Pursuing further our work on Born Reciprocal (Non-inertial) Relativity theory we find that the family of hyperbolic trajectories of particles living in a four-dim phase space of signature $ (2,2)$ automatically satisfy the maximal speed of light and the maximal proper force conditions. A direct link between the maximal proper force $ b $ and the minimum value of the initial positions $ (x_o)_{min}$ of these hyperbolic paths is found. Furthermore, when the maximal proper force is set to be equal to the Planck-mass squared $ b = m_P^2$ then $ (x_o)_{min} = 2 G {cal M } $ happens to coincide with the numerical value of the horizon radius of the Schwarzschild black hole in four spacetime dimensions. We proceed with a rigorous analysis of how open strings sweeping a two-dim world-sheet region inside a Rindler wedge in a $ D = 1 + 1 $ spacetime can be seen as a continuum of point masses that are being collectively accelerated along the Rindler hyperbolic trajectories. The upshot of this analysis reveals that these strings can naturally acquire a $variable$ tension which is consistent with the dynamical tension model of strings and branes proposed by Guendelman in the past years cite{Eduardo}. We finalize by analyzing strings moving in phase space backgrounds within the context of non-inertial relativity theory.