A block A of mass m is in equilibrium after being suspended from the ceiling with the help of a spring of…

Let the velocity of the combined system just after the collision is $v\text{'}$, then using conservation of linear momentum we get

$mv=2mv\prime$,

$\Rightarrow v\prime =\frac{v}{2}$

Initial elongation of the spring is

$x =\frac{mg}{k}$

Now, using the conservation of mechanical energy we get

$\frac{1}{2}2m{\left(v\text{'}\right)}^2+\frac{1}{2}k{x}^2=2mgx$

$m\left(\frac{v^2}{4}\right)+\frac{m^2{g}^2}{2k}=\frac{2m^2{g}^2}{k}$

$v=\sqrt{\frac{6m{g}^2}{k}}$