What is the Maximum Angle of Deflection in a Particle Collision?

[fedecolo]

Homework Statement

A particle of mass $M$ and speed $v_1$ bumps a particle of mass $m$, $(M >m)$, at rest. Find the maximum angle of deflection of $M$ after the bump.

The Attempt at a Solution

I would like to solve this problem in the reference frame of the center of mass, because it seems that the counts are very little :D
By definition, if I call $u_1,u_2$ the velocities of the particles in the CdM reference frame, I have $p_{CdM}=Mu_1 -mu_2=0$. Then I call $u_1 ',u_2 '$ the velocities after the bump (the quantity of motion of Cdm is 0 also after the bump). In the image there is a sketch of the situation, with $\phi$ the scattering angle. But now I'm struggling in relating the velocities $u_1,u_2,u_1 ',u_2 '$ with the end velocities of the particles $v_1 ', v_2'$ in the frame of the laboratory. My question is: how to sketch this four vectors?

 

[Orodruin]

In the CoM frame the masses can go in any direction. The problem asks for the frame where the larger mass is initially at rest.