The $30$ lb block $B$ lies on a smooth horizontal floor and is connected to a $20$ lb block $C$ through an inextensible

cable as shown. The system is initially at rest, the $k=3ki\frac{p}{f}t$ spring attached to $B$ is compressed $\frac{1}{3}$ in$.$

Assuming the cable remains taut for the subsequent motion, determine

$($a$)$ Velocity of collar $B$ after block $C$ has moved $\frac{1}{3}in(darr)$

$($b$)$ Maximum velocity of block $B$

$($c$)$ Maximum extension of the spring