Infrared Dark Clouds (IRDCs) host the initial conditions under which massive stars and stellar clusters form. We have obtained high sensitivity and high spectral resolution observations with the IRAM 30m antenna, which allowed us to perform detailed analysis of the kinematics within one IRDC, G035.39-00.33. We focus on the 1-0 and 3-2 transitions of N2H+, C18O (1-0), and make comparison with SiO (2-1) observations and extinction mapping. Three interacting filaments of gas are found. We report large-scale velocity coherence throughout the cloud, evidenced through small velocity gradients and relatively narrow line widths. This suggests that the merging of these filaments is somewhat "gentle", possibly regulated by magnetic fields. This merging of filaments may be responsible for the weak parsec-scale SiO emission detected by Jimenez-Serra et al. 2010, via grain mantle vaporization. A systematic velocity shift between the N2H+ (1-0) and C18O (1-0) gas throughout the cloud of 0.18 +/- 0.04 kms^{-1} is also found, consistent with a scenario of collisions between filaments which is still ongoing. The N2H+ (1-0) is extended throughout the IRDC and it does not only trace dense cores, as found in nearby low-mass star-forming regions. The average H2 number density across the IRDC is ~ 5 x 10^4 cm^{-3}, at least one order of magnitude larger than in nearby molecular clouds where low-mass stars are forming. A temperature gradient perpendicular to the filament is found. From our study, we conclude that G035.39-00.33 (clearly seen in the extinction map and in N2H+) has been formed via the collision between two relatively quiescent filaments with average densities of ~ 5 x 10^3 cm^{-3}, moving with relative velocities of ~ 5 kms^{-1}. The accumulation of material at the merging points started > 1 Myr ago and it is still ongoing. ; Comment: 20 pages, 19 figures, accepted for publication in MNRAS