Types of collisions

1.

Inelastic collision: $\vec{p}$ conserved, but not KE. Example - rubber ball on a hard surface (ball deforms $\rightarrow$ internal elastic PE ).

2.

Perfectly inelastic collision: Two objects stick together $\rightarrow$ v_1f = v_2f $\equiv$ v_f . $\vec{p}$ conserved, but not KE. Conservation of $\vec{p}$ gives,

m₁v_1i + m₂v_2i = (m₁ + m₂)v_f. (6)

Example - two lumps of clay.

3.

Elastic collision: $\vec{p}$ and KE are conserved. Example - two billiard balls (no deformations). We have,

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f, (7)

$${\textstyle\frac{1}{2}} {\textstyle\frac{1}{2}} {\textstyle\frac{1}{2}} {\textstyle\frac{1}{2}}$$ = (8)

By combining these two equations we obtain a third (dependent) equation that tells us that the relative velocity before collision is the negative of the relative velocity after collision:

v_1i - v_2i = - (v_1f - v_2f). (9)