REF-126 · Boswell, 2012 · Journal of Sports Sciences, · Paper

Power Variation Strategies for Cycling Time Trials: A Differential Equation Model

Cweak evidence: opinion or lab dataMeasured on a mixed groupWritten before October 2026, being re-checked

Boswell formulates a differential-equation model of a cyclist that includes continuous changes in velocity (accelerations), applies it to theoretical courses and athletes, and constrains it to a fixed mean work rate so that pacing strategies can be compared at equal average power.

Two findings: (1) on courses with many gradient changes the continuous accelerations matter, and the steady-state approximation previously used to judge pacing is not suitable; (2) at the same mean power, a rider saves substantial time by RAISING work rate on uphill sections and lowering it elsewhere.

The size of the saving is highly course- and athlete-dependent — greatest on courses with the longest continuous ascents and for heavier cyclists. The paper also states a result on the mathematically optimal solution under the mean-power constraint. It says nothing about negative splits or late-race surges.

Protocol

Modelling study with no human participants (from the abstract): a differential equation model of a cyclist that includes continuous changes in velocity, applied to a selection of theoretical courses and athletes.
The model is constrained to a mean work rate and various pacing strategies are compared under that constraint; a result on the mathematically optimal solution under the mean-power constraint is formulated.
Outcome: course time, compared across courses (length of continuous ascents) and athletes (mass).

Key Results

Variable-power pacing beats constant power at equal mean work rate: more power uphill, less elsewhere.
Gains scale with the length of continuous climbs and with rider mass.
Steady-state (velocity-constant) pacing models are inadequate on courses with many gradient changes.
Mathematically optimal solution under the mean-power constraint is formulated (no numeric values in the abstract).

Practical Application

For a time trial on rolling or hilly terrain, holding the same average power but pushing harder on the climbs and easing on flats and descents saves time — substantially so on courses with long continuous ascents and for heavier riders — because time lost climbing slowly is not recovered by extra speed where air resistance dominates.
On a flat course with few gradient changes the model offers little; even pacing at the target mean power is the default there.
Inference, not tested: the model constrains mean power, not physiological cost; surging uphill at the same average power costs more fatigue than the equation sees, so the real-world uphill surplus should be moderate and rehearsed in training.
Not from this paper: the energy-flow model for 1000 m and 4000 m track events (all-out start, then even pacing) is REF-127.

Limitations

Theoretical courses and athletes; mean-power constraint ignores the physiological cost of power variation; no validation against real rides in the abstract; abstract reports no numeric time savings.
How this entry was checked and changed
Why this level

regraded 2026-09-27 from B (was marked unverified): modelling study on theoretical courses, not validated against measured performances in the abstract

Content correction 2026-09-27
Entry said the model shows "even pacing minimizes fatigue", "negative-split pacing is optimal", "3-5% TTE advantage", "power reserve in the final 15-20%", "hold 90% FTP for 60% then 95-100% for the final 30 min" -> the paper is a mean-power-constrained model showing that raising power UPHILL and lowering it elsewhere saves time, with gains depending on climb length and rider mass; it contains none of the negative-split claims or numbers.Title, summary and lead rewritten from the abstract.
DOI line added from the PMID already in the entry; Level B [pending] -> C (unvalidated modelling); Sport "cycling, sprint" -> cycling.
2026-09-28: added Protocol (modelling design; sample size not applicable) from the abstract (Europe PMC PMID 22348315).