This paper develops a novel computational method for the falsification of safety properties specified by syntactically safe linear temporal logic (LTL) formulas φ for hybrid systems with general nonlinear dynamics and input controls. The method is based on an effective combination of robot motion planning and model checking. Experiments on a hybrid robotic system benchmark with nonlinear dynamics show significant speedup over related work. The experiments also indicate significant speedup when using minimized DFA instead of non-minimized NFA, as obtained by standard tools, for representing the violating prefixes of φ.