You don't really need much math to understand this. Imagine that a person is a perfect cube. The amount of rain hitting the top of this cube is constant-- does not depend on the speed of the cube. The amount of rain hitting the front of the cube is proportional to how fast the cube is moving. From this insight, everything else is algebra.

I was referring to coming with the differential equation first presented.

Understanding it is one thing, but coming up with it independently is different.

I wonder if the multivariate nature of this problem is throwing you off more than the differential equation aspect of it. If you know multivariable calculus you can think about this problem at a higher level of abstraction (vector fields and flux). The integral you see is the output of the thinking at that higher level of abstraction.

I bet with proper background and practice you could set things like this up rather easily. You just need to learn to think about the problems at the proper abstraction level and then develop some intuition by actually doing lots of problems.

Your CS degree required "math", but not much calculus or analysis. If your degree was in, say, mechanical engineering instead (or anything in the family of applied physics), you'd be much more comfortable with the differential equations. But then, you'd probably have a harder time with discrete stuff like statistics, and the number theory behind RSA would completely baffle you.

Yes, basically. You just need more practice.

I think there is a simpler way to picture this. Imagine a volume of water suspended in the air - the height and width is the person's cross section, and the length is distance to be travelled. All of that water is going to get onto you regardless of how fast you travel. So speed is not relevant there. But for the water hitting the top of the cube, it gets wet only with respect to time, and not to speed.

Conclusion: if the rain is falling straight down, it always makes sense to run.