Problem $4$ Elastic Collision

A rod of length $\backslash (2$ L $\backslash )$ and mass $\backslash (2$ m $\backslash )$ rests on a frictionless table $($the figure is a birds$-$eye view$).$ A billiard ball of mass $\backslash ($ m $\backslash )$ strikes one end of the rod with velocity $\backslash ($ v$\_\{0\}\backslash ).($a$)$ If the ball continues in the same direction after the collision, show that the velocity of the ball after the collision is $\backslash ($ v$=\backslash$frac$\{1\}\{3\}$ v$\_\{0\}\backslash )($note$,$ that objects tend to rotate about their center of mass$).($b$)$ Everything else being the same, find the velocity of the ball after the collision if the opposite end $($bottom in the figure$)$ of the rod was nailed to the floor instead.