Expert: Daniel Mazur - 5/26/2009

Question
When rain (with constant velocity), if you want walk from one place to another place how can you go to bit less with rain drop? With maximum speed , minimum speed or with a constant speed?
Please,Please proof it mathematically. It is not a homework.
Thanks for your answer
Hooman Mirrahimi  

Answer
Hi,

generally the fastest speed possible should minimize the wetness, but there are complications with human body geometry: the cross section from top view is different (smaller) from the cross section of face view.
Let the speed of falling rain be vr, your speed be v, top cross section A and face body cross section B. When you stand still (v=0) for a time interval dt, your cross section will be hit by A*vr*dt. There will be no contribution by cross section B. This is a looser, of course, because you never move our of your spot, so time spent on the way is infinite.

For non-zero speed v both cross sections will get hit. The total amount of rain will be Vtot = B*v*dt + A*vr*dt. In this equation we assume constant A, B and vr. As you are interested in minimizing the total wetness when walking a constant path s = v*dt = const, you rewrite the equation thus

Vtot = B*s + A*s*(vr/v)

This equation gives you the answer: The faster you walk the less wet you get. The reason is that the amount of rain hitting your front cross section B is a constant for a given distance, it does not depend on your speed at all. The wetness of your top cross section is inversely proportional to your walking speed.

Cheers!
Daniel