  We consider how dynamical friction acts on black holes that receive a
velocity kick while located at the center of a gravitational potential,
analogous to a star cluster, due to either a natal kick or the anisotropic
emission of gravitational waves during a black hole-black hole merger. Our
investigation specifically focuses on how well various Chandrasekhar-based
dynamical friction models can predict the orbital decay of kicked black holes
with $m_{bh} \lesssim 100 M_\odot$ due to an inhomogeneous background stellar
field. In general, the orbital evolution of a kicked black hole follows that of
a damped oscillator where two-body encounters and dynamical friction serve as
sources of damping. However, we find models for approximating the effects of
dynamical friction do not accurately predict the amount of energy lost by the
black hole if the initial kick velocity $v_{k}$ is greater than the stellar
velocity dispersion $\sigma$. For all kick velocities, we also find that
two-body encounters with nearby stars can cause the energy evolution of a
kicked BH to stray significantly from standard dynamical friction theory as
encounters can sometimes lead to an energy gain. For larger kick velocities, we
find the orbital decay of a black hole departs from classical theory completely
as the black hole's orbital amplitude decays linearly with time as opposed to
exponentially. Therefore, we have developed a linear decay formalism which
scales linearly with black hole mass and $\frac{v_{k}}{\sigma}$ in order to
account for the variations in the local gravitational potential.
