The impulse delivered by a force is $$\mathbf{J}=\int_{t_1}^{t_2}\mathbf{F}(t)\,dt,$$ and it equals the change in momentum $\Delta\mathbf{p}$. This is the correct framework for short-duration impacts where force is large but localized in time.

In isolated systems, momentum is conserved in all collisions, while kinetic energy is conserved only for perfectly elastic collisions. For inelastic impacts, the coefficient of restitution parameterizes the relative speed after collision along the line of impact.

For central scattering, the impact parameter $b$ and scattering angle $\theta$ determine the differential cross section. A key result is $$\frac{d\sigma}{d\Omega}=\frac{b}{\sin\theta}\left|\frac{db}{d\theta}\right|.$$ This formula connects trajectory geometry to measurable angular distributions.