Binary Candies

The other day, a few of my running friends were talking about running in the rain and were worried about the oncoming storm. I jokingly told them that running faster would get them less wet. Now of course there was a Mythbusters episode that went over this, but in that episode, they only experimented and went over the statistical data, to show that running faster makes you less wet, and given their experiment and sample size it is really hard to say the results were conclusive. While statistics is a very reliable method of proving things, often times it is hard to see why things occur, and it only shows that things do occur. Instead I decided to write up a mathematical proof that shows exactly why it is better to run rather walk in the rain.

Now this proof does make some assumptions.
The person who is running is shaped like a Cuboid with each face a rectangle.
The rain drops are all spread uniformly through the entire area.
Each drop of rain hit by the person adds the same amount of water to the person.
And the rain falls at a uniform rate through the entire area.

Given that the the rain is uniformly distributed, we are going to treat the drops of water that the person hits and the area of space that they occupy as the same thing.

Let’s say that…
x = the width of the person
y = the depth of the person
z = the height of the person
t = time passed
w = distance traveled
v = velocity

And let’s split the problem into 2 parts.

Drops of water that the person gets hit from above…
How wet the person gets is dependent upon the number of drops hit, which is obviously proportional to the top surface area of the person, and how long the person was exposed to the water.

x * y * t

In this case, obviously the longer the person stays out in the rain, the wetter they will become. Idealistically the person will be in the rain 0 sec, or is super skinny square shaped such that they would get hit by 0 drops of water.

Drops of water that the person gets hit by from the front…
With each unit of movement the amount of area the person goes through from the front is proportional to their front surface area.
While the rain is coming down so the space that collides with the front of the person changes shape but the space is in the shape of a Cuboid with 2 sides of a parallelogram and the other 4 of rectangles.
To get the volume of this area, we simply take the front surface area of the person and multiply it by the distance traveled.

x * z * w

Or alternatively we could take do this as a function of speed(v) in which case we get

x * z * v

which indicates the faster we go the more we get hit by the rain.
It is true that the faster we go the more distance we cover and therefore the more we do get hit by rain, but we are not out in the rain for a constant period of time but a constant distance ran.
Velocity (v) may be variable but so is time (t). Distance (w) is constant in our case because we need to get some place and because v * t = w the entire area we go through is also constant.

Hence then the total amount of rain we get hit by is

x * y * t + x * z * w

All variables constant besides t, which indicates the longer you are in the rain, the more wet you become.

Alternatively, we can do this with a bit of calculus instead.
If you know calculus, and you followed my logic above, you should be able to figure out how I came to the following equation…

x * y * dt  + z * x * v * dt

Tags: Geek, Math, Nerd, Running