The problem involves two concepts -

1. the conservation of linear momentum.

2. the conservation of energy .

But first we say that the system is our ball + barrel ....now as there is no external force acting on the system ( no friction) so we can conserve the linear momentum :

( letting the final velocity of the system be v)

1/2  mu²  = 1/2 (m+M) v² + Z .........................................(2)

(letting the energy store in spring be denoted by Z)

so putting the values from eqn 1 to eqn 2 we can get ...