referred to as The Oregon Rule. Analytical vs. numerical approaches towards resolving collisions Relatively few problems involving collisions can be solved analytically; the remainder require numerical methods. An important problem in simulating collisions is determining whether two objects have in fact collided. This problem is called collision detection. Examples of collisions that can be solved analytically Billiards Collisions play an important role in cue sports. Because the collisions between billiard balls are nearly elastic, and the balls roll on a surface that produces low rolling friction, their behavior is often used to illustrate Newton's laws of motion. After a zero-friction collision of a moving ball with a stationary one of equal mass, the angle between the directions of the two balls is 90 degrees. This is an important fact that professional billiards players take into account, although it assumes the ball is moving without any impact of friction across the table rather than rolling with friction. Consider an elastic collision in two dimensions of any two masses m1 and m2, with respective initial velocities u1 and u2 where u2 = 0, and final velocities V1 and V2. Conservation of momentum gives m1u1 = m1V1 + m2V2. Conservation of energy for an elastic collision gives (1/2)m1|u1|2 = (1/2)m1|V1|2 + (1/2)m2|V2|2. Now consider the case m1 = m2: we obtain u1 = V1 + V2 and |u1|2 = |V1|2 + |V2|2. Taking the dot product of each side of the former equation with itself, |u1|2 = u1•u1 = |V1|2 + |V2|2 + 2V1•V2. Comparing this with the latter equation gives V1•V2 = 0, so they are perpendicular unless V1 is the zero vector (which occurs if and only if the collision is head-on). Perfect inelastic collision In a perfect inelastic collision, i.e., a zero coefficient of restitution, the colliding particles coalesce. It is necessary to consider conservation of momentum: where v is the final velocity, which is hence given by The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy