Run or Walk in the Rain?

It's an age old question, the most obvious answer is to run, it makes sense that you spend less time in the rain therefore less time getting wet. However, there's always that one person, who likes to be different who suggests it's otherwise. If maths is ever going to be useful its here, its a problem of massive importance. Or at least it is to me.

Okay as with all models we shall make assumptions, first of all, humans are full of squiggly lines and strange shapes so lets just say out test dummy is just a cuboid with the rough dimensions of a human. Secondly rain is all psuedo-random and complicated but I can say with relative surety that in any given metre cubed at any time there is a constant amount of water (law of large numbers). As this is the case, its only a small leap to say that the water is continuous, like a mist of droplets. Also, journeys are all windey and complicated (humans are so awkward) so lets just say its a straight line, as any journey can be split up reasonably into a series of straight lines.

Are you siting comfortably with my assumptions? Then we shall begin. The main body of this problem lies with the fact that however fast you go, you will cut the same amount of water mist stuff. So actually the main body of the water that hits you is independant of speed. However this cuboid human is 3D, and the amount of water that hits his head is directly proportionate to the amount of time he spends in the rain (as it is falling straight down). So as far as this model goes, with rain that drops straight down, youll be slightly dryer if you run. Now when you factor in slanted rain...