Shooting an electron past a positive nucleus (trajectory)

[phantomvommand]

TL;DR Summary

An electron is shot horizontally. There is a proton located somewhere else, but not in the horizontal path of the electron. Is there a distance of closest approach, and how do you derive it? A physical explanation would be appreciated too. Feel free to use any variables.

An electron is shot horizontally. There is a proton located somewhere else, but not in the horizontal path of the electron. Is there a distance of closest approach, and how do you derive it? A physical explanation would be appreciated too.

 

[ergospherical]

It's straightforward as a one-body problem but a little more subtle as a two-body problem. For the one-body (fixed proton) scenario it's enough to conserve energy and angular momentum about the proton. If you want the scattering angle/trajectory then you can relate these to the hyperbolic parameters $a$ and $b$ with the standard formulae $L^2 = \gamma b^2 / a$ and $E = \gamma / 2a$ [where $\gamma = e^2/m$]; the proton is at the far focus.

 

[PeroK]

You could search for Rutherford scattering. There's quite a bit online about this.