The kinematics and dynamics of young stellar populations enable us to test theories of star formation. With this aim, we continue our analysis of the SDSS-III/APOGEE IN-SYNC survey, a high resolution near infrared spectroscopic survey of young clusters. We focus on the Orion A star-forming region, for which IN-SYNC obtained spectra of $sim2700$ stars. In Paper IV we used these data to study the young stellar population. Here we study the kinematic properties through radial velocities ($v_r$). The young stellar population remains kinematically associated with the molecular gas, following a $sim10:{rm{km:s}}^{-1}$ gradient along filament. However, near the center of the region, the $v_r$ distribution is slightly blueshifted and asymmetric; we suggest that this population, which is older, is slightly in foreground. We find evidence for kinematic subclustering, detecting statistically significant groupings of co-located stars with coherent motions. These are mostly in the lower-density regions of the cloud, while the ONC radial velocities are smoothly distributed, consistent with it being an older, more dynamically evolved cluster. The velocity dispersion $sigma_v$ varies along the filament. The ONC appears virialized, or just slightly supervirial, consistent with an old dynamical age. Here there is also some evidence for on-going expansion, from a $v_r$--extinction correlation. In the southern filament, $sigma_v$ is $sim2$--$3$ times larger than virial in the L1641N region, where we infer a superposition along the line of sight of stellar sub-populations, detached from the gas. On the contrary, $sigma_v$ decreases towards L1641S, where the population is again in agreement with a virial state.
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