# Caught In The Rain II

I was rather proud of my last post about being caught in the
rain. In
that post, I concluded that you were better off running in the rain, but
that the net effect wasn’t incredibly great. However, when I told people
about it, the question I inevitably got asked was: What if the rain
isn’t vertical? That’s what I’d like to look at today, and it turns out
to be a much more challenging question. I’m still going to assume that
the rain is falling at a constant rate. Furthermore, I’m going to assume
that the angle of the rain doesn’t change. With those two assumptions
stated, let me remind you of the definitions we used last time.

Fig. 1 - The rain, and our angle.

**Case 1: Running Against The Rain** This is the easier of the two
cases. After thinking about it for a while, I believe that it is the
same as when the rain is vertical. Let me explain why. If you are moving
with some velocity v, in a time t you will cover a distance x=v*t. Now,
suppose we paused the rain, so it is no longer moving, then moved you a
distance x, turned the rain back on, and had you wait for a time t. And
repeated this over and over until you got to where you were going. This
would result in an *average* velocity equal to v, even though it is not
a smooth motion. However, my claim is that in the limit that t and x go
to zero, this is a productive way of considering our situation. We note
that v=x/t, and in the limit that both x and t go to zero, that is the
*definition* of instantaneous velocity. The recap is, that my ‘pausing
the rain’ scheme of thinking about things is fine, as long as we
consider moving ourselves only very small distances over very short
times. Using this construction, we have an additional amount of rain
absorbed by moving the distance delta x of:

Fig. 2 - How wet you get vs. how fast you run for various wind angles.

Fig. 3 - How wet you get vs. how fast you run for various wind speeds in mph.

**Case 2: Running With The Rain**
This is the potentially harder case. We’ve got two obvious limiting
cases. If you run with the exact velocity of the rain and the rain is
horizontal, you shouldn’t get wet. If the rain is vertical, it should
reduce to the result from my first post. We’ll start with the stationary
case. This should be identical to case 1, if you’re stationary it
doesn’t matter if the rain is blowing on your front or back. That means
that for v=0, we should have

Fig. 4 - Geometry for small delta x.

Note that in front of us there is a rainless area, which we’ll be
advancing into. Consider a delta x less than the length of the base of
that triangle. If we advance that delta x, we’ll carve out a triangle of
rain as indicated, which, by some simple geometry, contains an amount of
rain

Fig. 5 - Geometry for large delta x.

We’ll carve out an amount of rain equal to the indicated triangle plus
the rectangle. From the diagram we see this gives an amount of water

We get two terms. There’s the triangle of rain that moves down and hits
our back, shown above. Hopefully it is apparent that this is the same as
the triangle of rain we carved out with our front, and so will
contribute a volume of water

Fig. 6 - How wet you get vs. how fast you run for various wind speeds in mph.

**Comparison**
I will conclude with a comparison of the two results, to each other and
to the vertical case. First, lets take the appropriate limits.

Fig. 7 - Solid lines are running with the rain, dashed lines are running against the rain.

**Conclusions**
Hopefully this has been an interesting exercise for you. I know it
certainly took me longer to work and write than I initially thought.
While you can’t see it in the post, there were a lot of scribblings and
thinking going on before I came to these conclusions. Most of it went
something like: “No, that can’t be right, it doesn’t have the right
(zero velocity/zero angle) limit!”. I think this concludes all of the
running in the rain that I want to do, but if you have more followup
questions, post them below, and I’ll do my best to answer. Also, I admit
that my analysis may be a bit rough, so if you have other approaches,
let me know. Finally, note that everything I’ve found favors running in
the rain, so get yourself some exercise and stay dry!

## Comments !