Do you get less wet if you run in
the rain?
By Nick Allen
The Beano comic character Billy
Whizz famously runs so fast that he doesn't get wet in the rain. Is
this possible? Inspired by an 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:
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.) ñ 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:
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
Here W is the "total wetness" (the total mass of rainwater on your
body), ñ 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:
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 =
ñAD.
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. Nick continues to teach physics to advanced
students at Valentine's High School, Ilford.
==============================================
Mod: The ñ is not the true symbol
for density. The true symbol is "roll". Too bad, this forum is not
able to show its true symbol. 
