# Distance of Closest Approach of Two Particles

• nothilaryy
In summary: Is x a position? If so, relative to what?In summary, the conversation discusses a problem involving finding the distance of closest approach (b) between two particles with given momenta and possibly charges. The equations of E=mc^2 and E^2 = E'^2 - (pc)^2 are mentioned, along with the suggestion to use Lorentz transformations. The speaker expresses a desire to understand the math behind the problem and mentions difficulty in applying mechanics concepts to relativity. There is also mention of a reference frame of the velocity of the center of mass, but uncertainty on how to approach this. The problem is not clear and the symbols x and q are not defined.
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.

Last edited by a moderator:
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?

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.

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.

## 1. What does "Distance of Closest Approach of Two Particles" mean?

The distance of closest approach of two particles refers to the minimum distance between the two particles when they are moving towards each other and are at their closest point of interaction.

## 2. How is the distance of closest approach calculated?

The distance of closest approach is calculated using the equations of motion and the initial positions, velocities, and masses of the particles. It takes into account the attractive and repulsive forces between the particles as well as their relative velocities.

## 3. What factors affect the distance of closest approach?

The distance of closest approach is affected by the masses and velocities of the particles, as well as the strength of the forces between them. It can also be impacted by external factors such as the presence of other particles or fields.

## 4. Why is the distance of closest approach important in physics?

The distance of closest approach is important in physics because it helps us understand the interactions between particles and how they affect each other. It is also a crucial factor in determining the outcomes of collisions and other physical processes.

## 5. Can the distance of closest approach be measured experimentally?

Yes, the distance of closest approach can be measured experimentally using various techniques such as particle accelerators or detectors that can track the positions and velocities of particles. These measurements can provide valuable insights into the behavior of particles and their interactions.

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