An Unconventional Game of Billiard

You attempt a trick shot by striking the cue ball below its center of mass, aiming away from the striped ball. Your hope is to give the cue ball enough backspin so that it will return and knock the striped ball into the corner pocket.

How can you guarantee that the cue ball will always return to hit the striped ball into the corner?

Details and Assumptions:

• The cue ball has mass \(m\) and radius \(r,\) the initial velocity you give it is \(v_0,\) and the backspin you give it is \(\omega_0.\)
• Treat the cue ball as a perfect solid sphere.
• The collision between the cue stick and cue ball is perfectly elastic.
• You hit the ball such that it is always in contact with the ground.