A particle of mass mi and initial velocity ui collides elast

A particle of mass mi and initial velocity ui collides elastically with a particle of unknown mass m_2 coming from the opposite direction. After the collision, the particle of mass m_1 has velocity nu_1 = u_1/2 at a right angle to the incident direction and the particle of mass m_2 moves at a 45 degree angle, as shown in the figure. Find the mass m_2 of the second particle.

Solution

In vertical direction :

Total momentum of the masses before collision P = 0

Since initial velocities of m1 and m2 before colliison in vertical direction are zero.

Total momenutm of the masses after collision P \' = m1(-v1) + m2v2 sin 45

From law of conservation of momentum , P \' = P

m1(-v1) + m2v2 sin 45 = 0

m2v2 sin 45 = m1v1

             0.7071 m2v2 = m1v1

                                = m1(u1/2)

                               = 0.5 m1u1

From this, m2 = 0.5 m1u1 /(0.7071 v2)

                     = (0.7071 m1u1) / v2