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There are three primary forces that you, as a cyclist, must overcome in order to move forward:
Gravity: If you're cycling uphill, you're fighting against gravity,but if you're cycling downhill, gravity works for you. This page measures the steepness of a hill in terms of percentage grade G: rise divided by run, multiplied by 100. The heavier you and your bike are, the more energy you must spend to overcome gravity. The combined weight of you (the cyclist) and your bike is W (kg). The gravitational force constant g is 9.8067 (m/s2).
The formula for gravitational force acting on a cyclist, in metric units, is:
Fgravity=9.8067â
sin(arctan(G100))â
W$Fgravity=9.8067â
sinâĄ(arctanâĄ(G100))â
W$
Rolling resistance: Friction between your tires and the road surface slows you down. The bumpier the road, the more friction you'll experience; the higher quality your tires and tube, the less friction you'll experience. As well, the heavier you and your bike are, the more friction you'll experience. There is a dimensionless parameter, called the coefficient of rolling resistance, or Crr, that captures the bumpiness of the road and the quality of your tires.
The formula for the rolling resistance acting on a cyclist, in metric units, is:
Frolling=9.8067â
cos(arctan(G100))â
Wâ
Crr$Frolling=9.8067â
cosâĄ(arctanâĄ(G100))â
Wâ
Crr$
Aerodynamic drag: As you cycle through the air, your bike and body need to push the air around you, similar to how a snowplow pushes snow out of the way. Because of this, the air exerts a force against you as you ride. There are a few things that dictate how much force the air exerts against you. The faster you ride, velocity V (m/s), the more force the air pushes against you. As well, you and your bike present a certain frontal area A (m2) to the air. The larger this frontal area, the more air you have to displace, and the larger the force the air pushes against you. This is why cyclists and bike manufacturers try hard to minimize frontal area in an aerodynamic position. The air density Rho (kg/m3) is also important; the more dense the air, the more force it exerts on you.
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=0.5â
Cdâ
Aâ
Rhoâ
V2$Fdrag=0.5â
Cdâ
Aâ
Rhoâ
V2$
The total force resisting you, the cyclist, is the sum of these three forces:
Fresist=Fgravity+Frolling+Fdrag$Fresist=Fgravity+Frolling+Fdrag$
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 distanceD(m) against this force is called theWork(Joules) that you do:
Work=Fresistâ
D$Work=Fresistâ
D$
If you are moving forward at velocityV(m/s), then you must supply energy at a rate that is sufficient to do the work to moveVmeters each second. This rate of energy expenditure is calledpower, and it is measured in watts. The powerPwheel(watts) that must be provided to your bicyle's wheels to overcome the total resistive forceFresist(Newtons) while moving forward at velocityV(m/s) is:
Pwheel=Fresistâ
V$Pwheel=Fresistâ
V$
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 2%, assuming you have a clean and nicely lubricated drivetrain. Let's call the percentage of drivetain lossLossdt(percent).
So, if the power that your legs provide is Plegs (watts), then the power that makes it to the wheel is:
Pwheel=(1âLossdt100)â
Plegs$Pwheel=(1âLossdt100)â
Plegs$
Putting it all together, the equation that relates the power produced by your legs to the steady-state speed you travel is:
Plegs=(1âLossdt100)â1â
[Fgravity+Frolling+Fdrag]â
V$Plegs=(1âLossdt100)â1â
[Fgravity+Frolling+Fdrag]â
V$
or, more fully:
Plegs=(1âLossdt100)â1â
[(9.8067â
Wâ
[sin(arctan(G100))+Crrâ
cos(arctan(G100))])+(0.5â
Cdâ
Aâ
Rhoâ
V2)]â
V$Plegs=(1âLossdt100)â1â
[(9.8067â
Wâ
[sinâĄ(arctanâĄ(G100))+Crrâ
cosâĄ(arctanâĄ(G100))])+(0.5â
Cdâ
Aâ
Rhoâ
V2)]â
V$
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!
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