We explore how high-speed robot arm motions can dynamically manipulate cables to vault over obstacles, knock objects from pedestals, and weave between obstacles. In this paper, we propose a self-supervised learning framework that enables a UR5 robot to perform these three tasks. The framework finds a 3D apex point for the robot arm, which, together with a task-specific trajectory function, defines an arcing motion that dynamically manipulates the cable to perform tasks with varying obstacle and target locations. The trajectory function computes minimum-jerk motions that are constrained to remain within joint limits and to travel through the 3D apex point by repeatedly solving quadratic programs to find the shortest and fastest feasible motion. We experiment with 5 physical cables with different thickness and mass and compare performance against two baselines in which a human chooses the apex point. Results suggest that a baseline with a fixed apex across the three tasks achieves respective success rates of 51.7%, 36.7%, and 15.0%, and a baseline with human-specified, task-specific apex points achieves 66.7%, 56.7%, and 15.0% success rate respectively, while the robot using the learned apex point can achieve success rates of 81.7% in vaulting, 65.0% in knocking, and 60.0% in weaving. Code, data, and supplementary materials are available at https: //sites.google.com/berkeley.edu/dynrope/home.