You must show that the initial vertical acceleration of the end of the rod (book or plank or whatever is used) based on it's angular acceleration is higher than g.

You must use the formula for angular acceleration for a rod based on the moment of inertia and the applied torque on the rod. The torque is at right angles to the rod. The initial vertical acceleration will depend on the geometry of the situation, in this case at what angle the plank is placed. The higher the angle, the lower the initial vertical acceleration. At an angle of about 35.3 degrees is is just about equal to g. It needs to be higher than g for the ball to depart from the plank.

The rest is just geometry, the cup needs to be in the right position to be under the ball when the ball lands.