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The Science
A brief explanation of the forces acting on a cyclist and the resulting relationship between power output (watts) and velocity (kph).
The Segment Parameters and Rider & Equipment Parameters subpages give more detailed information about the variables used by the Power Calculator to determine the required power.
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Gravity
When cycling uphill, the rider fights against gravity. When cycling downhill, gravity works with the rider. This doc calculates the steepness of a segment in terms of percentage grade G: rise/run * 100. The heavier the rider and the bike are, the more energy is required to overcome gravity. The combined weight of the cyclist and the bike is W (kg). The gravitational force constant g is 9.8067 (m/s^2).
Rolling Resistance
Friction between the tires and the road surface slows the rider down. The bumpier the road, the more friction; the higher quality the tires and tube, the less friction. As well, a heavier rider and bike experience more friction. The dimensionless parameter, the coefficient of rolling resistance, or Crr, captures the bumpiness of the road and the quality of the tires.
Aerodynamic Drag
A rider’s bike and body need to displace the air, similar to how a snowplow pushes snow out of the way. The air exerts a force against the rider as they ride. Higher velocity V (m/s) creates more force the air pushes against the rider. The rider and bike present a certain frontal area A (m^2) to the air. The larger this frontal area, the more air must be displaced, and the larger the force the air exerts. The air density Rho (kg/m^3) is also important; the more dense the air, the more force it exerts.
Other effects, like the slipperyness of clothing and the degree to which air flows laminarly rather than turbulently around the rider and bike, are captured in another dimensionless parameter called the drag coefficient, or Cd. Aerodynamic drag is the drag coefficient Cd multiplied by the frontal area A or “CdA”.
Total Resisting Force
The sum of the forces acting against the rider.
Gravity
Explanation
When cycling uphill, the rider fights against gravity. When cycling downhill, gravity works with the rider. This doc calculates the steepness of a segment in terms of percentage grade G: rise/run * 100. The heavier the rider and the bike are, the more energy is required to overcome gravity. The combined weight of the cyclist and the bike is W (kg). The gravitational force constant g is 9.8067 (m/s^2).
Force Formula
Fgravity = 9.8067 * sin(arctan(G/100)) * W
Overcoming the resisting forces: Work
With every meter traveled, the rider spends energy overcoming the resistive forces highlighted above. The total amount of energy expended to move a distance D (m) against this force is the Work (Joules) done by the cyclist:
Work = Fresist * D
Moving forward at velocity V (m/s), the rider must supply energy at a rate that is sufficient to do the work to move at V. This rate of energy expenditure is called power, measured in watts. The power Pwheel (watts) that must be provided to the bicycle’s wheels to overcome the total resistive force Fresist (Newtons) while moving forward at velocity V (m/s) is:
Pwheel = Fresist * V
The cyclist is the engine providing this power. Not all of the power that a rider’s legs deliver make it to the wheels. Friction in the drive train (chains, gears, bearings, etc.) causes a small amount of loss, usually around 2%. The percentage of drivetrain loss is Lossdt (%).
So, with power Plegs (watts), the power that makes it to the wheel is:
Pwheel = (1 − Lossdt/100) * Plegs
All together, the equation that relates the power produced by the rider to steady-state speed is:
Plegs = (1 − Lossdt/100)^-1 * [Fgravity + Frolling + Fdrag] * V
or, more fully:
Plegs = (1 − Lossdt/100)^−1 * [(9.8067 * W * [sin(arctan(G/100)) + Crr * cos(arctan(G/100))]) + (0.5 * Cd * A * Rho * V^2)] * V
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