First, though, we must figure out how the ball's velocity changes at each impact. Consider two different reference frames: the ball's motion as seen from the ground (the ground's reference frame) and the ball's motion as seen from the table (the table's reference frame). Begin by considering the simple case where the table is stationary and the two reference frames are identical. As we will show shortly, understanding the stationary case will solve the nonstationary case.
Let 
be the ball's velocity right before the kth impact, and let 
be the
ball's velocity right after the kth impact. The prime notation indicates a
velocity immediately before an impact. If the table is stationary and the
collisions are elastic, then 
: the ball
reverses direction but does not change speed since there is no energy loss. If
the collisions are inelastic and the table is stationary, then the ball's speed
will be reduced after the collision because energy is lost: 
, where 
is the
coefficient of restitution . The constant
is a measure of the energy loss at each impact. If 
,
the system is conservative and the collisions are elastic. The coefficient of
restitution is strictly less than one for
inelastic collisions.