\(\overrightarrow{ P }_{ i }=\overrightarrow{ P }_{ f }\)

\(m ( ui )+3 m ( D )= mvj +3 m \overline{ v }_{2}\)

\(mui - mvj =3 m \overline{ v }_{1}\)

\(\bar{v}_{1}=\frac{u i-v j}{3}\)

or \(\left| v _{1}\right|=\frac{\sqrt{ u ^{2}+ v ^{2}}}{3}\)

or \(v_{1}^{2}=\frac{u^{2}+v^{2}}{9} \ldots .(1)\)

As collision is perfectely elastic hence

\(k_{i}=k_{j}\)

\(\frac{1}{2} m u^{2}+\frac{1}{2} 3 m 0^{2}=\frac{1}{2} m v^{2}+\frac{1}{2} 3 m v_{1}^{2}\)

\(\Rightarrow u ^{2}= v ^{2}+3 v _{1}^{2}\)

\(u ^{2}= v ^{2}+3 \frac{\left( u ^{2}+ v ^{2}\right)}{9}\)

\(\Rightarrow 3 u ^{2}=3 v ^{2}+ u ^{2}+ v ^{2}\)

\(\Rightarrow 2 u ^{2}=4 v ^{2}\)

\(v =\frac{ u }{\sqrt{2}}\)