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# Bike Speed and Power Calculator

## Calculate Speed and Power

 Mode of calculation Power Speed Speed km / h Body weight kg Weight bike & apparel kg Air density ρ kg / m3 cw⋅a m2 Friction coefficient Distance km Drop m Time h:mm:ss Slope % Total power W Power for wind resistance W Power for friction W Power for slope W Relative power W / kg

The default values can be used for a road cyclist on the hoods. The air density ρ depends on temperature and altitude. The preset value is used for 20 °C at sea level. The product of $$c_{W}$$ and front face depends on the bike, the apparel and the position on the bike. The more aero the position is, which means you try to hide from the wind, the lower this value is and the fewer watts need to be done at the same speed. The friction coefficient for a mountain bike is higher thant the preset value. All values can be freely changed.

## Calculate Air Density

Air density can be calculated with the following calculator. The result is automatically used for the calculation of power in the above calculator.

 Temperature °C Altitude m Air density kg / m3

## Formulas

All values for the calculation need to be in SI-units. First, speed and slope needs to be converted into SI units.

$$v_{m/s} = \frac{v_{km/h}}{3.6}$$
$$slope_{decimal} = k = \frac{slope_{\%}}{100}$$

The necessary power to overcome air resistance and rolling resistance can be calculated using the following formulas.

$$P_{air} = \frac{1}{2} \cdot c_{w} \cdot A \cdot \rho \cdot v^3$$
$$P_{roll} = F_{roll} \cdot v = m \cdot g \cdot \mu_{R} \cdot v$$

To calculate the power for the slope, the climb rate has to be calculated first.

$$v_{climb} = \frac{k \cdot v}{\sqrt{1+k^2}}$$
$$P_{climb} = m \cdot g \cdot v_{climb}$$

The sum of these three portions of power results in the total power. To compare the power with people of different body weight objectively, this power is often based on the weight. This results in a relative Power with the unit W/kg.

$$P_{total} = P_{air} + P_{roll} + P_{climb}$$
$$P_{rel} = \frac{P_{total}}{m}$$

### Air Density

The air density $$\rho$$ can be calculated with the temperature $$T$$ and the air pressure $$p$$. The air pressure itself is dependend on the altitude $$h$$.

$$\rho = \frac{p}{R_{S} \cdot T}$$

The specific gas constant for air is:

$$R_{S} = 287,058 \frac{J}{kg \cdot K}$$

It is only valid for dry air and changes with increasing humidity. The temperature $$T$$ needs to be present as absolute temperature in Kelvin.

The air pressure is calculated using the barometric elevation formula. The following numeric values equation uses the altitude $$h$$ in metres and the standard measure air pressure $$p_{0}$$ of 101325 Pa.

$$p = p_{0} \cdot (1 - \frac{6,5 \cdot h}{288000})^{5.256}$$

## References

Internation System of Units, Retrieved 19 October 2019

Desnity of air, Retrieved 19 October 2019