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Cycling power calculation

The physics behind cycling

To move forward with constant speed \(V\) you have to provide energy (power) to overcome total resistive force:

\[P = F_r \cdot V = ( F_{downhill} + F_{rolling} + F_{drag} ) \cdot V\]


Cycling uphill or downhill force:

\[F_{downhill} = m \cdot g \cdot sin(\theta)\]


  • \(m\) - weight of cyclist and bike;
  • \(g = 9.80665~m/s^2\) - earth-surface gravitational acceleration.

Rolling resistance

\[F_{rolling} = C_{rr} \cdot m \cdot g \cdot cos(\theta)\]


  • \(C_{rr}\) - coefficient of rolling resistance.

The coefficient of rolling resistance of the air filled tires on dry road:

\[C_{rr} = 0.005 + \frac 1 p \left( 0.01 + 0.0095 \left(\frac V {100}\right)^2 \right)\]


  • \(p\) - the wheel pressure (Bar);
  • \(V\) - the velocity (km/h).

The angle \(\theta\) can be calculated using elevation gain and total distance:

\[tan(\theta) = \frac H L \Rightarrow \theta = arctan\left(\frac H L\right)\]


  • \(H\) - height (opposite side);
  • \(L\) - length (adjacent side).

Aerodynamic Drag

Drag force:

\[F_{drag} = \frac 1 2 \cdot \rho \cdot (V - V_w)^2 \cdot C_d \cdot A\]


  • \(\rho\) - the density of the air;
  • \(V\) - the speed of the bike;
  • \(V_W\) - the speed of the wind;
  • \(A\) - the projected frontal area of the cyclist and bike;
  • \(C_d\) - the drag coefficient.

Approximated body surface area can be estimated from the measurement of the body height and body mass (Du Bois & Du Bois, 1916; Shuter & Aslani, 2000):

\[A = 0.00949 \cdot (H/100)^{0.655} \cdot m^{0.441}\]


  • \(H\) - the body height in \(m\);
  • \(m\) - the body mass in \(kg\).

Drag coefficient in cycling can be related to the body mass also and depends on cyclist position.


The density of the air is its mass per unit volume:

\[\rho = \frac m V\]


  • \(m\) - the mass;
  • \(V\) - the volume.

It decreases with increasing altitude and changes with variation in temperature or humidity.

The density of dry air:

\[\rho = \frac {p_0 M} {R T_0} \left(1 - \frac {Lh}{T_0}\right)^{gM/RL-1}\]

where air specific constants:

  • \(p_0 = 101325~Pa\) - sea level standard pressure;
  • \(T_0 = 288.15~K\) - sea level standard temperature;
  • \(M = 0.0289654~kg/mol\) - molar mass of dry air;
  • \(R = 8.31447~J/(mol \cdot K)\) - ideal gas constant;
  • \(g = 9.80665~m/s^2\) - earth-surface gravitational acceleration;
  • \(L = 0.0065~K/m\) - temperature lapse rate.

Density close to the ground is:

\[\rho_0 = \frac {p_0 M} {R T_0}\]

At sea level and at 15℃, air has \(1.225~kg/m^3\).

Using exponential approximation:

\[\rho = \rho_0 e^{(\frac {gM}{RL} - 1) \cdot ln(1 - \frac {Lh}{T_0})} \approx \rho_0 e^{-(\frac {gMh} {R T_0} - \frac {Lh} {T_0})}\]


\[\rho \approx \rho_0 e^{-h / H_n}\]


\[\frac 1 H_n = \frac {gM} {R T_0} - \frac L T_0\]

So \(H_n = 10.4~km\).

Coefficients Table

Rolling resistance coefficient:

Tire type\(C_{rr}\)
Road bike0.004

Surface area and drag coefficient of cyclist:

Position\(A~m^2\)\(C_d\)\(C_d A\)
Back Up0.4230.6550.277
Back Horizontal0.3700.6380.236
Back Down 10.3390.6550.222
Back Down 20.3340.6410.214
Top Tube 10.3710.6440.239
Top tube 20.3550.6110.217
Top Tube 30.3450.5880.203
Top Tube 40.3330.6040.201
TT* & TT Helmet0.3700.6410.237
TT Top Tube0.3310.5680.188
TT & Helmet0.3740.6790.254

* Time trial


Based on all this formulas we are able to calculate power effort, burned calories and fat loss of bike ride activity.

  • http://bikecalculator.com
  • http://thecraftycanvas.com/library/online-learning-tools/physics-homework-helpers/incline-force-calculator-problem-solver/
  • https://www.gribble.org/cycling/power_v_speed.html
  • https://www.researchgate.net/publication/51660070_Aerodynamic_drag_in_cycling_Methods_of_assessment
  • https://www.sciencedirect.com/science/article/pii/S0167610518305762
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