HSC Physics Elastic Collision

A sheet of harder mechanics questions appears below.

An elastic collision occurs between an alpha partice of mass m moving at a speed u and a stationary nucleus of mass M. After collision the alpha particle moves at speed v at an angle 𝜃 to its initial direction and the nucleus moves at speed V at an angle ϕ to the initial direction of the alpha particle.

1. Show that v = u sinϕ/sin(ϕ + 𝜃).
2. Show that V = m u sin𝜃/( M sin(ϕ + 𝜃)).
3. Show that m/M = sin(2ϕ + 𝜃)/sin𝜃.