 Solution

Choose the cylinder's velocity to be $v\hat{\mathbf{x}}$, and the ball's laboratory velocity to be $u(\cos \theta ,\sin \theta )$. By Galilean velocity addition, the ball's velocity relative to the cylinder isFor a suitably centred entry without hitting a wall, the cylinder's axis must be parallel to this relative velocity. Its oriented angle $\alpha$ to the cylinder's direction of travel is thereforeEquivalently, $\tan \alpha =u\sin \theta /(u\cos \theta -v)$, with the quadrant chosen from the two components. For $v/u\ll 1$,Thus the extra tilt relative to the ball's original direction is approximately $(v/u)\sin \theta$. The use of $atan2$ also covers nearly transverse incidence, where taking an unqualified inverse tangent could select the wrong axis direction.