A thin rod with mass $m$ and length $L$ hangs from a frictionless axle through one end. A small ball with mass $m$ is thrown at the bottom of the rod so that it hits the end of the rod with speed $v$ at an angle of $$ below the horizontal as shown below. If the ball sticks to the rod and they swing up to horizontal, how fast is $v?$

$I_r=\frac{1}{3}M{L}^2,I_B=M{r}^2$

A$)\sqrt[\phantom{2}]{3gL}$

B$)\frac{2\sqrt[\phantom{2}]{gL}}{cos()}$

C$)\frac{\sqrt[\phantom{2}]{gL}}{cos()}$

D$)\frac{2\sqrt[\phantom{2}]{gL}}{sin()}$

E$)\frac{4\sqrt[\phantom{2}]{3}gL}{cos()}$