A particle of mass m with an initial velocity  ui^ collides perfectly elastically with a mass 3m at rest. It moves with a velocity vj^  after collision, then, v is given by:

Using momentum conservation, P→i = P→f ⇒m(ui^)+3m(0)=m(vj^)+3m(v'→) ⇒v'→=(u3)i^ −(v3)j^  Using energy conservation, 12mu2=12mv2+123m{(u3)2+(v3)2}  ⇒u2=v2+u23+v23 ⇒2u23=4v23  ⇒v=u2