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The exposition you give is fine in classical physics. Note though that in classical physics a particle cannot be a point particle, because something has to carry the charge in classical physics formulations. So the fact that one finds infinity at r=0 just hits on this constraint. One could use the argument as a proof by reductio ad absurdum that particles ...

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Well, I'm no expert on this, but a quick literature search suggests that you are sort of correct. Here's a paper on arXiv which is about this. You shouldn't trust everything you read on arXiv because it is not peer reviewed but this paper has subsequently been published in Phys. Rev. B as 1/N expansion in correlated graphene By: Kotov, Valeri N.; Uchoa, ...

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The way you do calculations like this is to work in the centre of mass frame. The setup you describe looks like this (the convention here is that velocities to the left are positive, and velocities to the right are negative): But suppose we view the collision from a frame moving to the left at $v/2$ i.e. we have to subtract $v/2$ from all the velocities. ...

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