Power pacing in a time trial: the physics behind every watt

pacingJuly 11, 20266 min readBorja Alfaraz

Physics

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Doubling speed multiplies aerodynamic power eightfold: it grows with the cube of v.

This non-linearity is why constant-power pacing is almost never optimal.

Abstract. This article develops the physics of power pacing in a time trial: the decomposition of power into its three resistances, the cubic dependence of the aerodynamic term on speed, and its implications for the optimal distribution of effort segment by segment.

‘Go full gas and hang on’ is the worst possible advice for a time trial. A TT is a physics problem with an optimal solution, and that solution is almost never constant power.

The equation that governs your speed

At every instant, your power is split to overcome three resistances plus drivetrain losses:

P = P_air + P_rolling + P_gravity

Air = ½·ρ·CdA·v³ — dominates on the flat and grows with the cube of speed.
Rolling = Crr·m·g·cosθ·v — nearly constant, matters more than it seems on bad surfaces.
Gravity = m·g·sinθ·v — rules on climbs, is zero on the flat and negative descending.

Because the air term scales with v³ and the gravity term with the gradient, the same power produces very different speeds depending on the terrain. That is the whole key to the split.

Figure 1. Power required to hold each speed on the flat (CdA 0.24, Crr 0.004, 78 kg). The curve is cubic: going from 40 to 45 km/h costs almost 100 W more than going from 20 to 25 km/h.

Why constant power is not optimal

Picture one extra watt. On a climb, that watt turns into a decent chunk of speed because you are mostly fighting gravity. On a fast descent, the same watt barely accelerates you: air resistance (v³) swallows almost all of it.

A watt is ‘worth’ more time on the climb than on the descent. That is why you should spend more uphill and less downhill.

The optimal strategy, provable mathematically, is variable: push above your target on climbs and headwinds, and ease on descents and tailwinds — where watts are wasted. Distributing like this can save meaningful time on the same total effort.

The limit: your physiology

If it were only physics, the answer would be ‘infinite power on every climb’. But your W′ is finite (see our article on W′bal). The real optimal split is the one that wins time with physics without emptying your anaerobic tank before the line. It is a constrained optimisation problem, not a rule of three.

From theory to plan

Solving this by hand is infeasible: you have to combine the power equation (with your CdA, your Crr and the real air density), the route profile segment by segment and your CP/W′ model. That is exactly what the AthletePro Super Record engine does: it solves speed by bisection in each segment and uses dynamic programming to find the watt split that minimises your time while respecting your reserve. The result is a per-segment power target you can actually follow, not a flat number impossible to hold.

Stop going ‘full gas’. Ride the right power in the right place.