  We consider a non relativistic quantum system consisting of $K$ heavy and $N$
light particles in dimension three, where each heavy particle interacts with
the light ones via a two-body potential $\alpha V$. No interaction is assumed
among particles of the same kind. Choosing an initial state in a product form
and assuming $\alpha$ sufficiently small we characterize the asymptotic
dynamics of the system in the limit of small mass ratio, with an explicit
control of the error. In the case K=1 the result is extended to arbitrary
$\alpha$. The proof relies on a perturbative analysis and exploits a
generalized version of the standard dispersive estimates for the
Schr\"{o}dinger group. Exploiting the asymptotic formula, it is also outlined
an application to the problem of the decoherence effect produced on a heavy
particle by the interaction with the light ones.
