[wordup] Do you get less wet if you run in the rain?

[Adam Shand]

Interesting, I obsessed about this as a child and finally reached the  
conclusion that the simple answer (not accounting for wind, human  
body shape etc) was that you got progressively less wet the faster  
you ran, until you started running faster then the rain was falling;  
at which point you started to get wetter the faster you ran.

Adam.

From: http://news.bbc.co.uk/1/hi/magazine/4562132.stm

Do you get less wet if you run in the rain?
By Nick Allen
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The Beano comic character Billy Whizz famously runs so fast that he  
doesn't get wet in the rain. Is this possible? Inspired by BBC Online  
Magazine's occasional Formula Won feature, which highlights equations  
of dubious value, here's a brief analysis of the problem.

FORMULA BIT
The formal solution looks something like this:
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Which looks a bit (Dennis the) menacing. dW/dt is the rate you're  
getting wet (mass of rain per time incident on your body.) ? is the  
density of the rain shower (mass of water in unit volume of  
atmosphere.) V is the velocity of the rain relative to you, and dA  
represents a little bit of your body surface. The large "S" shape  
tells you to add together all the rain falling on these little bits  
of body surface to calculate the total amount of wetness per time.  
(Picky note: the summation should only be done over body surfaces  
facing the rain, or the equation will accidentally calculate  
"negative" rain that it thinks has passed right through your body.)

The relative velocity of the rain depends on the rain's velocity, and  
your own velocity. This is where we can introduce the possibility of  
someone running around in the rain. The relative rain velocity, V, is  
equal to the true velocity of the rain minus the velocity of your  
body. We can now put these in the above equation and write:
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Where VP is the velocity of the person and VR the velocity of the  
rain. (They're not the wrong way round, because we dropped the minus  
sign.)

SO WHAT?
Precisely! The problem with a solution like this, is that although it  
is designed to be exactly correct, it is far too complicated to be of  
much use - because it can't easily be calculated.

For a start, the shape of a human body is too complex, and all parts  
of it are in different states of motion when running. To get some  
answers the formal solution must be simplified by making some  
assumptions and approximations. Physicists do this all the time - it  
is called "cheating".

AN APPROXIMATE SOLUTION
This is where the fun starts. To get some idea of how running around  
in the rain affects wetness, we'll need to make some fairly  
significant simplifications.

We will assume that the rain is falling vertically and also that the  
person is running horizontally. To get around the problem of our  
complex body shape, we'll imagine our person as a rectangular block -  
like a house brick standing on end. The smaller top surface of this  
"brick" is of area a and represents all our own top surfaces (head  
and shoulders.) The larger front surface of this brick is of area A  
and represents our front surfaces (chest, stomach, front of arms,  
front of legs etc.) This approximation won't give us the complete  
truth - but it might provide some insight into what is going on.

This enables us to produce our first "total wetness" equation. It can  
be derived from the formal solution above, or worked out by other  
reasoning. Anyway, here goes:

THE (SIMPLIFIED) TOTAL WETNESS EQUATION

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Here W is the "total wetness" (the total mass of rainwater on your  
body), ? is the rain shower density as before, a our top surface area  
and A our front surface area. VR and VP are the velocities of the  
rain and person respectively and t is the time spent out in the rain.

Looking at the equation, it's clear that there is little we can do  
about the rain velocity, rain density and the size of our bodies  
(except by dieting.) The only quantities we can directly control in  
the total wetness equation are t (the time spent in the rain) and VP  
(how fast we're running.)

The equation tells us quite clearly that we get most wet if we:

1) stay out in the rain for a long time (no surprise there)
and / or
2) run very fast

So running fast actually makes us wetter according to this analysis -  
the reason being that you are moving your front surface through the  
"rain field", scooping up water as you go.

By the way, should you ever want to get really wet, the equation  
suggests you should stay out in the rain for a long time whilst  
running around like a maniac.

There's more to it than this though. Although running fast looks like  
a bad idea, what if we are running towards shelter - surely by  
running we will minimise the time spent in the rain? This is a fair  
point, and makes the first equation look incorrect - but in fact it  
is fine.

This is because the equation "knows" nothing about the possibility of  
shelter. It simply tells us that if you're in the rain, the best  
thing to do is stand still. However, we can introduce the idea of  
shelter into it to get some further advice.

Let's assume that when it starts to rain, you identify the nearest  
shelter and run towards it. If the distance to the shelter is D, then  
the time spent in the rain (t in the above equation) will be D/VP.

If we insert this into the "total wetness" equation to replace t, we  
get the "modified simplified total wetness equation" which now  
includes the distance to the shelter D:
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So here we have it - more mathematical advice to avoid getting wet.  
Because we divide by VP in this equation, maximising our velocity now  
emerges as a good idea, assuming there is a shelter available.

SO SHOULD I RUN IN THE RAIN OR NOT?
When it starts to rain, first identify the nearest shelter, and then  
run to it as quickly as you can.

This is remarkable, because that is precisely what most people do!  
The power of mathematics has finally given us the reassurance that,  
when we run for that bus shelter, store canopy or random shop (and  
start pretending to browse), we are getting it exactly right!

Bad news for Billy Whizz though - the equation shows that you get wet  
no matter how fast you run, with a minimum value of W = ?AD.

PS: If the rain is falling at an angle it is possible to decrease  
your total wetness by running in the correct direction. Unfortunately  
this may not coincide with the nearest shelter direction. If you are  
worried about this, we may be able to deal with it in a further  
instalment.

PPS: Alternatively, ignore the maths and get an umbrella.

Nick Allen is a Master of Science in astrophysics and a fellow of the  
Royal Astronomical Society. An entrepreneur and inventor, he was also  
co-developer of MouseCage, a disability software www.mousecage.org.  
Nick continues to teach physics to advanced students at Valentine's  
High School, Ilford.