Distance of Closest Approach of Two Particles

[nothilaryy]

Homework Statement

Given the momentum p₁ and p₂, find the distance of closest approach (b) also known as the impact parameter. I think we are also given charges q because I later have written solve for b² in terms of x², q², P², p$$\bullet$$x, q$$\bullet$$x, and p$$\bullet$$q. [those are supposed to be dot products but I think I fail at latex] I believe x is supposed to be relative position.

Homework Equations

I'm not really sure which equations I'm supposed to be using at all, but my best guess is to use

E = mc²
E²= E'² - (pc)²

and maybe the lorentz transformations?

The Attempt at a Solution

Ok so here's the deal. As much as I'd like an answer to this problem, I'd like to really understand what is going on much more. I've essentially been thrown into the world of special relativity without so much as a textbook for a side project I'm doing for a professor, and I understand the theory behind this weird stuff like time dilation and etc, but I don't understand most of the math behind it. So if anyone happens to know of a good online resource for this sort of stuff, I would greatly appreciate it.

That being said, I have actually given the problem a try! According to the hint my professor gave us, I'm supposed to solve it in the reference frame of the velocity of the center of mass, but I'm not really sure how to go about that. I thought about finding the center of mass like I would in mechanics, but not only can I not find enough information to be able to do that, I routinely mess up when I try to apply mechanics ideas when working with relativity.

I also looked at https://www.physicsforums.com/showthread.php?t=161767", because it seems really similar, but I cannot seem to convert between the information I was given and the information they were given. I did gather that I should be conserving total energy and momentum, but again, I'm not sure where to go from there.

Sorry for this wall of text- even the littlest bit of help or nudge in the right direction would be appreciated.

 

[kuruman]

Can you be more specific with the description of the problem? You are looking for the impact parameter, but what kind of particles do you have and what are the initial conditions? Also you seem to imply that the calculation needs to be relativistic. What is the evidence that it is?

 

[nothilaryy]

Uhm all I was definitely given were that there are two particles each with some momentum (and possibly some charge). The only reason why I think its supposed to be relativistic is because that's what we were going over before the problem was assigned? I really wish I could be less vague, but truth be told I'm not all that clear on the problem myself. Sorry if this problem is just way too not concrete.

 

[kuruman]

It is "way to not concrete". It is also not clear what the symbols x and q stand for. If q is a charge (a scalar) then a term like q ^. x makes no sense as a dot product.