Physics Question #100

A. D., a 17 year old female from the Internet asks on December 3, 1999,

I am interested in the age-old debate on whether a person gets wetter by walking in the rain or by running. I would like to compare the results of various studies on the subject and read how they arrived at their results.

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The answer

Barry Shell answered on December 3, 1999

The Mad Scientists Network is a very good source for this sort of thing: It states: Re: Does a person get wetter if they walk or if they run through rain.

Assume the rain is in the form of a homogeneous flow of a liquid with volume density D, with units of mass per volume. Assume also that the flow of rain has reached terminal velocity v, straight down.. So, in an amount of time t the mass of rain, R, crossing an area A is R = D x A x t x v Checking the units in the above equation we have [mass/(length^3)] x [length^2] x [time] x [length/time] which gives, properly, mass. Now assume that the person walks with velocity s, which we will assume to be perpendicular to the flow of rain. The person will experience the rain at a velocity w which is the vector sum of v and s', where s' = -s. The magnitude of w is w = sqrt( v*v + s x s). Let's let the vertical, or Top, cross-sectional area of a person be T, and the frontal cross-sectional area be F. Normally F will be greater than T by about a factor of 4 or 5, more or less. The amount of rain collected by each of T and F will be proportional to R times the cosine of the angle which the vector w makes with the normals to these two surfaces. When the person's speed, s, is zero w makes an angle of zero with T and 90 degrees with F, so the cosine term will be 1.0 for T and 0.0 for F. When the person's speed is infinite w makes an angle of 90 degrees with T and zero degrees with F, so the cosine terms will be 0.0 and 1.0, respectively. In general for T the cosine term is v/w and for F the cosine term is s/w. The total rain collected, RC, will then be RC = D x T x t x w x (v / w) + D x F x t x w x (s / w). Now, the time t is the time it takes for the person to cross from start to finish, so if the distance to be crossed is X we get t = x / s Putting this all together, and remembering that F = 5 * T, we get RC = D x T x v x x / s + 5 x D x T x X Now, consider the extremes of speed s. If s is zero we find that the rain collected, RC, is equal to infinity!! So don't stand in the rain! If s is infinite we find that RC equals 5 x D x T x X. In between these extremes there are no maxima or minima, and the extremum at s = infinity is a minimum. So, this is why we run like mad to get out of the rain. It is our experience that we get less wet the faster we run, and this analysis bears out our experience. We have just shown that the faster we run the drier we stay.

[Posted By: John Link, Physics Date: Tue Sep 30 11:45:03 1997 Area of science: Physics ID: 873909296.]

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