Calculations of the Stark broadening of hydrogen lines treat the radiating atoms and the perturbing ions as quasistatic. The present experiment represents an attempt to determine whether the possible breakdown of this approximation near the center of the line can account, at least partially, for the existing discrepancies between theoretical and experimental profiles in the core of the Balmer lines. The central regions of ${H}_{\ensuremath{\alpha}}$, ${H}_{\ensuremath{\beta}}$, ${H}_{\ensuremath{\gamma}}$, and ${H}_{\ensuremath{\delta}}$ profiles have been measured in a wall-stabilized arc over a range of atom-ion relative velocities by varying the atom-ion reduced mass. The cores of all four lines exhibit a significant dependence on the reduced mass. With increasing reduced mass, the experimental profiles gradually show more structure, but still less than the theories predict. Extrapolation of the results for ${H}_{\ensuremath{\alpha}}$ and ${H}_{\ensuremath{\beta}}$ to infinite reduced mass, i.e., to the static case, gives results that agree quite well with recent calculations.