To use this page, you need to be able to estimate parameters like your coefficient of rolling resistance (Crr), your drag coefficient (Cd), and others. I'll provide you with reasonable initial estimates of these, but you should try to measure or estimate them for yourself, since they have a substantial effect on your speed/power relationship. Use the links at the top-right of the page to try out some of my estimation tools.a
If you are interested in the physics behind the model used by this page, read through the the physics section at the bottom of this page. Otherwise, start tweaking the various parameters in the form below, and the graphs and time tables will automatically update.
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This web page uses physical models of forces on a cyclist to help you estimate the relationship between power P (watts) and velocity V (kph or mph) of a cyclist. To do this, you need to estimate several parameters; reasonable defaults are given.
There are three primary forces that you, as a cyclist, must overcome in order to move forward:
The formula for gravitational force acting on a cyclist, in metric units, is:
Fgravity (Newtons) = 9.8067 (m/s2) · sin(arctan(G/100)) · W (kg)
The formula for the rolling resistance acting on a cyclist, in metric units, is:
Frolling (Newtons) = 9.8067 (m/s2) · cos(arctan(G/100)) · W (kg) · Crr
Finally, there are other effects, like the slipperyness of your clothing and the degree to which air flows laminarly rather than turbulently around you and your bike. Optimizing your aerodynamic positions also help with this. These other effects are captured in another dimensionless parameter called the drag coefficient, or Cd. Sometimes you will see people talking about "Cd · A", or CdA. This is just the drag coefficient Cd multiplied by the frontal area A. Unless you have access to a wind tunnel, it is hard to measure Cd and A separately; instead, people often just measure or infer Cd · A as a combined number.
The formula for the aerodynamic drag acting on a cyclist, in metric units, is:
Fdrag (Newtons) = 0.5 · Cd · A (m2) · Rho (kg/m3) · (V (m/s))2
Fresist (Newtons) = Fgravity + Frolling + FdragFor each meter that you cycle forward, you spend energy overcoming this resistive force. The total amount of energy you must expend to move a distance D (m) against this force is called the Work (Joules) that you do:
Work (Joules) = Fresist (Newtons) · D (m)If you are moving forward at velocity V (m/s), then you must supply energy at a rate that is sufficient to do the work to move V meters each second. This rate of energy expenditure is called power, and it is measured in watts. The power Pwheel (watts) that must be provided to your bicyle's wheels to overcome the total resistive force Fresist (Newtons) while moving forward at velocity V (m/s) is:
Pwheel (watts) = Fresist (Newtons) · V (m/s)You, the cyclist, are the engine providing this power. The power that must be provided to your bicycle's wheels comes from your legs, but not all of the power that your legs deliver make it to the wheels. Friction in the drive train (chains, gears, bearings, etc.) causes a small amount of loss, usually around 3%, assuming you have a clean and nicely lubricated drivetrain. Let's call the percentage of drivetain loss Lossdt (percent).
So, if the power that your legs provide is Plegs (watts), then the power that makes it to the wheel is:
Pwheel (watts) = (1 - (Lossdt/100)) · Plegs (watts)Putting it all together, the equation that relates the power produced by your legs to the steady-state speed you travel is:
Plegs (watts) = (1-(Lossdt/100))-1 · (Fgravity + Frolling + Fdrag) · V (m/s)One of the scary implications of this equation is that at high speed, the power you have to produce is proportional to the cube of your velocity. So, to increase your speed by 25%, you need to nearly double your wattage!
or, more fully:
Plegs (watts) = (1-(Lossdt/100))-1 · ( ( 9.8067 (m/s2) · W (kg) · ( sin(arctan(G/100)) + Crr · cos(arctan(G/100)) ) ) + ( 0.5 · Cd · A (m2) · Rho (kg/m3) · (V (m/s))2 ) ) · V (m/s)