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PART 1: Is it Worth Running in the Rain?

May 2000

It seems that as soon as rain starts to fall, people speed up their pace. There could be many reasons for this but one is the expectation that by running - or at least walking faster - they will arrive drier at their destination. The physics needed to analyze this presumption covers many topics from your introductory course: kinematics, relative velocity, terminal velocity, flux and vector components to name a few.

TERMINAL VELOCITY
As an object falls through air, it starts to experience drag forces. The higher its speed, the more resisting drag. It will eventually reach a "terminal velocity" where the drag force equals its weight, hence there is zero net force and it stops accelerating. The value of the terminal velocity depends on the size, shape and mass of the object.

For raindrops at sea level, the range of terminal velocities is 2 m/s (0.5 mm drop - drizzle) up to 9 metres per second (5 mm drop - downpour). (See the question about rainfall in The Last Word).

Kinematics
v=d/t or t = d/v ... the faster you run (speed v) to get to your destination (distance d), the less time (t) it will take. It might seem obvious if you get to your destination in half the time, only half the rain will have hit you - but by changing your speed you change your relative velocity compared to the rain.

Relative Velocity
Suppose the rain is falling straight down. If you are standing still you will only get wet on the top of your head/shoulders and anything else unsheltered and pointing straight up. However, there is a certain amount of water between you and your destination, and by moving you are going to start colliding with it... you are changing the relative velocity between you and the rain. Now instead of seeing the rain coming straight down, you see it as blowing into you, getting your face/legs/etc wet as well. You now have to deal with components of the rain's velocity.

Vector Components
You now treat the rain as having two components one vertical and one horizontal. The vertical component is independent of your running speed - so this part of the answer truly does predict that by doubling your speed you halve your "wetness". However, the horizontal is the difference between the horizontal speed the rain has, and your running speed. It will be easiest, however, to deal not with components of the speed, but components of the flux.

Flux
Flux is used in nearly every topic area covered in introductory physics, but is seldom explained by itself. Flux is the amount per unit area of something against a surface. In our case, the flux will be the (number of raindrops/time) per unit area. So suppose you watch a 1 m2 area of the ground, and 50 drops hit in 10 seconds. The flux is 50 drops/10 seconds per square meter, or 5 drops/(s m2). So estimate your vertical target area (top area of head and shoulders etc), and multiply by 5 to get the number of drops hitting per second. This number is independent of your speed.

The components of the flux vary in the same proportion as the speeds. So, if the rain is falling vertically at 7 m/s with a wind in your face of 2 m/s and you are running at 5 m/s your relative velocity horizontally is 7 m/s - the rain appears to be falling at an angle of 45o. So the horizontal flux is (vertical flux)*tan(45). Multiply this by the area presented by your face/body/legs to get the rate that droplets hit your front.

Putting it all together
If you add the rate the droplets hit your top and front areas, you now have droplets/second - so multiply this by the amount of time you will be in the rain - d/v. A few comparisons might lead you to worry less about how fast you run, but there are a lot of cases to consider: wind in your face, wind from behind, wind from the side (we haven't covered this component, but if you understand the above argument you should be able to work it out), drizzle vs downpour, and the various possible running speeds. For perspective however, remember that the "World's Fastest Man" runs 100 m in approximately 10 s - so don't think you're going to go faster than 10 m/s.

Running in the Rain Calculator
In June we'll be presenting a javascript calculator to perform the above calculations for you.

See more at http://seattletimes.nwsource.com/html/localnews/134370003_rain23m.html

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