It's times like this I wish...

... that I had more than high school physics.

I was solving a nifty little optimization problem which occurred to me as a continuous version of a discrete problem that used to come up in an old computer game I used to play long ago. The problem boiled down to finding a best route of travel given two different speeds in different kinds of terrain.

Anyway, after a page of scribbling around, I came up with a formula for a simple case of the problem.

Then I realized that my simple case was (in a modified form) essentially going to be solved by Snell's Law (also called the Law of Sines). And sure enough, my simple formula was Snell's law (but changed about a bit so it wasn't instantly obvious, like having a ratio of cosecants of angles to the normal instead of sines - which is just a matter of inverting both sides...).

This is essentially what I was doing, but less formally and with a lot more faffing about and a few false starts. (It's easy to find on-line if you already know what to look for, eh?)

I felt like a bit of a dope. On the other hand, at least I was definitely on the right track. Since I'd had a headache all day, that was all I could manage in the time available, so I didn't go on to derive the problem I was actually interested in (the shape made by traveling as far as you could in a given time, given a particular boundary between the two regions), but with Snell's Law it should become a somewhat more straightforward calculation for the situations I was playing with (since it tells me "where to head" after striking a smooth boundary, so for the simpler cases it's a matter of computing where you end up given each boundary point).