Suppose we have two masses, $m_1$ and $m_2$, where $m_2$ is at rest, and $m_1$ is headed directly towards $m_2$. I would like to write the ratio of the masses as a function of the angle.
Using Conservation of Energy, Conservation of Momentum, and applying the Law of Cosines to the the triangle that the initial momentum vector and final momenta vectors form, I obtain these three equations:
(1) $ m_1^2 v_1^2 = m_1^2 v_1^2~' + m_2^2 v_2^2~' - (m_1 v_1)(m_2 v_2) \cos \theta$ (Law of Cosines)
(2) $m_1 v_1^2 = m_1 v_1^2~' + m_2 v_2^2~'$ (Cons. of Energy)
(3) $m_1 \mathbf{v}_1 = m_1 \mathbf{v}_1~' + m_2 \mathbf{v}_2~' $ (Cons. of Mom.)
Taking the magnitude of the Conservation of Momentum equation, and squaring the result, yields
(4) $m_1^2 v_1^2 = m_1^2 v_1^2~' + m^2 v_2^2~' + (m_1 m_2)(v_1~' v_2~') \cos \theta$
I would like to try to eliminate the variables $v_1~'$ and $v_2~'$, seeing as in a experimental setting they might be a little more difficult to measure; however, I would like to keep the variable $v_1$, because it is a little more easy to determine--suppose the moving particle is an electron, then $m_1$ is known, and $v_1$ can be easily determine by knowing the accelerating voltage.
I was hoping that someone might guide me towards the correct path, I certainly would appreciate it.
