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Problem statement:

A sphere is rolling in rain from point A to point B. The vertical speed of rain is V and horizontal speed of rain is v, as shown in the picture. Angle between horizontal component of rain's velocity and sphere's velocity is $\varphi$. What is the optimal speed of sphere so that it would be as dry as possible?

Problem picture

So here's what I tried to do:
The velocity of rain in respect to the sphere is $\vec{v} = \vec{v_s} - \vec{v_r}; v=\sqrt{v_s^2 + v_r^2} = \sqrt{v_s^2+(\sqrt{V^2+v^2})^2}=\sqrt{v_s^2+V^2+v^2}$. I thought that maybe if I found the derivative of this function in respect to $v_s$ and equated it to 0, that could be the answer. But I don't know how to find the derivative of such function and the solution doesn't seem right to me, though I couldn't come up with anything better.. Any help appreciated!

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  • Your equation is not 100% correct, since it would be correct if each component would be orthogonal to each other. However $v$ and $v_s$ are both horizontal, so the expression for the relative velocity would be: $$\sqrt{\left(v+v_s\right)^2+V^2}$$ – fibonatic Dec 14 '13 at 12:44
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    I remember I have solved it for rectangular cabinet and the answer is that you must run as quickly as possible since it does not matter at which speed you move, the amount of water you'll get from the front is a constant but the water from the top integrates over time. I also do not understand what does fly to the upper right corner, at $v$? – Val Dec 14 '13 at 12:45
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    @Val note: this doesn't generalize to other shapes, however. Particularly, if you have a very thin object, you get the best results when the rain falls parallel to its plane. – John Dvorak Dec 14 '13 at 17:19
  • More on optimization under the rain: http://physics.stackexchange.com/q/19499/2451 and links therein. – Qmechanic Jan 13 '14 at 19:18

1 Answers1

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This is a variation of an old chestnut. The simplest analogy is that of a shower stall: Suppose you need to walk thru a shower (at least at my old high school, they still had a long row of showers sans walls we had to get thru). What strategy would you take to stay as dry as possible? Clearly the answer is to get out of the rain as soon as possible. All that stuff about "rain falling on you" vs "running into raindrops" is just there to confuse the situation.

Carl Witthoft
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  • Hmm... but running as fast as possible does not equate getting out as fast as possible... unless you mean feet first, of course. – John Dvorak Dec 14 '13 at 17:15