When the table is in motion, the ball's velocity immediately after an impact will have an additional term due to the kick from the table. To calculate the change in the ball's velocity, imagine the motion of the ball from the table's perspective. The key observation is that in the table's reference frame the table is always stationary. The ball, however, appears to have an additional velocity which is equal to the opposite of the table's velocity in the ground's reference frame. Therefore, to calculate the ball's change in velocity we can calculate the change in velocity in the table's reference frame and then add the table's velocity to get the ball's velocity in the ground's reference frame. In Figure 1.5 we show the motion of the ball and the table in both the ground's and the table's reference frames.
Figure 1.5: Motion of the ball in the reference frame of the ground (a) and the
table (b).
Let 
be the table's velocity in the ground's
reference frame. Further, let 
and 
be the velocity in the
table's reference frame immediately before and after the kth impact,
respectively. The bar denotes measurements
in
the table's reference frame; the unbarred coordinates are
measurements in the ground's
reference frame. Then, in the table's reference frame,
since the table is always stationary. To find the ball's velocity in the ground's reference frame we must add the table's velocity to the ball's apparent velocity,
or equivalently,
Therefore, in the ground's reference frame, equation (1.4) becomes
when it is rewritten using equation (1.5).
Rewriting equation ({1.6) gives the velocity 
after the kth
impact as
This last equation is known as
the impact relation .
It says
the kick from the table contributes 
to the
ball's velocity.
