$4\mathrm{.}53($a$)$ Consider an electron $($charge $$e and mass m$)$ in a circular orbit of radius r around a fixed

proton $($charge $+$e$).$ Remembering that the inward Coulomb force ke$2/$r$2$ is what gives the electron its

centripetal acceleration, prove that the electron$$s KE is equal to $1$ times its PE; that is$,$ T $=1$ U and $22$

hence E $=1$ U $.($This result is a consequence of the so$-$called virial theorem. See Problem $4\mathrm{.}41.)$ Now $2$

consider the following inelastic collision of an electron with a hydrogen atom: Electron number $1$ is in a circular orbit of radius r around a fixed proton. $($This is the hydrogen atom.$)$ Electron $2$ approaches