> OK, let me see if I have something straight here. Does this mean that there is a discontinuity at the speed at which the rain falls directly on you? Obviously, in the first case, there is an asymptote at this point as the amount of rain striking you approaches infinity as your velocity decreases, but in the second case, the best action is to move at the speed which causes the rain to fall vertically relative to you. Since in the first case, you want to go as fast as possible, clearly there is an inverse relationship between your speed and the amount of rain hitting you. In the second case, though, if the system were continuous, then obviously, the highest speed possible would stil be desirable. The fact that this is NOT your final conclusion leads me to believe that V = R Sin (A) must be a local minimum which is lower than the asymptotic minimum which would be reached as your speed approaches infinity.
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> What made this occur to me is that from the frame of reference of the walker, there is no difference between the optimum speed in the second case and standing still in the first case, since the rain falls straight down in either case. The only real difference is that the walker is making progress towards his goal in the second case, which makes this solution feasible. This seems to lead me to the conlusion of a discontinuous system (which seems reasonable based on your equation).
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> So, is my reasoning sound? Did the way I said it make sense?
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> Don "I would figure this out much more quickly with a visible plot to work from..." Monkey

I'm not sure what you mean by a discontinuous system, but here's what happens. You find the function that gives the total amount of water hitting you as a function f(A,V) as a function of V with fixed angle A, and then try varying the parameter A. If I set it up so that negative values of A represented the rain going in the same direction as you, then there is a discontinuity of sorts when you hit negative values of A. For negative A, f(A,V) has a corner at V = R Sin(A). It's still continuous, just not differentiable at that one point, so it's the derivative that has a discontinuity. And it IS a global minimum as well as a local one, since to the right of this point f(A,V) is increasing as a function of V (though it levels off to a horizontal asymptote).

You're right to say that for the walker, standing still relative to the horizontal minimizes the raindrops hitting you per second. However, if the rain were coming in from ahead, you'd have to move backward for this to happen and you'd never reach your goal. That's the main difference.