  In this paper we describe a new data structure that supports orthogonal range
reporting queries on a set of points that move along linear trajectories on a
$U\times U$ grid. The assumption that points lie on a $U\times U$ grid enables
us to significantly decrease the query time in comparison to the standard
kinetic model. Our data structure answers queries in $O(\sqrt{\log U/\log \log
U}+k)$ time, where $k$ denotes the number of points in the answer. The above
improves over the $\Omega(\log n)$ lower bound that is valid in the
infinite-precision kinetic model. The methods used in this paper could be also
of independent interest.
