Question regarding Rutherford's atomic model

[daselocution]

Homework Statement

I am basically wondering how they got to the assumption that the V_e = 2V_α. I've tried it a few times and I keep on getting that based off of their other assumptions, Ve should =0, even though I know that this cannot possibly be the case.

Homework Equations

Conservation of momentum: M_av_a = M_av'_a + m_ev_e

Conservation of energy: .5M_av_a² = .5M_av'_a² + .5M_ev_e²

The Attempt at a Solution

I really don't know what to do. I've tried starting under the assumption as given that Va≈Va', which leads to getting Ve=0. I tried to do a binomial expansion using the fact that Me<<Ma, but again I got zero. I'm really not sure how to approach this. Any help would be appreciated.

 

[tms]

I am basically wondering how they got to the assumption that the V_e = 2V_α.

It's not an assumption, it's a conclusion.

Set up the equations for an elastic collision in one dimension, then get a general solution for the final velocities of both particles. Don't put in any numbers or the like, just get a general solution. Then assume that $m_1 >> m_2$, and see what you get for $v_{2f}$.

 

[daselocution]

Thank you very much. That makes much more sense, I'm not really sure why I didn't just do that the first time. My mistake was not solving for Valpha' in terms of the other equations. Once I realized that I had to do that, it was much clearer.