Example Of Billiard Ball Collision
In a game of billiards, a player wishes to sink a target ball in the corner pocket, as shown below.  If the angle q₁ to the corner pocket is 45^o, at what angle is the cue ball deflected?  Assume that friction and rotational motion are unimportant and that the collision is elastic.

Solution:

Because the target ball is initially at rest, conservation of energy gives:

But all balls have the same mass, thus m₁ = m₂, so that:

(1)     v_1i² = v_1f² + v_2f²

Applying conservation of momentum to the two-dimensional collision gives:

(2)     v_1i = v_1f + v_2f

Note that because m1 = m2, the masses also cancel.  if we square both sides and use the definition of the dot product of two vectors, we get:

Because the angle between v_1f and v_2f is q₂ + 45^o, v_1fv_2f = v_1fv_2fcos(q₂+45^o), and so

(3)     v_1i² = v_1f² + v_2f² + 2v_1fv_2fcos(q₂+45^o)

Subtracting (1) from (3) gives

0 = 2v_1fv_2fcos(q₂+45^o)

0=cos(q₂+45^o)

q₂+45^o = 90^o or = 45^o

This result shows that whenever two equal masses undergo a glancing elastic collision and one of them is initially at rest, they move at right angles to each other after the collision.