Several weeks ago, my officemates and I had a discussion on the question, “do you get wetter walking or running in the rain?” My guess is that the walker gets more wet, primarily because he spends more time in the rain. But intuitively, you could also say that the runner gets more wet because he is running into rain, rain that would not hit you if you were walking more slowly.

So I had the time of my life reading this Acts of Volition weblog entry tackling the same problem (link via Kottke’s Remaindered Links). Read the comments for some of the geekiest discussions on a such seemingly simple problem.

My personal take on the problem is to analyze it in terms of volume. The unspoken assumption is that the walker or runner has to cover the same distance. First, let’s assume that rain is evenly distributed in space and that it falls with a constant velocity (assume that the direction is straight down). Now imagine shifting the frame of reference of space from earth into the inertial frame of reference (how relativity!) where the rain is not moving. Got that? Then a person walking/running in the rain would both have a horizontal and a vertical component of velocity in this new inertial frame of reference. The slower the person moves, the larger the vertical component.

Now let’s idealize a walker/runner by imagining that he is a simple rectangular solid (a box if you may). Therefore, the amount of rain this “person” gets is proportional to the volume that the box carves out in this frame of reference. The diagram below shows the side view of a runner (left) and a walker’s (right) volume occupied in our frame of reference. The walker/runner is moving left to right. The large light-blue areas are the rain-filled space, while the walker and the runner are the dark rectangles (the left position is the person just going into the rain, the right one is the person just coming out of the rain). The medium-blue filled areas are the volume occupied by the walker and runner.

Without going into the details, we can see that the volume (or side-view area—we can ignore depth since this is constant) of the walker is larger. While the height of the box carves out identical areas in both diagrams, the width of the box swipes out a much larger area when the person is moving more slowly. Therefore, a person gets wetter walking in the rain rather than running. Q.E.D.

Of course the real solution to the problem would be to bring an umbrella.