In the field of control engineering, the connection between Signal Temporal Logic (STL) and time-varying Control Barrier Functions (CBF) has attracted considerable attention. CBFs have demonstrated notable success in ensuring the safety of critical applications by imposing constraints on system states, while STL allows for precisely specifying spatio-temporal constraints on the behavior of robotic systems. Leveraging these methodologies, this paper addresses the safety-critical navigation problem, in Socially Responsible Navigation (SRN) context, presenting a CBF -based STL motion planning methodology. This methodology enables task completion at any time within a specified time interval considering a dynamic system subject to velocity constraints. The proposed approach involves real-time computation of a smooth CBF, with the computation of a dynamically adjusted parameter based on the available path space and the maximum allowable velocity. A simulation study is conducted to validate the methodology, ensuring safety in the presence of static and dynamic obstacles and demonstrating its compliance with spatio-temporal constraints under non-linear velocity constraints.