Power variation strategies for cycling time trials: a differential equation model

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In cycling time trials, competitors aim to ride a course in the fastest possible time and the implementation of a pacing strategy is therefore essential. In this study, a differential equation model of a cyclist incorporating continuous changes in velocity is formulated and applied to a selection of theoretical courses and athletes. The model is augmented with a constraint corresponding to a mean work rate and various pacing strategies are considered. The inclusion of continuous accelerations experienced by the cyclist forms an essential component in a model for courses comprising many changes of gradient, and a steady-state approximation, which has previously been used to assess pacing strategies, is not suitable. In addition to formulating a result on the mathematically optimal solution of the model equations subject to the mean power constraint, it is also shown that substantial time savings can be realized by cyclists increasing their work rates on uphill sections and suitably reducing their work rates elsewhere. However, the amount of time saved is highly course- and athlete-dependent with the greatest gains arising on courses with the longest continuous ascents by cyclists of greatest mass.
Original languageEnglish
Pages (from-to)651 - 659
Number of pages8
JournalJournal of Sports Science
Issue number7
Publication statusE-pub ahead of print - 26 Mar 2012


  • mathematical model
  • time trial
  • pacing strategy
  • power output


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