Collisions between rigid bodies are impulsive events, meaning they create near-instantaneous, finite discontinuities in the velocity state of the dynamic system. If velocity is assumed to be continuous, that requirement may come into conflict with the desire for bodies to not interpenetrate, or with Coulomb friction (creating a condition known as the Painlevé paradox). This letter presents a contact-implicit framework for trajectory optimization using direct collocation that combines an unplanned impulsive contact model with an implicit, high-order numerical integration scheme. We demonstrate that this framework can resolve problems that other implicit formulations cannot, namely elastic or partially elastic collisions, and impacts without collision – the established resolution of the Painlevé paradox. We then evaluate its applicability to legged robotics problems and test its ability to discover the intricate sequence of varied collision types necessary to execute a skateboarding trick.