Two balls of equal mass, with equal but opposite velocities, approach each other for a head-on elastic collision. (a) After collision, the balls will (1) move off stuck together, (2) both be at rest, (3) move off in the same direction, or (4) recoil in opposite directions. (b) Prove your answer explicitly.
(a) CONCEPTUAL REASONING. Make a sketch of the situation. Then, looking at the choices, (1) is eliminated because if they stuck together it would be an inelastic collision. If they come to rest after collision, momentum would be conserved (why?), but not kinetic energy, so (2) is not applicable for an elastic collision. If they both moved off in the same direction after collision, (3), the momentum would not be conserved (zero before, nonzero after). The answer must be (4). This is the only option by which momentum and kinetic energy could be conserved. To maintain the zero momentum before collision, the objects would have to recoil in opposite directions with the same speeds as before collision