We study the spreading of the wave function of a Bose-Einstein condensate accelerated by a constant force both in the absence and in the presence of atom-atom interactions. We show that, despite the initial velocity dispersion, the local velocity dispersion defined at a given position downward can reach ultralow values and be used to probe very narrow energetic structures. We explain how one can define quantum mechanically and without ambiguities the different velocity moments at a given position by extension of their classical counterparts. We provide a common theoretical framework for interacting and non-interacting regimes based on the Wigner transform of the initial wave function that encapsulates the dynamics in a scaling parameter. In the absence of interaction, our approach is exact. Using a numerical simulation of the 1D Gross-Pitaevskii equation, we provide the range of validity of our scaling approach and find a very good agreement in the Thomas-Fermi regime. We apply this approach to the study of the scattering of a matter wave packet on a double barrier potential. We show that a Fabry-Perot resonance in such a cavity with an energy width below the pK range can be probed in this manner. We show that our approach can be readily transposed to a large class of many-body quantum systems that exhibit self-similar dynamics.