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 * A * t * v

Checking the units in the above equation we have [mass/(length^3)] * 
[length^2] * [time] * [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*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 * T * t * w * (v / w)    +  D * F * t * w * (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 * T * v  * x / s    +  5 * D * T * 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 * D * T * 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.