A spherical ball of radius r initially at rest on a rough horizontal surface is hit horizontally at a point at a distance x above the central line. Due to this sharp impulse the centre of the ball acquires a velocity v_c. After some time, the ball will start pure rolling with a velocity equal to 

Initially ball will gain both linear and angular velocity.

$\begin{array}{l}\text{ Linear impulse: }{I}_0=m{v}_C\\ \text{ Angular impulse: }{I}_{0}x=I{\omega }_0\Rightarrow m{v}_{C}x=I{\omega }_0\end{array}$

I is moment of Inertia about C.
Apply conservation of angular momentum about lowest point.

$I{\omega }_0+m{v}_{C}r=I\omega +mvr$

$\text{ where }\omega =\frac{v}{r}\text{ at the time of pure rolling }$

$\begin{array}{l}\Rightarrow m{v}_{C}x+m{v}_{C}r=\frac{2}{5}m{r}^2\frac{v}{r}+mvr\\ \Rightarrow v_C(x+r)=\frac{7}{5}rv\\ \Rightarrow v=\frac{5}{7}{v}_C\left[\frac{x+r}{r}\right]\end{array}$