Road cycling is a captivating sport that demands both physical prowess and strategic decision-making. Behind every cyclist's powerful pedaling lies a fascinating world of calculations and equations that help determine the energy required for optimal performance. In this blog post, we'll delve into the science behind power calculation for road cycling.

When a cyclist pushes through the wind, the first force they encounter is air resistance. This force increases exponentially with speed, demanding more power as velocity rises. By utilizing the equation for air resistance,

\begin{equation} P_{drag} = 0.5 \: \text{CdA} \: \rho \: |v_c| \: (v_c - v_a)\:|v_c - v_a| \label{eq:pdrag} \end{equation}

where $P_{drag}$ is the power required to overcome air resistance, $CdA$ is the drag coefficient multiplied by the frontal area of the cyclist and the bicycle, $\rho$ is the air density, and $v_c$ and $v_a$ are the velocity of the cyclist and air velocity respectively. With help of Eq. (\ref{eq:pdrag}) the power required to overcome this formidable resistance can be estimated.

Additionally, the power needed to conquer rolling resistance cannot be overlooked. As the bicycle's tires roll over the road, friction opposes the motion, necessitating the expenditure of energy. The equation for rolling resistance,

\begin{equation} P_{rolling} = c_{rr} \: m \: g \: \cos(\theta) \: |v_c| \end{equation}

where $P_{rolling}$ is the power required to overcome rolling resistance, $\theta$ is the angle of the incline, $c_{rr}$ is the coefficient of rolling resistance, $m$ is the total mass of the cyclist and bike, $g$ is the acceleration due to gravity, and $v_c$ is the velocity of the cyclist. Now by taking these factors into account an estimation of the power required is obtained readily.

Confronting an incline poses yet another challenge to cyclists. Against the pull of gravity, they must summon additional power to conquer the slope. By incorporating the angle of the incline, the cyclist's mass, acceleration due to gravity, and velocity into the equation for power on gradients,

\begin{equation} P_{gradient} = m \: g \: \sin(\theta) \: |v_c| \end{equation}

where $P_{gradient}$ is the power required to overcome the gradient, $\theta$ is the angle of the incline, we can determine the extra energy expenditure required to conquer hills. The total power delivered by a cyclist is obtained by summation of the separate power contributions,

\begin{equation} \sum P = P_{gradient} + P_{rolling} + P_{drag} \end{equation}

it must be noted that the largest contribution to the power on flat terrain is the drag due to air resistance. Whereas in hilly or mountanous areas the largest contribution in the power balance riginates from the pull of gravity caused inclination angle of the road.

While the physical effort exerted by the cyclist is paramount, the efficiency of the drivetrain plays a crucial role in power transfer from pedals to the rear wheel. Calculating drivetrain power involves considering the total torque applied at the pedals, the angular velocity of the pedals, and the drivetrain's efficiency. The equation for drivetrain power is:

\begin{equation} P_{drivetrain} = \frac{{T_{total} \omega}}{\eta} \end{equation}

where $P_{drivetrain}$ is the power needed for the drivetrain, $T_{total}$ is the total torque applied at the pedals, $\omega$ is the angular velocity of the pedals or cadance, and $\eta$ is the overall drivetrain efficiency. Typical values of the drivetrain efficiency is $0.95<\eta<0.98$

By unraveling the complex equations that underpin power calculation for road cycling, we gain a deeper understanding of the factors influencing performance. From overcoming air resistance and rolling resistance to tackling gradients , each element contributes to the total power required. Armed with this knowledge, cyclists can make informed decisions for optimizing (training) sessions during rides.