In Example $44.3$, it was shown that a proton beam with an $800$-GeV beam energy gives an available energy of $38.7$ GeV for collisions with a stationary proton target. In a colliding-beam experiment, what total energy of each beam is needed to give an available energy of $2(38.7$ GeV$)=77.4$ GeV?

Calculate the minimum beam energy in a proton-proton collider to initiate the $p+p\to p+p+{\eta }^0$ reaction. The rest energy of the ${\eta }^0$ is $547.3$ MeV (see Table $44.3$).

You work for a start-up company that is planning to use antiproton annihilation to produce radioactive isotopes for medical applications. One way to produce antiprotons is by the reaction $p+p\to p+p+p+\bar{p}$ in proton-proton collisions. You first consider a colliding-beam experiment in which the two proton beams have equal kinetic energies. To produce an antiproton via this reaction, what is the required minimum kinetic energy of the protons in each beam?

How much energy is released when a ${\text{µ}}^{-}$ muon at rest decays into an electron and two neutrinos? Neglect the small masses of the neutrinos.

A $K^{+}$ meson at rest decays into two $p$ mesons. What are the allowed combinations of ${\pi }^0$ , ${\pi }^{+}$, and ${\pi }^{-}$ as decay products?

A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is $16.0$ GeV?

What is the speed of a proton that has total energy $1000$ GeV?