Question

Two students stand at rest, facing each other on frictionless skates. They then start tossing a heavy ball back and forth between them. Describe their subsequent motion?

Answer 1

Short answer: Students "A" and "B" will move away from each other, and a point between them ("C"), moving at a higher combined velocity after each toss of the ball, and accelerating away from the other during the pushing of the ball toward the other student.
When the student "A" (initially holding the ball) tosses the ball to student "B", A will have to apply a force to move the ball from A to B. That force will also push A away from B and also a fixed point "C" on the ground between them. Because each student is assumed to be more massive than the ball, A will initially move away from B (and C) with an acceleration that is smaller than that of the ball during the push.
The ball (now having kinetic energy) will be caught by student B. In the process of B slowing the ball (horizontally) to a stop, the force required to stop the ball will accelerate B from B's initial position.
As the students continue to toss the ball, back and forth, they will move away from each other at a higher velocity. As they move farther apart, they will need to toss the ball with more force to cover a ever greater distance. This will cause a greater acceleration with each toss, resulting in a greater change in velocity.
If one plotted a curve showing the change in combined velocity over time, the slope of the velocity would become steeper, looking somewhat like a parabola.