This study adopts a curvature dynamics approach to understand and predict the trajectory of an idealized depth-averaged barotropic outflow onto a slope in shallow water. A novel equation for streamwise curvature dynamics was derived from the barotropic vorticity equation and applied to a momentum jet subject to bottom friction, topographic slope, and planetary rotation. The terms in the curvature dynamics equation have a natural geometric interpretation whereby each physical process can influence the flow direction. It is shown that a weakly spreading jet onto a steep slope admits the formulation of a 1D ordinary differential equation system in a streamline coordinate system, yielding an integrable ordinary differential equation system that predicts the kinematical behavior of the jet. The 1D model was compared with a set of high-resolution idealized depth-averaged circulation model simulations where bottom friction, planetary rotation, and bottom slope were varied. Favorable performance of the 1D reduced physics model was found, especially in the near field of the outflow. The effect of nonlinear processes such as topographic stretching and bottom torque on the fate of the jet outflow is explained using curvature dynamics. Even in the tropics, planetary rotation can have a surprisingly strong influence on the near-field deflection of an intermediate-scale jet, provided that it flows across steep topography.