MHB What is the optimal running speed in the rain to minimize getting wet?

[Ackbach]

My apologies for not getting to this on time. I will try harder next week!

Here is this week's POTW:

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Imagine a man running from his parked car to a building. He runs a distance $d \, \text{m}$ in the rain. The rain is falling at a terminal velocity of $v \, \text{m/s}$. What is the best speed for him to run so as to get as little rain on him as possible?

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[Ackbach]

No one answered this week's University Problem of the Week. You can see my solution below.

Spoiler

Just run as fast as possible without slipping and falling. Justification: imagine the person as a cylinder, and assume the rain is falling down vertically. The cylinder presents a rectangle cross-section moving forward, and a circle (really an ellipse) on top. First consider the rectangle presented forward: for every drop of water that falls on the ground, there will be another drop of rain entering the top of the rectangle. Therefore, from a forward perspective, it doesn't matter how fast you run. Next consider the top. Not only does a moving circle present a smaller target (the ellipse presented to the rain becomes more eccentric the faster you run), but the total amount of time you are exposed to the rain decreases as you run faster. However, if you slip and fall, you'll be delayed and get wetter, not to mention injured. Therefore, you should run as fast as you safely can.