I have a problem with demonstrating a relation in a Compton scattering exercise:

A low energy photon collides with an electron at rest. After the shock, the photon is backscattered.

It is supposed that the energy of the photon is very small compared with the energy of the electron at rest

I have to show that the ration between the energy of the backscattered photon and the kinetic energy acquired by the electron in a compton scattering is approximately c / v

where v is the speed of the electron

I use the energy conservation:

\(\displaystyle h\mu +m c^2=h\mu' +Ec +mc^2\) and the relation \(\displaystyle \lambda'-\lambda=\frac{2h}{mc}\)

I have arrived on this \(\displaystyle \frac{E_\gamma}{Ec}=\frac{2h\mu c^2}{v^2(2h\mu+mc^2)}\)but there is no way i can obtain c/v...

Thank you so much for your help(Sorry for my english...)