Action Principle

n_1 \sin{\theta_1} = n_2 \sin{\theta_2}

n = \frac{c}{v}

\frac{c}{v_1} \sin{\theta_1} = \frac{c}{v_2} \sin{\theta_2}

\frac{\sin{\theta_1}}{v_1} = \frac{\sin{\theta_2}}{v_2}

v \propto \sqrt{2gy}

\frac{\sin{\theta_1}}{\sqrt{y_1}} = \frac{\sin{\theta_2}}{\sqrt{y_2}}

\frac{\sin{\theta_1}}{\sqrt{y_1}} = \frac{\sin{\theta_2}}{\sqrt{y_2}}

\frac{\sin{\theta_1}}{f(y_1)} = \frac{\sin{\theta_2}}{f(y_2)}