This paper focuses on the development and use of a nonlinear observer for tracking of vehicle motion trajectories on highways while using a radar or laser sensor. Previous results on vehicle tracking have typically used an interacting multiple model filter that needs different models for different modes of vehicle motion. This paper uses a single nonlinear vehicle model that can be used for all modes of vehicle motion. A corresponding exponentially stable nonlinear observer is needed. Previous nonlinear observer design results do not work for the nonlinear system under consideration. Hence, a new nonlinear observer that utilizes better bounds on the coupled nonlinear functions in the dynamics is developed. The observer design with the developed technique is implemented in both simulations and experiments. Experimental results show that the observer can simultaneously estimate longitudinal position, lateral position, velocity and orientation variables for the vehicle from radar measurements during highway driving.