The computational cyclist.
Steve Gribble   ·   gribble [at] gmail [dot] com

Interactive cycling power and speed calculator

This page contains an interactive calculator that lets you tweak the various physical parameters that dictate how fast you can ride. Using it, you can explore the relationship between the power you produce (wattage) and the steady-state speed you travel under it, and how the different physical parameters affect the power vs. speed relationship.

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.

I've tested this page on Safari, Chrome, and Firefox. The graphs work on IE9, but not on earlier versions of IE. Upgrade, or download Chrome; it's a wonderful browser!

The interactive power model

Provide estimates by modifying the default values in the editable fields below. As you complete an update to a field, the model will update itself. Remember that the speeds involved are steady-state, i.e., the speed you will be able to sustain under the given power, assuming all other parameters stay constant.

Be sure to move your cursor over the graph to the right; it is interactive!

Units:   metric   imperial

Rider and bike parameters

Weight of rider (kg)
Weight of bike (kg)
Total weight W (kg):

Frontal area A(m2)
Drag coefficient Cd
Cd · A (m2):

Drivetrain loss Lossdt (%)

Environmental parameters

Percent grade of hill G (%)
Coefficient of rolling resistance Crr
Air density Rho (kg/m3)
Use the following fields to (a) type in a particular velocity, and see the power required to produce it, or (b) type in a particular power, and see the velocity the model predicts.
Power P (watts)
Velocity V (km/h)
Velocity V (km/h)
Power P (watts)

The physics affecting cycling at constant speed

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 total force resisting you, the cyclist, is the sum of these three forces:
Fresist (Newtons) = Fgravity + Frolling + Fdrag
For 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)

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)

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!