Light is in a hurry! 

It was fun to derive the law of refraction using a simple fact about light: it tries to move from one point to the other as fast as it can (time-wise).

I can write the details if someone would like, but here is what I did.

The first medium can be thought of in plane {y > 0} and second one in the plane {y < 0}. The light source is at (0,1) and the light is trying to get to (1, -1). Postulate that the light enters the second medium at (x,0) and try to minimize the resulting time taken.

We will find an equation like:

2x / (v_1 * sqrt(1 + x^2)) = 2 * (1-x) / (v_2 * sqrt(1 + (1-x)^2))

You can verify that one of the roots of this equation satisfies

sin \theta_1 / v_1 = sin \theta_2 / v_2.

Thanks for Christelle for helping me work out the Math! 🙂