How fast to ride a bike when it’s cold

Last updated on 27/01/2024

Summary

Riding a bike is not very pleasant when it is cold, because the relative speed of air intensifies the thermal power subtracted from the body. On the other hand, speed reduction would increase the duration of the trip, the time of exposure to cold air and finally the total heat subtracted from the body.
So it makes sense to ask: what is the optimal speed that minimizes the heat lost by the body during the trip?

We will demonstrate that, under certain hypotheses, the optimal speed is the one at which the intensity of perceived cold doubles compared to the static case, i.e. at rest.

Analysis

Let $v$ be the cyclist’s speed and $d$ be the distance to be travelled. The duration of the trip is $\Delta t = \frac{d}{v}$ .

Let’s assume that the thermal power released by the human body to the environment is comprised of two terms: one constant and one dependent on speed. The constant term takes into account the thermal dispersion that would occur when $v=0$ ; the second term, for which we will assume a quadratic dependence on velocity, takes into account turbulent convective cooling.

In summary, we are assuming that the thermal power $P$ transferred from the body to the environment can be described by

$P = \alpha + \beta v^2$

The total heat lost by the body is $P \Delta t$ , i.e.:

$Q = \frac{d}{v}\left( \alpha + \beta v^2\right)$

The optimal speed is the one that minimizes the total heat lost by the body, i.e. the value of $v$ for which the following function is minimum:

$f(v) =\frac{\alpha + \beta v^2}{v }$

Deriving:

$\frac{df}{dv} = \frac{2\beta v^2-(\alpha + \beta v^2)}{v^2}=\frac{1}{v^2} (\beta v^2-\alpha)$

The optimal speed is

$v_m = \sqrt{\frac{\alpha}{\beta}}$

In practice

An estimate of $\alpha$ and $\beta$ is needed to determine $v_m$ . Several approaches are possible here. For simplicity, we assume the following psychometric hypothesis: the sensation of cold is proportional to the thermal power.

At rest ( $v=0$ ) the sensation of cold is proportional to $\alpha$ . Now let’s imagine that the intensity of the perceived cold increases by a factor $k$ when moving at speed $\bar{v}$ . That is

$\alpha + \beta \bar{v}^2 = k \alpha$

i.e

$\beta = (k-1)\frac{\alpha }{\bar{v}^2}$

According to the psychometric hypothesis (cold sensation $\propto$ thermal power) we can determine the optimal speed as $v_m = \sqrt{\frac{\alpha}{(k-1)\frac{\alpha}{\ bar{v}^2}}}$ , from which

(1) $\begin{equation*} v_m=\frac{\bar{v}}{\sqrt{k-1}} \end{equation*}$

Caution: At higher pedaling intensities the body needs to dispose of excess heat. Physiological cooling mechanisms guarantee body thermostatation but alter the temperature perception. For this reason it is necessary to carry out the estimates in quasi-rest conditions, i.e. before prolonged physical effort.

Numerical example

A cyclist estimates that the sensation of cold increases by $5$ times when traveling at $20\, km/h$ . Then the optimal speed will be $\frac{20}{\sqrt{5-1}}=10\, km/h$ .