  We consider a one-dimensional system consisting of a granular tracer particle
of mass $M$ in a bath of thermalized particles each of mass $m$. When the mass
ratio, $M/m$, is equal to the coefficient of restitution, $\alpha$, the system
maps to a a one-dimensional elastic gas. In this case, Boltzmann equation can
be solved exactly. We also obtain expressions for the velocity autocorrelation
function and the diffusion coefficient. Numerical simulations of the Boltzmann
equation are performed for $M/m\neq \alpha$ where no analytical solution is
available. It appears that the dynamical features remain qualitatively similar
to those found in the exactly solvable case.
