A hard ball of mass M subjected to the gravitational force Mg bounces off of the floor at z=0. The potential energy experienced by the ball is sketched in the figure below. Note that the ball cannot go through the floor to regions where z<0. (See Figure 3.)
Due to the uncertainty principle, even in its minimum energy state,
the ball can never be completely at rest at z=0. Instead, the ball
always has some characteristic non-zero kinetic energy T>0 and
height 
above the ground. Your task is to estimate the
minimum average total energy of the system E and the
minimum characteristic height 
.
