  Quasar winds can shock and sweep up ambient interstellar medium (ISM) gas,
contributing to galactic quenching. We combine and extend past models of
energy-conserving shock bubbles around quasars, investigate model implications
from an observational standpoint, and test model predictions using new
high-resolution spectroscopic observations of the broad absorption line quasar
SDSS J030000.56+004828.0 (J0300). Even with constant energy input from the
wind, a bubble's expansion decelerates over time as more ISM gas is swept up.
Our new observations enable a direct search for this deceleration. We obtain
the tightest reported 3-sigma limit on the average rest-frame deceleration (or
acceleration) of a quasar outflow: |a|$<$0.1 km s$^{-1}$ yr$^{-1}$ ($<3 \times
10^{-4}$ cm s$^{-2}$) in the relatively low-velocity Ca II outflow of J0300
over 9.65 rest-frame years. We can satisfy these limits with certain parameter
choices in our model, but the large velocity range of the Ca II absorption in
J0300 rules out the hypothesis that such gas shares the velocity of the
swept-up ISM gas in a self-similar shock bubble. We investigate the possibility
of ram-pressure acceleration of preexisting ISM clouds and conclude that the
velocity range seen in Ca II in J0300 is potentially consistent with such an
explanation. The Ca II-absorbing gas clouds in J0300 have been inferred to have
high densities by Choi et al., in which case they can only have been
accelerated to their current speeds if they were originally at least an order
of magnitude less dense than they are today.
