We investigate the behavior of a finite-momentum impurity immersed in a weakly interacting three-dimensional Bose-Einstein condensate (BEC) of ultra-cold atoms, giving a detailed account of the dynamical quantum Cherenkov transition discussed in Ref. [arXiv:2101.00030]. Using a time-dependent variational approach, we identify a transition in the far-from-equilibrium dynamics of the system after the attractive short-range impurity-boson interaction is quenched on. The transition occurs as the impurity's velocity crosses an interaction-dependent critical value, and manifests in the long-time behavior of the Loschmidt echo and average impurity velocity. This behavior is also reflected in the finite momentum ground state of the system, where the group velocity of the interaction-dressed impurity loses it's dependence on the total momentum of the system as the critical point is crossed. The transition we discuss should be experimentally observable via a variety of common protocols in ultracold atomic systems such as time-of-flight imaging, RF spectroscopy, Ramsey interferometry, and absorption imaging.