Dynamic modeling of folding joints is critical for predicting dynamic behavior, optimizing design parameters, and developing control strategies for origami robots and machines. Although kinematics of the folded joints exists, little research describes their dynamics. Currently, prevailing models neglect the stiffness of the hinges by making zero-thickness assumption or ignore physical factors such as gravity and friction. In this work, we focus on the dynamic modeling of an origami prismatic joint with rotary-to-translational transmission by using the Newton-Euler method and pseudo-rigid-body-approximation. We provide a comprehensive dynamic model by including gravity, friction, and hinge parameters. We validate the model in an all-inclusive experimental setup addressing static, quasi-static, and dynamic conditions. Our proposed model successfully predicts the dynamics and structural stiffness of the joint. This novel model can be combined with other origami-joint models, such as pin and spherical joint models, to allow model-based design and control strategies for desired output performance of origami robots.