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With your assumption the distance is the same (if I understand you correctly), then my answer is you will be dryer when running because you will get inside more quickly than if you walked.

 

Assuming an equal time in the rain, running will get you wetter because instead of the rain only falling straight down on your head and shoulders, the raindrops will contact the front part of your body (chest, stomach, etc.) due to the forward motion in addition to the raindrops falling on your head and shoulders.

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Also, running gets more of you wet, rather than mostly your head and shoulders. Running against the wind will get you wet even quicker, but running away from the wind will keep you drier. When you run you don't have as fine control over what angle to hold your body at. I don't think that this would be a simple physics problem without some simplifying assumptions.

 

Biking in the rain gets you soaked very quickly, as the wheels of the bike will pick up water off the ground and throw it at you, in addition to the extra rain you hit due to your speed.

 

We've all been caught in the rain, so this really isn't a dumb question.

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I know the mythbusters tackled this (twice, I think). I believe the idealized solution is that they are equal, up to the point where your clothes are saturated, but several of the assumptions aren't good ones in reality. Once you reach saturation, the differences don't matter, because you can't get any wetter.

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The amount of water that hits you will be proportional to the time you spend in the rain, the size of the cross-section of your body perpendicular to the relative velocity of the raindrops, and the magnitude of that velocity.

 

The first factor is obviously inversely proportional to your velocity relative to your destination. Hence running will decrease it.

 

The second factor is the tricky one. If the rain is falling straight down, and you are standing still, your cross section is just the top of your head, your shoulders, your beer gut, whatever. If you're moving forward, this is equivalent to you standing still and the rain slanting towards you. Your cross section is larger, because humans are tall and skinny. It doesn't increase linearly, though, and approaches only your vertical sillhouette as your velocity approaches infinity.

 

And the final factor, the magnitude of the velocity, is just the velocity of the falling raindrops minus your velocity (keeping in mind that velocity is a vector, of course). Since it's perpendicular, it can be pythagorean: (rain falling speed)^2+(your speed)^2 = (relative speed)^2

 

To simplify, let's just say your body shape is a perfect sphere, thus having the same cross-section from any angle, so we can remove that messy term. So the wetness = time in rain * magnitude of incoming velocity.

 

So let's say the rain falls at a speed of 1, and you walk at a speed of 1, and it takes you 1 time unit to get to shelter. Wetness = 1 time * sqrt(2) = ~1.41

 

When you double your speed to 2, your wetness = 0.5 time * sqrt(5) = ~1.12

 

It's less, because as you increase speed, time decreases linearly, but rain hitting you per unit time increases less than linearly. So run.

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