# Which candidate is most washed up?

Donovan Clump and Hermione Timmon are political candidates in the electoral race for "Most Thankless Job in Modern History".

Both candidates set out into the wilderness at the same time. Rains are pouring down, hence both candidates quickly get their feet wet. There are no winds of change, hence the rain is falling straight down at terminal velocity.

Ms. Timmon sets out walking in a straight line at a constant speed (significantly less than terminal velocity), consistent with her "go slow with what you know™" campaign slogan.

Mr. Clump, concerned about the rain's effect on his hair, tarries in the rain just outside the door for ten seconds, contemplating turning back. Ultimately he decides to run too. He takes off on the same straight path as Ms. Timmon, accelerating at a uniform rate and gaining campaign momentum, until catching up with Ms. Timmon ten seconds after he starts running.

All the while, the rains are falling at a constant rate.

Assuming the two candidates are identical in all respects except their differing motion paths (which one might consider their campaign trails), at the exact moment Mr. Clump catches up to Ms. Timmon, which candidate will have been pelted by the greater number of rain droplets?

In other words: which candidate is most washed up?

Modeling:

• assume both candidates are cuboids of identical dimensions ($2$ m tall by $\tfrac{3}{10}$ m side-to-side and front-to-back)
• all rain droplets hitting the candidates count, including their fronts
• all physical effects/considerations aside from the candidates' differing $v\left( t\right) \equiv \dot{x}\left( t\right)$ should be ignored
• Ms. Timmon's (constant) forward velocity is $0 < v_{\rm tim} < v_{\rm t}$, where $v_{\rm t}$ is the terminal velocity of a rain droplet

This is fundamentally a problem of constructing a very basic physical model, and then good old fashioned math.

• My gut tells me that Mr Clump gets hit with more since his surface area hit from water droplets is larger (his whole front will be receiving water, whereas Ms Timmon only gets hit on the head and shoulders). – Ian MacDonald Jul 17 '15 at 18:43
• Do we assume they are cuboids? – Puzzle Prime Jul 17 '15 at 18:54
• This assumes that we can only hit Trump from the top and not the front correct? – qwertylpc Jul 17 '15 at 18:59
• @qwertylpc: I've clarified modeling in the problem statement. Indeed we must consider all raindrops. As you point out, considering only those on the top renders the answer to the puzzle trivial. ;) – COTO Jul 17 '15 at 19:44
• Providing some details about their dimensions would be crucial as assuming unrealistic sizes Will allow us to make the answer sway either way – CodeNewbie Jul 18 '15 at 3:32

They will both be hit by the same amount of rain. Assuming the OP doesn't tweak the slight ambiguity in the question, they have both been outside for the same amount of time. Furthermore they have both traveled the same distance.

It can be seen that the amount of rain that hits you from the front is based solely on the distance you travel, because if you move slowly you will get barely hit over a long period of time, or if you move quickly you will get hit by a lot more but for a shorter period of time.

The amount you will be hit on your head is also only a function of the amount of time you are outside, because for every drop that misses you because you took a step forward, there is another drop that hit you because you took a step forward.

Because they have both gone the same distance and been in the rain just as long, they are equally wet.

• I was holding out for a more mathematically rigorous proof, but your solution is correct. Essentially, the only variable is the amount of water picked up by the candidates' fronts, which is proportional to their velocity. Hence integrating $cv\left( t\right)$ for some constant $c$ over time yields the total "excess" water. But of course $\int {v\; dt}$ is just distance, hence total excess water is proportional to total distance, which is equal for both candidates. Hence both are equally washed up. – COTO Jul 19 '15 at 7:53
• For a visual proof/demonstration, check out this video – KuCoder Jul 21 '15 at 13:27

If you accept Mythbusters as a relevant source they did an episode with running in the rain versus walking and discovered that if you run in rain rather than walk you will end up drier.

• A +1 from me for thinking outside the box, but I'm looking for an analytical proof with a more basic model. I've included the more rigorous modeling specs in the problem statement. – COTO Jul 17 '15 at 19:45
• +1 from me because I remember that episode! Though it did not (I don't think) include the acceleration aspect. – Duncan Jul 17 '15 at 23:37