Can you always win in a billiard game?

Abstract

While substantial efforts have already been spent to understand the dynamics of hard sphere collisions in a plane, finding the exact positions of the interacting spheres at any given time has proven to be a non-trivial problem that has only been addressed in detail recently. The difficulty of the problem arises because the unknown quantities, which are the final velocities of two colliding spheres, have four degrees of freedom (V_1fx, V_1fy, V_2fx, V_2fy), while the use of conservation principles of momentum {S₁= ∑P_x = ∑m_iV_ix = constant; S₂ = ∑P_y = ∑m_iV_iy= constant} and energy {S₃ = ∑m_iV_i²/2= constant} could only provide for three independent equations.
In this study we demonstrate that hard-sphere interaction can be accurately modeled by introducing a geometrical constraint that determines the direction of colliding spheres. In particular, we investigate whether a billiard game can be won always.