Recent advances in sensing and machine learning technologies have paved the way for the belief that safe, accessible, and convenient autonomous vehicles may be realized in the near future. Despite tremendous advances within this context, the fundamental challenge generally regarded as limiting the arrival and comprehensive adoption of autonomous systems are challenges around safety and reliability. Autonomous vehicles are often tasked with operating in dynamic and uncertain environments. As a result, they often make use of highly complex components such as machine learning components to handle the nuances of sensing, actuation, and control. While these methods are highly effective, they are notoriously difficult to assure. Moreover, within uncertain and dynamic environments, design time assurance analyses may not be sufficient to guarantee safety. Thus, it is critical to monitor the correctness of these systems at runtime. One approach for providing runtime assurance of systems with components that may not be amenable to formal analysis is the simplex architecture, where an unverified component is wrapped with a safety controller and a switching logic designed to prevent dangerous behavior. In this paper, we propose the use of a real-time reachability algorithm for the implementation of such an architecture for the safety assurance of a 1/10 scale open source autonomous vehicle platform known as F1/10. The reachability algorithm (a) provides provable guarantees of safety, and (b) is used to detect potentially unsafe scenarios. In our approach, the need to analyze the underlying controller is abstracted away, instead focusing on the effects of the controller's decisions on the system's future states. We demonstrate the efficacy of our architecture through a vast set of experiments conducted both in simulation and on an embedded hardware platform.