  We study the motion of an elastic object driven in a disordered environment
in presence of both dissipation and inertia. We consider random forces with the
statistics of random walks and reduce the problem to a single degree of
freedom. It is the extension of the mean field ABBM model in presence of an
inertial mass m. While the ABBM model can be solved exactly, its extension to
inertia exhibits complicated history dependence due to oscillations and
backward motion. The characteristic scales for avalanche motion are studied
from numerics and qualitative arguments. To make analytical progress we
consider two variants which coincide with the original model whenever the
particle moves only forward. Using a combination of analytical and numerical
methods together with simulations, we characterize the distributions of
instantaneous acceleration and velocity, and compare them in these three
models. We show that for large driving velocity, all three models share the
same large-deviation function for positive velocities, which is obtained
analytically for small and large m, as well as for m =6/25. The effect of small
additional thermal and quantum fluctuations can be treated within an
approximate method.
