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### 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
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?

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/
```
Associated Topics:
College Physics
High School Physics/Chemistry

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