Minimize Final Energy in 2D and 3D

It is very obvious that in 2D the largest probability of the energy distribution is at zero energy and results in a simple exponential energy distribution. It is equally obvious that in 3D the probability of having zero energy is zero. For simple translational motion, this results directly from the scattering process discussed here. The object of the present animation is to show why a result of zero energy from collisions in 3D is much more unlikely than in 2D. It is perfectly general, for either 2D or 3D, to have the initial velocity vectors in the (x,y) plane. But then for 3D we must have the initial colliding positions of the 2 particles with at least one z component. To compute the final velocities and energies, we need to have only the unit vector that points along the line between the centers of the particles, not the actual particle positions. For 3D the unit vector will generally have a non-zero z component. If it doesn't, then the particular collision is the same as a 2D one. In 2D it is quite easy to find a set of initial velocities and unit vector that results in zero or near zero energy for one of the particles. And, of course, that same set will give zero energy for the 3D case but only when the z component of the unit vector is zero. So the real reason for not expecting large probability of zero energy in 3D is that there is an infinity of cases where the z component of the unit vector is NOT zero and these almost always prevent the 3D energy from being zero. The learners may adjust the sliders and view the resulting energies (depicted by mumbers and arrows) to convince themselves of the above statements.