A novel reactive method for robot trajectory planning within unknown 2D workspaces is presented in this letter. The trajectories provided by this method stem from an underlying potential field and are provably safe, with asymptotic convergence to the desired position. Given an initially unknown workspace, and a sensing process for obtaining boundary information, the scope of this work is to provide an underlying vector field that can be used as a reference signal for higher order, non-linear dynamical systems (robots). This can be achieved by tracking, either the field itself, or the trajectories that result from the latter. In this spirit, our method is advantageous due to: a) its reactivity, as open-loop approaches may exhibit large errors during the transient phase within unknown workspaces, whereas our method provides a velocity profile for each position of the robot within the workspace, b) its smoothness, and c) its safety and convergence guarantees. We prove the asserted claims and provide rigorous comparative simulations in various benchmark workspaces, as well as in a high-fidelity ROS-Gazebo implementation.