Cycling Power to Speed Calculator

Power to Speed Calculation Explained

This calculator estimates the cycling speed based on the power output and various environmental and physical factors. The calculation involves solving the equation that balances the forces acting on the cyclist.

Key Factors:

• Air Resistance: Depends on air density, drag area (C_dA), and speed.
• Rolling Resistance: Depends on the rolling resistance coefficient (C_rr) and weight.
• Drivetrain Efficiency: Accounts for power losses in the bike's mechanical system.

Calculation Steps:

1. Calculate Air Density (ρ):

   Air density is calculated based on temperature and altitude using the International Standard Atmosphere model.

2. Adjust Power for Drivetrain Efficiency:

   The power at the wheel is calculated by multiplying the input power by the drivetrain efficiency:

   Pwheel = Pinput × (Efficiency ÷ 100)

3. Define the Forces Acting on the Cyclist:

   The total opposing forces are air resistance and rolling resistance:

   Ftotal = Fair + Frolling

   Where:

   Fair = 0.5 × ρ × CdA × v²
Frolling = m × g × Crr

4. Set Up the Power Equation:

   The power required to overcome these forces at speed v is:

   Pwheel = Ftotal × v

   Substituting the expressions for the forces:

   Pwheel = (0.5 × ρ × CdA × v² + m × g × Crr) × v

   Simplify to get a cubic equation in terms of v:

   Pwheel = a × v³ + b × v
   Where:
   a = 0.5 × ρ × CdA
b = m × g × Crr

5. Solve for Speed (v):

   The equation is solved numerically using the bisection method to find the speed that satisfies the power equation.

Constants and Variables:

• ρ (Air Density): Calculated from temperature and altitude.
• C_dA (Drag Area): Product of drag coefficient and frontal area.
• C_rr (Rolling Resistance Coefficient): Depends on tire material and pressure.
• m (Mass): Total mass of the rider and bike.
• g (Gravity): Acceleration due to gravity (9.80665 m/s²).
• P_wheel: Power at the wheel after drivetrain losses.

Final Speed Calculation:

By solving the cubic equation for v, we find the estimated cycling speed given the input power and other parameters.

Note: This calculation assumes flat terrain and steady-state conditions without wind. Real-world conditions may vary.