Hello guys I have homework that has tasked me with connecting the effect of the scattering parameter to the energy transfer in a 2d elastic collision of two arbitrary spheres with one of them standing still and I feel like I'm really struggling in connecting the two without feeling like it's circle logic like. what I have so far are the following things that I understand: for this problem lets say that $\theta$ is the scattering angle of the moving particle in the lab frame and $\phi$ is the same in the CoM frame then also let $v'$ denote a velocity in the CoM frame as well then what I have figured is:
$\theta=2\cos^{-1}(\frac{b}{r_1+r_2})$
$ \phi=\tan^{-1}(\frac{v_{f,1}\sin(\theta_{f,1})}{v_{f,1}\cos(\theta_{f,1})+\frac{m_1}{m_1+m_2}v_{i,1}}) $
$v_{f,1}^2=(\frac{m_2}{m_1+m_2})^{2}v_{i,1}^{2}+2\frac{m_2m_1}{(m_1+m_2)^2}v_{i,1}\cos(\phi)+(\frac{m_1}{m_1+m_2})^2 v_{i,1}^2$
$ \frac{m_{2}v_{f,2}^2}{2}=\frac{2m_2m_1(1-\cos(\phi))}{(m_1+m_2)^2} m_1 v^2_{i,1}$
My idea was to define $\phi$ in terms of $\theta$ using b which is the scattering angle and show the connection as such but given that $\phi$ requires $v_{1,f}$ to be defined and $v_{1,f}$ requires $\phi$ to be defined this doesn't seem possible. I really don't know what to do about this and as such would appreciate any type of hint.
