With a relatively simple basic power equation, it is possible to accurately describe how much power a cyclist must produce at a given speed to overcome opposing forces. This equation helps estimate, for instance, how much power is needed to maintain a certain speed, how much time can be saved by reducing weight on a climb, or whether your aerodynamic position on the bike is efficient compared to others. However, before addressing these practical questions, let’s cover the theoretical basics. I promise to keep it short!
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Three main parts are regarded when calculating the total power needed to ride at a certain speed: Climbing resistance, rolling resistance and aerodynamic drag. All three yield forces that are opposing the cyclists forward movement.
Gravitational Resistance and Climbing Power
Cycling uphill is more challenging, as everyone knows. The steeper the gradient, the harder it becomes, primarily due to gravity. The corresponding gravitational force always acts directly downward and is calculated as the product of the gravitational constant g and the mass m. The so-called downhill force, which causes you to roll backward (or makes climbing harder), is a component of this force and can be determined using the sine of the slope angle α.
$F_{down} = F_{g} \cdot \sin{\alpha} = m \cdot g \cdot \sin{\alpha}$
In practice, we don’t usually deal with this angle directly but instead refer to the slope (or incline) in percentage terms, known as the gradient. The slope angle α and the gradient grad can be easily converted into one another. See? High school trigonometry is unexpectedly useful!
$\alpha = \arctan{grad}$
While the gravitational force remains constant as long as the total mass and gradient do not change, cycling faster requires more power to overcome it. In physics, power is defined as work (energy) per unit of time. Work is the product of force and distance. Climbing power, therefore, can be calculated by multiplying the gravitational force by speed (distance divided by time).
$P_{climb} = m \cdot g \cdot \sin{\alpha} \cdot v$
Rolling Resistance and Its Coefficient
Resistance is everywhere in life, even when you’re casually cycling on flat terrain. This resistance arises primarily from energy losses due to wheel bearing friction and, more significantly, the deformation of tires as they roll. At the point of contact between tire and ground, the tires are compressed under the weight force during every rotation. This constant compression consumes energy, which is drawn from the rider’s kinetic energy.
Additionally, the nature of the surface plays a significant role. The rougher it is, the greater the vibrations generated, which also consume energy. Of course, cycling adheres to the laws of physics, particularly the principle of energy conservation. The energy lost due to tire deformation and vibrations does not disappear but is converted into heat.
Unlike gravitational resistance, rolling resistance cannot be derived as elegantly from theory. It increases linearly with weight (a heavily loaded bike rolls worse) and thus depends on the slope angle or gradient. To calculate it, multiply the cosine of the angle by an experimentally determined coefficient $C_{rr}$, which varies based on tire type and width.
$P_{roll} = F_{g} \cdot \cos{\alpha} \cdot C_{rr} \cdot v = m \cdot g \cdot \cos{\alpha} \cdot C_{rr} \cdot v$
Since the cosine of the slope angle is used here instead of the sine (as in climbing power), rolling resistance becomes less significant on steeper inclines. The steeper the slope, the less power is required to overcome rolling resistance. However, the gravitational resistance becomes so significant that this reduction isn’t particularly noticeable.
Aerodynamic Drag
While casually cycling or riding with a motorized assist, the wind can be pleasant. On hot days, it cools you down and provides a sense of speed—perhaps even a sense of freedom for the romantics. For road cyclists, however, aerodynamic drag is often the greatest challenge. It is the “queen” of resistances and a real nuisance because it is highly nonlinear!
To double your speed, you don’t just need twice as much power to overcome aerodynamic drag, as is the case with gravitational and rolling resistance—you need significantly more. Why? Because this is not a constant force; instead, the opposing force itself depends on speed.
Since we (typically) don’t ride in a vacuum, we must constantly displace air molecules as we move forward, essentially claiming our space cubic millimeter by cubic millimeter. These molecules don’t simply disappear; instead, they are pushed forward and sideways, displacing other molecules in turn. You can think of this process as akin to compressing a large spring: the harder we push, the greater the resistance force becomes. In this case, it’s even more challenging because we’re dealing with a progressive spring, where the opposing force rises disproportionately (not just the power required) with the square of the speed.
Similar to rolling resistance, the force of aerodynamic drag cannot be easily calculated but depends heavily on the size, shape, and surface texture of the object in question (in this case, the rider and the bike). To address this complexity, experimentally determined coefficients, known as $C_dA$ are used. But more on that another time.
Naturally, more air molecules result in greater resistance, meaning that the opposing force also depends on the air density $\rho_{Air}$, which is influenced by factors such as temperature and altitude. For now, we’ll skip further details on this as well.
$P_{drag} = \frac{1}{2}\rho_{Air} \cdot C_dA \cdot v^3$
In the case of headwinds w, cycling becomes even more challenging, as the effective airspeed increases accordingly. To calculate the power needed to overcome aerodynamic drag, multiply the resistance force (which depends on the square of the relative airspeed) by the ground-relative speed.
$P_{drag} = \frac{1}{2}\rho_{Air} \cdot C_dA \cdot v \cdot (v+w)^2$
The Complete Power Equation
When you combine all three components – power required to overcome gravitational resistance, rolling resistance, and aerodynamic drag – you get the cycling power equation. Even without calculating, it’s clear that aerodynamic drag plays a special role due to its cubic dependence on speed, becoming increasingly significant as speed increases. To relate this to the actual power produced by the rider – typically measured at the crank or pedals – efficiency losses in the drivetrain can be accounted for using a factor ε (epsilon for drivetrain efficiency). This results in the full cycling power equation in all its glory:
$P=\epsilon\cdot[ P_{climb}+P_{roll}+P_{drag}]$
$P=\epsilon\cdot[ m \cdot g \cdot v \cdot (\sin{\alpha} + C_{rr} \cos{\alpha}) +\frac{1}{2}\rho_{Air} \cdot C_dA \cdot v \cdot (v+w)^2]$
$\alpha = \arctan{grad}$