A smooth sphere of mass $m$is moving on a horizontal plane with a velocity $\left(3\hat{i}+\hat{j}\right)$, when it collides with a vertical wall which is parallel to the vector $\hat{j}$. If the coefficient of restitution between the sphere and the wall is $\frac{1}{2}$, then

If we resolve the initial velocity along $x-and y-axis$, then it is clear that $u_y$ will remain constant after collision but $u_x$ will change. Also, $m_2>>m_1$, therefore $m_1$ can be neglected. Hence, 

${\left(v_x\right)}_1=\left(\frac{-e{m}_2}{m_2}\right){\left(u_x\right)}_1=-e{\left(u_x\right)}_1=-\frac{1}{2}\times 3=-\frac{3}{2}$

∴ Velocity after collision is

$\overrightarrow{v}=-\frac{3}{2}\hat{i}+\hat{j}$

Similarly we can find that

 $$\begin{gathered} loss in K.E. =\frac{27m}{8}\hat{j} \\ and impulse=-\frac{9m}{2}\hat{i} \end{gathered}$$