  Vortex motion is a complex problem due to the interplay between the
short-range physics at the vortex core level and the long-range hydrodynamical
effects. Here we show that the hydrodynamic equations of vortex motion in a
compressible superfluid can be solved exactly in a model "slab" geometry.
Starting from an exact solution for an incompressible fluid, the hydrodynamic
equations are solved with a series expansion in a small tunable parameter
provided by the ratio of the healing length, characterizing the vortex cores,
to the slab width. The key dynamical properties of the vortex, the inertial and
physical masses, are well defined and renormalizable. They are calculated at
leading order beyond the logarithmic accuracy that has limited previous
approaches. Our results provide a solid framework for further detailed study of
the vortex mass and vortex forces in strongly-correlated and exotic
superfluids. The proposed geometry can be realised in quantum-gas experiments
where high-precision measurements of vortex mass parameters are feasible.
