Suppose an electron approaches a proton with greater energy than the hydrogen ground state. Will the electron scatter? If so, how could an electron ever be captured given it would require exactly the right momentum?
If the electron is instead captured, what happens to the excess kinetic energy?
Is there a threshold at which the energy difference is too great, or will all electrons in the nucleus' vicinity be captured (provided there is a vacancy in the valence shell)?
The electron can be captured; the excess kinetic energy is carried off by the emitted photon. This, for example, is why spectrum of hydrogen recombination has "continua" such as the Lyman continuum: the capture of electrons with zero velocity produces photons with energy equal to the ionization potential, with a continuum of higher-energy photons from the capture of electrons with nonzero velocities (the individual photons have energies = ionization potential + electron's kinetic energy).
The cross-section for electron capture is, for hydrogen ions (i.e., bare protons) at least, an inverse function of the electron's velocity squared. So slow-moving electrons are more easily captured, but there isn't a strict threshold: some fast-moving electrons will be captured.
