How fast to ride a bike when it rains

Last updated on 31/01/2025

Rain catches you while you are cycling towards your destination, which is at a distance $L$ from your current position. There is no wind, and the raindrops fall with an average speed $V$ that is practically vertical.

The rain wets your shoulders (the average area of your footprint on the ground is $A_h$ ) but it also wets you by hitting you in front (the average area of the frontal projection of your body is $A_v$ ), because you are moving forward at a speed $v$ .

The natural question in this situation is: “how fast should I move to minimize the number of drops that will hit me before arriving?“

One possible idea could be to move slowly. In this way, the number of drops that impact you frontally is minimal. However, if you reduce $v$ , the time spent in the rain ( $\Delta t$ ) increases, since $\Delta t = \dfrac{L}{v}$ .

Vice versa, moving fast, i.e. increasing $v$ would reduce the travel time, but that would also increase the number of drops that would impact you frontally.

So what is the best strategy, i.e. the optimal speed? Let’s do some calculations.

Let $\sigma$ be the number of drops that hit the unit surface of the ground in the unit time. Let’s take a time interval $\delta t$ . During that time interval, a number of drops $N$ will have fallen to the ground on a unit surface. Such number is given by

$N = \sigma \delta t$

These drops were enclosed in a prism with a unit base and height $\delta H = V \delta t$ . We can then determine the spatial density of drops, -let’s indicate it with $\rho$ -, that is, the average number of drops present in the unit volume of air:

(1) $\begin{equation*}\rho = \dfrac{N}{\delta H} = \dfrac{\sigma \delta t}{V \delta t}= \dfrac{\sigma}{V}\end{equation*}$

In the time $\Delta t$ the number of drops that will have hit us from above will be

(2) $\begin{equation*} N_h = \Delta t A_h \sigma \end{equation*}$

while the number of drops that will have hit us from the front will be

(3) $\begin{equation*} N_v = L \, A_v \, \rho = \Delta t \, v \, A_v \dfrac{\sigma}{V}\end{equation*}$

Overall we will have been hit by a number $N_T$ of drops given by

(4) $\begin{equation*} N_T = N_h+N_v = \Delta t \, \sigma \left( A_h + A_v \dfrac{v}{V}\right) = L \, \sigma \left( \dfrac{A_h}{v} + \dfrac{A_v}{V}\right) \end{equation*}$

The conclusion that simply emerges from inspecting 4 is that you need to pedal as fast as possible to minimize the amount of rain collected during the trip. The minimum number of drops that will hit you will be