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Another nifty calculation is the bevatron-design-energy
calculation. The objective is to manufacture antiprotons.
Consider the reaction where an accelerator smashes a
fast-moving proton #1 into a stationary proton #2 to
produce a proton/antiproton pair:
proton + proton --> proton + proton + proton + antiproton
The question is, what is the minimum voltage the accelerator
must produce in order to make this reaction go?
(Note that starting with the given reactants on the LHS,
the RHS is the simplest thing we can have that includes
an antiproton while upholding the conservation laws.)
Write down the [energy,momentum] 4-vector for the RHS
in the center-of-mass frame. Hint: the lowest energy
at which the reaction can take place is the case where
the four products on the RHS have zero velocity relative
to each other.
Then analyze the LHS in the lab frame.
Hint: both sides have the same invariant norm. Also
proton #2 has px=py=pz=0 in the lab frame.
At this
point you have one equation in one unknown, namely the
KE of proton #1. The whole calculation takes only a
few lines and a few minutes. The result is simple and
IMHO interesting.
Note that each of these examples is a pretty strong
advertisement for the 4-vector approach, demonstrating
that you can get interesting results from relativity
without having to write out any Lorentz transformations.
Also, to get back to the point of this thread: these
exercises, individually and (especially) collectively
serve to clarify the ideas of total energy, rest energy,
and kinetic energy.