  The kinematics on spatially flat FLRW space-times is presented for the first
time in co-moving local charts with physical coordinates, i. e. the cosmic time
and Painlev\' e-type Cartesian space coordinates. It is shown that there exists
a conserved momentum which determines the form of the covariant four-momentum
on geodesics in terms of physical coordinates. Moreover, with the help of the
conserved momentum one identifies the peculiar momentum separating the peculiar
and recessional motions without ambiguities. It is shown that the energy and
peculiar momentum satisfy the mass-shell condition of special relativity while
the recessional momentum does not produce energy. In this framework, the
measurements of the kinetic quantities along geodesic performed by different
observers are analysed pointing out an energy loss of the massive particles
similar to that giving the photon redshift. The examples of the kinematics on
the de Sitter expanding universe and a new Milne-type space-time are
extensively analysed.
