Elastic collisions in two dimensions: A billiard ball traveling at 3.00 m/s collides perfectly elastically with an identical billiard ball initially at rest on the level table. The initially moving billiard ball deflects 30.0° from its original direction. What is the speed of the initially stationary billiard ball after the collision?

Question

Quizplus · Accepted Answer

Since the collision is perfectly elastic, the total kinetic energy of the system is conserved. Let $m$ be the mass of the billiard ball. Before the collision, the initial kinetic energy is $(1/2)mv_i^2$, where $v_i$ is the initial speed of the moving billiard ball. After the collision, the moving billiard ball will be deflected to a velocity vector that makes an angle of $30.0^\circ$ with its initial velocity vector. Let $v_f$ be the speed of both balls after the collision. Therefore, the final kinetic energy is $(1/2)mv_f^2+ (1/2)mv_f^2=mv_f^2$ (since both balls have the same mass). Equating the initial and final kinetic energies, we get:

$$(1/2)mv_i^2=mv_f^2$$
$$v_f=\sqrt{(1/2)v_i^2}=\frac{v_i}{\sqrt{2}}$$

The initially stationary billiard ball will have a speed equal to $v_f$. Therefore, the answer is (C) 1.50 m/s.

Elastic Collisions in Two Dimensions: a Billiard Ball Traveling at 3.00