a
(a)

Initial momentum of the system

\(p _{ i }=m \times 2 v \hat{i}+2 m \times v \hat{j}\)

\(=\sqrt{(m \times 2 v)^2+(2 m \times v)^2} \quad \text { (magnitude) }\)

\(=2 \sqrt{2}\,m v\)

Final momentum of the system \(=3\,mV\)

By the law of conservation of momentum

\(2 \sqrt{2} m v=3 mV\)

\(\Rightarrow  \frac{2 \sqrt{2} v}{3}= V _{\text {combined }}\)

Loss in energy

\(\Delta E =\frac{1}{2} m _1 V _1^2+\frac{1}{2} m _2 V _2^2-\frac{1}{2}\left( m _1+ m _2\right) V _{\text {combined }}^2\)

\(\Delta E =3 mv ^2-\frac{4}{3} mv ^2=\frac{5}{3} mv ^2=55.55 \%\)