The Dynamics

The system consists of a ball which we consider to be a point mass, moving freely without friction on a horizontal 2-Dimensional flat table. The dynamics of the ball is trivial: it experiences no forces and thus no acceleration: it`s velocity is constant,

ie.

dx/dt=V_x,

dy/dt=V_y,

dV_x/dt=0,

dV_y/dt=0.

The interesting part comes from the elastic collisions of the ball with the hard edges of the table. The only twist is that we considered billiard tables which are different from the usual rectangular shape. We looked at

• [1.]a square shaped,
• [2.]a circular shaped and
• [3.]a stadium shaped billiard table,

and plotted out a single trajectory of the ball which rebounded one thousand times off the edges of the table.

First we have to understand how the impacts with the edges of the table alter the un-spinable ball`s velocity. One can think of the ball`s trajectory as a light ray and the inside edge (the cushion) of the table as a mirror. Then for an elastic collision, the angle of incidence equals the angle of reflection. To see how this translated into velocities, let

$$\overrightarrow{V_{i}}=(\overrightarrow{V}_{ix\: },\overrightarrow{V}_{iy})$$

which is the initial velocity of the ball just before it impacts with the ball, then we can decompose the ball`s incoming velocity into components parallel and perpendicular to the wall,

$$\overrightarrow{V}_{i}=\overrightarrow{V}_{i\parallel }+\overrightarrow{V}_{i\perp },$$

where

$$\overrightarrow{V}_{i\perp }=(\overrightarrow{V}_{i}\: .\overrightarrow{n})\overrightarrow{n},$$

$$\overrightarrow{V}_{i\parallel }=\overrightarrow{V}_{i}-\overrightarrow{V}_{i\perp }.$$

Now a mirror-like reflection does not change the component parallel to the mirror but reverses the component perpendicular. That is, only the velocity vector component perpendicular to the tangent (parallel to the normal) is changed. It is of the same magnitude of of opposite sign since the collision is elastic. So the expression for the velocity of the ball immediately after impact is

$$\overrightarrow{V}_{f}\, =\overrightarrow{V}_{f\parallel }+\overrightarrow{V}_{f\perp }$$

where

$$V_{f\parallel }=V_{i\parallel },$$

$$\overrightarrow{V}_{f\perp }=-\overrightarrow{V}_{i\perp }.$$

This type of reflection is sometimes called specular.