a
(a)Let mass \(A\) moves with velocity \(v\) and collides inelastically with mass \(B\), which is
According to problem mass \(A \) moves in a perpendicular direction and let the mass \(B \) moves at angle \(\theta\) with the horizontal with velocity \( v\).
Initial horizontal momentum of system
(before collision) \(= mv\) \(....(i)\)
Final horizontal momentum of system
(after collision) \(= mV\, cos\) \(\theta\)\( ....(ii)\)
From the conservation of horizontal linear momentum \(mv = mV \) \(cos\)\(\theta\)\(⇒\) \( v = V cos\)\(\theta\) \(...(iii)\)
Initial vertical momentum of system (before collision) is zero.
Final vertical momentum of system \(\frac{{mv}}{{\sqrt 3 }} - mV\sin \theta \)
From the conservation of vertical linear momentum \(\frac{{mv}}{{\sqrt 3 }} - mV\sin \theta = 0\)==>\(\frac{v}{{\sqrt 3 }} = V\sin \theta \)\(...(iv)\)
By solving \((iii)\) and \((iv)\)
\({v^2} + \frac{{{v^2}}}{3} = {V^2}({\sin ^2}\theta + {\cos ^2}\theta )\)
\(⇒\) \(\frac{{4{v^2}}}{3} = {V^2}\) \(⇒\)\(V = \frac{2}{{\sqrt 3 }}v\).