Take the Rutherford scattering, as for example in this picture:
What is the easiest way to show that the impact parameter "b" (see picture) is the same for the ingoing and outgoing trajectories? Are there different arguments to show this?
A summary of the answer by Mike (see below for more details):
Since the scattering is elastic, the magnitude of the momentum does not change: $$ |\vec p_1|=|\vec p_2|, $$ and because the deflecting force is a central force, the angular momentum is conserved: $$ L_1=L_2. $$ From $L_1=|\vec p_1|b_1$, and $L_2=|\vec p_2|b_2$ we see immediately that $b_1=b_2$.