Correct option is (4) \(v = \frac u{\sqrt 2}\)
The mass of particle 1 is m, and its initial velocity is \(u\hat i\). The mass of particle 2 is 3m, and its initial velocity is 0; the velocity of particle 1 after collision is \(v\hat j\).
Let velocity of particle -2 after collision be v'
By law of conservation of linear momentum:
\(mu\hat i + 0 = mv\hat j + 3mv'\)
\(v' = \frac u3 \hat i - \frac v3 \hat j\)
By law of conservation of kinetic energy:
\(\frac 12 mu^2 + 0 = \frac 12mv^2 + \frac 12 (3m)v'\)
\(u^2 = v^2 + 3 \left[(\frac u3)^2 + (\frac v3)^2\right]\)
\(u^2 - \frac{u^2}3 = v^2 + \frac{v^2}3\)
\(\frac 23 u^2 = \frac 43 v^2\)
\(u^2 = 2v^2\)
\(v = \frac u{\sqrt 2}\)