Rain - Run or Walk?

Date: 08/05/2003 at 23:53:58
From: Simon Wood
Subject: Rain

Is there an optimum speed at which to walk through the rain to 
minimise the number of drops you get hit by?

I'm not very good at calculus and I think it is necessary to work it 
out.

Say you want to travel a distance in the rain. If you stand still 
(velocity=0), you get an infinite amount of rain on the surface of 
your head. As your velocity increases, the number of drops on your 
head decreases while the number of drops on the front of your body 
increases. If you run really fast (velocity = infinite), you get a 
fixed number of drops on the front of your body based on the density 
of the rain.

I believe there is an optimum speed based on the proportion of the 
areas of the front of the body to the head, and the speed of travel 
versus the speed of the rain which minimises the number of drops 
that hit the body, but I can't prove it. I've thought about this for 
about 10 years, but am no closer to an answer. Help, please!

Date: 08/06/2003 at 09:56:20
From: Doctor Edwin
Subject: Re: Rain

Dear Simon,

I'm so glad you asked, because this problem has been niggling at me 
for years and I have never taken the time to think it through.

Here's what it looks like to me. There are, as you pointed out, two 
sources of wetness - the rain that runs into you (on your head and 
shoulders), and the rain that you run into (on your front).

The rain that runs into you is a constant amount per unit time. The 
longer you're out there, the wetter you get. If you're out 30% 
longer, you get 30% wetter.

But the rain that you run into is a constant amount per unit distance 
travelled. I know it seems like you're getting more water if you run 
into it. But if you don't move at all you get zero water on your 
front. If you move very slowly, you get almost no water on your front 
per minute, but you're out there for many minutes. In other words, 
the amount of water on your front depends on entirely on how far you 
go, not on how fast.

Here's another way to look at it. You're going to pass through a 
volume of watery air, and you're going to absorb all the water that's 
in the air that you pass through. The amount of water per unit volume 
doesn't change over time - the number of drops falling out of that 
volume are the same as the number falling in - so it doesn't matter 
when you get to a particular piece of that volume or how fast you're 
going when you get there. Imagine if we could stop gravity for a few 
minutes so the raindrops would just stay where they are. Would it 
matter how fast you went? 

So running faster means you run into just as much rain, but you're 
out there less time, so less rain runs into you. 

Here are some more things to think about: 

 - What if the wind is moving in the same direction you are? If you 
can run at the same speed as the wind, you don't run into any rain. 
But if it's blowing slowly, then the amount that falls on your head 
during your slow walk might be greater than the amount you ran into 
if you went faster than the wind. The best case would be if the wind 
is moving along with you, exactly as fast as you can run. If it's 
going slower than that, then you have to bite the bullet and do the 
calculus problem.

 - What if you're going fast enough that you can lean your body into 
the wind? If you were leaning at exactly the right angle, you'd get 
no rain on your front or back, but you'd take more rain on your head. 
Would you get more wet or less?

 - What if you'd just rather get the water on your head and shoulders 
than down your whole front? 

I just did a Web search and came up with this link:

To stay drier, do you walk or run in rain? If you walk, researchers 
say, you're all wet - Eric Sorensen, Seattle Times

   http://seattletimes.nwsource.com/html/localnews/134370003_rain23m.html

Apparently someone actually did this as a physical experiment.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/