A proton of mass m collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of 90° with respect to each other. The mass of the unknown particle is:

Apply principle of conservation of momentum along x-direction, mu = mv₁cos45° + Mv₂cos45°
mu = (1/√2)(mv₁ + Mv₂).. (1)
along y-direction, 0 = mv₁sin45° - Mv₂sin45°
0 = (mv₁ - Mv₂)(1/√2).. (2)
Coefficient of restitution e = 1 = (v₂ - v₁cos90°)/ucos45°
Restitution ⇒ v₂/u√2 = 1 ⇒ u = √2v₂.. (3)
solving eqn (1), (2), (3) we getM = m