Rainfall on a Moving Object

Date: 09/10/2002 at 12:13:24
From: Susan Christy
Subject: Rainfall on a moving object

I would like to figure out the relation between a moving object (a 
person) and the amount of rain that will hit him given a fixed 
distance. If I'm running and you're walking 1 mile in a rainstorm, who 
will get wetter?

Date: 09/10/2002 at 13:45:21
From: Doctor Rick
Subject: Re: Rainfall on a moving object

Hi, Susan.

Here are some things to consider.

The rain is coming down at some rate. How do we measure this rate? We 
could measure the volume of water that comes down per second, but this 
depends on the ground area over which you are measuring. If you double 
the area, you'll find that twice as many cubic inches of water hit 
that area per second. The correct measure, then, is something like 
cubic inches per square inch of surface area per second. (These units 
reduce to inches per minute; that's why the total rainfall during a 
storm is measured in inches. It's really cubic inches per square 
inch.)

Suppose we could know how big the raindrops are, and we could take a 
snapshot and count the raindrops in 1000 cubic inches of the air at 
one moment. Then, to find the rate of rainfall, we could multiply the 
amount of water per cubic inch of air by the SPEED at which the 
raindrops fall, in inches per second. If the amount of water is 
measured in cubic inches, we'll have the rate in inches per second.

The rain is also coming down at a certain angle. You could simplify 
the problem if you require that there be no wind, so the rain comes 
straight down. But in doing the calculations for the moving object 
(person), you'll find that it helps to think about rain coming down at 
other angles. That's because we can apply "Newtonian relativity" - 
which isn't as difficult to understand as Einstein's relativity. If an 
object is moving at velocity v with no wind, it's the same as if the 
object is stationary and there is a wind of velocity -v (that is, just 
as fast in the opposite direction).

If rain isn't falling vertically, this doesn't affect the rate at 
which it hits the ground, as long as we use the VERTICAL COMPONENT of 
the rain in calculating the rate. The horizontal component only causes 
the rain to fall at a different place; it doesn't affect WHEN a 
raindrop hits the ground.

Now back to the measurement of the rainfall. I referred to a certain 
area of ground -- the area of a horizontal flat surface. When you 
consider the rain hitting the object (person), you'll need to consider 
surfaces that aren't horizontal. Just as in the case of non-vertical 
rainfall hitting a horizontal surface, what counts is the COMPONENT of 
the rain's velocity PERPENDICULAR to the surface.

Putting it all together, here are the factors you'll need to consider 
in calculating the amount of water that hits a moving object. First, 
consider the object as a bunch of flat surfaces at different angles. 

Find the angle at which the rain would be falling, if the object were 
stationary and there was a wind as fast as the object is really 
moving. (Note: This will require knowing the vertical speed of 
raindrops; I'm sure this information is out there somewhere on the 
Internet.) 

Then for each of the object's surfaces, find the component of the 
rain's velocity perpendicular to the surface, and use this to 
calculate the rate at which water hits the surface.

Add up the rates for each surface, and you'll have the rate at which 
the object gets wet. Finally, multiply by the TIME it takes for the 
object to go the specified distance, and you'll have how wet it gets 
in the course of the trip.

One final hint. You don't need a realistic division of the object into 
surfaces; for a person, that would be really tough. All you really 
need is one horizontal surface representing the area of the object's 
shadow when the sun is directly overhead, and a vertical surface 
representing the area of the object's shadow when a light is directly 
in front of it.

See what you can do with this. I'd like to hear what you come up with. 
Once you get the concepts involved, the math isn't all that 
complicated. Don't worry about coming up with estimates for the three 
numbers - rainfall velocity and the two shadow areas. You can get a 
formula first and worry about the numbers later, if you find you need 
them.

For discussions of who gets wetter in a rainstorm, a runner or a 
walker, see:

   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

   Do you get wetter if you run or walk in the rain?
   http://www.physlink.com/Education/AskExperts/ae212.cfm 

   Which will keep you drier, running through the rain or walking?
   http://www.straightdope.com/classics/a3_395.html 

- Doctor Rick and Sarah, The Math Forum
  http://mathforum.org/dr.math/