Rainfall on a Moving ObjectDate: 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/ |
Search the Dr. Math Library: |
[Privacy Policy] [Terms of Use]
Ask Dr. Math^{TM}
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/