  We study a Paveri-Fontana type model, which describes the evolution of the
mesoscopic distribution of vehicles through a combined effect of adjustment of
the velocity with respect to nearby vehicles, and slowing down and speeding up
of the vehicles arising as a result of exchange of velocity with the vehicles
on the same location on the road. We first prove the global-in-time existence
of weak solutions. The proof is via energy, $L^p$, and compact support
estimates together with velocity averaging lemma. The combined effect of
alignment nature of $Q_r$, which keeps the characteristic from spreading, and
the dissipative nature of $Q_i$, which gives the uniform control on the size of
the distribution function, is crucially used in the estimates. We also
rigorously establish a hydrodynamic limit to the presureless Euler equation by
employing the relative entropy combined with the Monge-Kantorovich-Rubinstein
distance.
