# The cycling power equation

These calculators are based on the equation in the paper detailed below. Aerocoach Australia thanks the authors for their contribution to our understanding of cycling power models.

**Validation of a Mathematical Model for Road Cycling Power**

*Authors: James C. Martin, Douglas L. Milliken, John E. Cobb, Kevin L. McFadden, Andrew R. CogganJournal of Applied Biomechanics Volume: 14 Issue: 3 Pages: 276-291*

## P_{total} = [ P_{at} + P_{ke} + P_{rr} + P_{wb} + P_{pe} ] / E_{c}

P_{total} - Total power demand (W)

P_{at} - Power required to overcome aero drag (W)

P_{ke} - Power from changes in kinetic energy (W)

P_{rr} - Power required to overcome rolling resistance (W)

P_{wb} - Power required to overcome wheel bearing drag (W)

P_{pe} - Power from changes in gravitational potential energy (W)

E_{c} - Efficiency of drivetrain (unitless)

P_{at} = ½ ρ.V_{a}^{2}.V_{g} . (CdA + F_{w})

P_{ke} = ½ (m+(I/r^{2})) . (V_{gf}^{2}-V_{gi}^{2}) / (T_{f}-T_{i})

P_{rr} = V_{g}.C_{rr}.m.g.Cos(i)

P_{wb} = 0.0087 V_{g}^{2} + 0.091 V_{g}

P_{pe} = V_{g}.m.g.Sin(i)

ρ - air density (kg.m^{-3})

V_{a} - air velocity relative to direction of travel (m.s^{-1})

V_{g} - ground velocity (m.s^{-1})

CdA - coefficient of drag x frontal area (m^{2})

F_{w} - wheel rotation factor, expressed as incremental frontal area (m^{2})

m - total mass of bike + rider (kg)

l - wheel moment of inertia (kg.m^{2})

r - outside radius of tyre (m)

V_{gf} - final ground velocity (m.s^{-1})

V_{gi} - initial ground velocity (m.s^{-1})

T_{f} - final time (secs)

T_{i} - initial time (secs)

Crr - coefficient of rolling resistance (unitless)

g - acceleration due to gravity (m.s^{-2})

Gr - gradient rise/run (unitless)

i - ARCTAN(Gr)

Notes about model as programmed:

F_{w} is ignored and included within the CdA value

P_{ke} - Power from changes in kinetic energy, i.e. accelerations and decelerations, are ignored and set at zero as the model only considers steady state cycling.

Cardano's method is used to solve the cubic equation required to derive speed from power. This closed form solution does however have its limitations, in particular this method is not suitable for some combinations of inputs, which generally occur with larger negative gradients.

Finally, the calculators have been programmed using a web based calculation service - calculoid.com

Aerocoach Australia notes there may be bugs in the calculoid tools and while much care has also been taken to set up the calculators, no warranties are provided as to the accuracy of the results displayed.