classical mechanics - Does more rain strike a vehicle while moving or while stopped (or neither)?

Assume there is a rainstorm, and the rain falling over the entire subject area is perfectly, uniformly distributed. Now assume there are two identical cars in this area. One is standing still, and one is traveling (at any rate, it doesn't matter).

In theory, does one get struck by more water than the other? I understand that the velocity at which the raindrops strike the moving car will be higher. But because the surface area of the vehicles is identical and the rain is uniformly distributed, shouldn't each get hit by the same amount of water at any given moment or over any span of time?

My intuition is pulling me in all sorts of different directions on this question.

Answer

My understanding of the question is that it's about minimizing the rate at which rain hits the car. That makes it different from this question, which assumes you want to minimize the total amount of water that hits you before you get to a certain destination.

First let's assume the rain is perpendicular to the road and the car is a sphere. Then by the following argument, more rain hits the moving car.

We have $\mathbf{v}_{cr}=\mathbf{v}_{ce}+\mathbf{v}_{er}$, where $\mathbf{v}_{cr}$ is the car's velocity relative to the rain, $\mathbf{v}_{ce}$ is the car's velocity relative to the earth, and $\mathbf{v}_{er}$ is the earth's velocity relative to the rain. Let the car have cross-sectional area $A$. In the rain's frame of reference, the car is moving at $\mathbf{v}_{cr}$, and in time $t$ it sweeps out a volume $V=At|\mathbf{v}_{cr}|$. This volume is maximized by maximizing $|\mathbf{v}_{cr}|$, and if $\mathbf{v}_{ce}$ is perpendicular to $\mathbf{v}_{er}$, then this is always maximized by mazimizing $|\mathbf{v}_{ce}|$.

In reality, the car isn't a sphere, so the cross-sectional area $A$ presented to the rain is a variable. In some cases, this could allow the car to hit less rain while moving. As an unrealistic example, suppose the car is a pancake tilted at an angle of 45 degrees, and the rain is falling at 10 km/hr. Then the car can minimize how much rain hits it by driving at 10 km/hr.

If the rain isn't perpendicular to the road, but the car is a sphere, then $|\mathbf{v}_{cr}|$ may be minimized for some nonzero value of $\mathbf{v}_{ce}$.

These examples show that in general, the result depends on both the shape of the car and the angle of the rain with respect to the road.