The Jumping Balls

Ups, that wasn't quite right.

Correct is this:

Both balls jump up to their original height after bouncing off of the ground.

In the moment the ball bounces off, the direction of its velocity is reversed. This also means that its momentum is reversed, as well. This change of momentum interacts with the ground (or the earth). However, the earth's mass is so huge that „in reality“ it doesn't change its velocity.
This means: the momentum is transferred between the ball and the ground; not energy. The ball keeps (nearly) all its energy.

In the next experiment you might get even closer to the solution of the problem.