation, that the ascent time strictly increases with path length, thereby favoring the shortest path. Numerical analyses with representative parameters reveal that the fastest ascent corresponds to the steepest feasible grade, subject to practical limits on cadence and balance, and quantify the trade-offs between slope, speed, and ascent time. The findings have implications for climbing strategy and performance metrics like VAM, highlighting that, under power constraints, steeper constant-grade ascents can maximize ascent rate within physical and ergonomic limits.

Abstract

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. The limit of steepness is imposed, among others, by such factors as the efficiency of pedalling, which is related -- among others -- to feasible cadence, maintaining balance and preventing skidding of the rear wheel. This article, however, is focused on the primary issue of 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.

Paper Structure

This paper contains 5 sections, 3 theorems, 22 equations, 3 figures.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Formulation
  4. Numerical insights
  5. Discussion and conclusion

Key Result

Proposition 1

For given start and end points, the curve of the minimum ascent time --- for any fixed average-power constraint --- is the straight line connecting these points.

Figures (3)

  • Figure 1: $V(\theta)$: Optimal constant ground speed as a function of slope
  • Figure 2: $V(\theta)\sin\theta$: Vertical speed as a function of slope
  • Figure 3: $T(\theta)$: Ascent time for the elevation gain of $1\,000\,\rm{m}$ as a function of slope

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof