  We study the dynamics of a heavy particle of mass $M$ moving in a
one-dimensional repulsively interacting Fermi gas. The Fermi gas is described
using the Luttinger model and bosonization. By transforming to a frame
co-moving with the heavy particle, we map the model onto a generalized
``quantum impurity problem". A renormalization group calculation reveals a
crossover from strong to weak coupling upon scaling down in temperature. Above
the crossover temperature scale $T^*=(m/M) E_F$, the particle's mobility,
$\mu$, is found to be (roughly) temperature independent and proportional to the
dimensionless conductance, $g$, characterizing the 1d Luttinger liquid. Here
$m$($<<M$) is the fermion mass, and $E_F$ is the Fermi energy. Below $T^*$, in
the weak coupling regime, the mobility grows and diverges as $\mu(T) \sim
T^{-4}$ in $T \to 0$ limit.
