Relativistic Antiproton Protion

[alex3]

This isn't really so much of a homework problem, but I'm reading this webpage, and there's one part of the derivation of the required proton KE needed to produce a static antiproton.

The part that's puzzling me is

"and using... m²_inc⁴ - p²_inc² = m²₀c⁴ "

I don't see where this formula is coming from. I can see the resemblance to the invariant, but the RHS deals with the moving proton, and the LHS only has the mass of a single proton, I'm confused as to why it's not (2m₀)²c⁴.

Thanks for any help.

 

[Doc Al]

The part that's puzzling me is

"and using... m²_inc⁴ - p²_inc² = m²₀c⁴ "

I don't see where this formula is coming from.

This is just a rearrangement of the usual Einstein formula:

$$E^2 = m_0^2c^4 + p^2c^2$$

$$m^2c^4 = m_0^2c^4 + p^2c^2$$

$$m^2c^4 - p^2c^2 = m_0^2c^4$$

where m is the 'relativistic mass'.

(This formula is derived on the page just before the one you linked.)

I can see the resemblance to the invariant, but the RHS deals with the moving proton, and the LHS only has the mass of a single proton,

I'm not sure what you mean. This is the invariant. It's true for any proton, including the incoming one.

 

[alex3]

Aha of course, elementary really. Thanks very much for the clarification!