Cycling power to speed calculator

How the speed is calculated

Your pedalling power, after drivetrain losses, is balanced against the three forces resisting you: air drag, rolling resistance and gravity. This is the equation of motion of a cyclist [1], which Martin and colleagues validated against power measured on the road [2]. There is no closed-form solution for speed, so the calculator solves it numerically:

η · P  =  ½ · ρ · CdA · (v + w) · |v + w| · v   (air drag)
       +  Crr · m · g · cos θ · v             (rolling resistance)
       +  m · g · sin θ · v                   (gravity)

P is your power, η the drivetrain efficiency, ρ the air density, CdA your drag area, v your ground speed, w the headwind, Crr the rolling resistance, m rider plus bike and θ the slope. Every term is written out with its source on the methodology page.

Worked example, step by step

A 75 kg, 178 cm rider on the hoods, on an 8 kg bike, pedalling 250 W on flat ground at sea level in still air:

1. Drag area: estimated from height, weight and position at 0.339 m² (see the CdA estimator).
2. Air density at sea level on a standard day: 1.225 kg/m³.
3. Power at the wheel: 250 W × 0.976 = 244.0 W.
4. Solving for the speed where the resistance uses exactly that power gives 35.8 km/h.
5. At that speed, 203.6 W (83 percent) goes into air drag and 40.4 W (17 percent) into rolling resistance. On the flat, gravity takes nothing.
6. Ten kilometres at that speed takes 16:47.

That split is why drag area matters so much on the flat: most of your power goes into pushing air, so a better position buys more speed than a lighter bike.

Air speed, not ground speed

Drag builds on your speed through the air, while the power it costs is set by your speed over the ground. In still air the two are the same and drag power grows with the cube of speed, which is the shortcut most calculators take. With wind they come apart, and the calculator keeps them apart. A 20 km/h headwind cuts the example rider from 35.8 km/h to 24.5 km/h; the same wind from behind lifts them to 49.3 km/h. A tailwind faster than the rider pushes them along rather than holding them back, because the sign of the air speed is kept.

Altitude

Thinner air means less drag, so the same watts go further at altitude. The calculator works out the air density from the International Standard Atmosphere [3] at the altitude you enter:

The same rider and bike as the worked example. The model holds power constant, so it does not include the power most riders lose at altitude from the thinner air.

Climbing

Uphill, gravity takes over from drag, and speed falls fast with the gradient. Here weight matters far more than on the flat, which the watts per kg calculator explores.

Limitations

The speed limits

The solver works between 1.8 km/h and 144.0 km/h. If your power cannot hold even the lower limit on a steep climb, the calculator says so instead of showing a speed. Descents are capped at a maximum speed you can set, 70 km/h by default, as in the planner, and the result says when that cap applies.

What it assumes

Your CdA is estimated from a population regression unless you enter a measured value, and it is the input that moves the answer most on flat roads. The speed is a steady state: the model does not include acceleration, drafting, cornering or a wind that changes, and it uses a standard-day atmosphere rather than the weather.

A real ride is not one gradient. To get a target wattage and a predicted time for every climb, flat and descent of an actual course, use the race power planner with your GPX file.

References

1. di Prampero PE, Cortili G, Mognoni P, Saibene F. Equation of motion of a cyclist. Journal of Applied Physiology. 1979;47(1):201–206.
2. Martin JC, Milliken DL, Cobb JE, McFadden KL, Coggan AR. Validation of a mathematical model for road cycling power. Journal of Applied Biomechanics. 1998;14(3):276–291.
3. U.S. Standard Atmosphere, 1976. NOAA, NASA and U.S. Air Force.