The behaviour of the angular velocity of a test particle moving in a stable circular orbit in the vacuum on the brane is considered. In the braneworld scenario, the four-dimensional effective Einstein equation acquires extra terms, called dark radiation and dark pressure, respectively, which arise from the embedding of the 3-brane in the bulk. A large number of independent observations have shown that the rotational velocities of test particles gravitating around galaxies tend, as a function of the distance from the galactic centre, towards constant values. By assuming a constant tangential velocity, the general solution of the vacuum gravitational field equations on the brane can be obtained in an exact analytic form. This allows us to obtain the explicit form of the projections of the bulk Weyl tensor on the brane, and the equation of state of the dark pressure as a function of the dark radiation. The physical and geometrical quantities are expressed in terms of observable/measurable parameters, such as the tangential velocity, the baryonic mass and the radius of the galaxy. We also analyse the dynamics of test particles by using methods from the qualitative analysis of dynamical systems, by assuming a simple linear equation of state for the dark pressure. The obtained results provide a theoretical framework for the observational testing at the extra-galactic scale of the predictions of the braneworld models.