Answer 5—The analysis is pretty straightforward, if you break it down into components that we’ll assume are independent of each other. There are two components to consider: your horizontal cross-section (the view looking down on the top of your head) and your vertical cross-section (the view looking at your front).

Your horizontal cross-section intercepts raindrops at a constant rate for as long as you are out, regardless of your (horizontal) speed. Your vertical cross-section intercepts raindrops that are in a certain volume of air at the moment you start moving. If you were to move infinitely fast, that volume would be horizontal and would extend from your starting point to the shelter. If you were to move somewhat more slowly, the volume would be skewed upward from your starting point at an angle that would depend on your speed relative to the rain’s downward velocity.

Now, the interesting thing is that this skewed shape has exactly the same volume as the horizontal one! (This follows from the area of a parallelogram, which is simply base × height.) This means that the amount of rain you catch on your front is only a function of your distance to the shelter and is completely independent of your speed. Therefore, your total wetness is minimized by minimizing the wetness of the horizontal cross-section, which is minimized by getting to shelter as soon as possible or running as fast as you can.

Contributor: David Tweed