A bullet of mass m travelling at a speed v strikes a ball of mass M...

By the principle of conservation of linear momentum, the total initial momentum of the system is equal to the total final momentum.

Initial momentum of the system is $p_i = mv + M(0) = mv$.

After the collision, a part of mass $m'$ breaks off from the ball and sticks to the bullet. The new mass of the bullet becomes $m + m'$. Let its final velocity be $v_2$.

The remaining mass of the ball is $M - m'$, and it moves with a velocity $v_1$ in the initial direction of the bullet.

Final momentum of the system is $p_f = (m + m')v_2 + (M - m')v_1$.

Equating initial and final momentum:

$mv = (m + m')v_2 + (M - m')v_1$

$(m + m')v_2 = mv - (M - m')v_1$

$v_2 = \dfrac{mv - (M - m')v_1}{m + m'}$

The velocity of the bullet following the collision is $\dfrac{mv - (M - m')v_1}{m + m'}$ in the initial direction.