The authors show that a (simple, nonholonomic) motorcycle can exactly track a large class of smooth trajectories in the plane. Instability and nontrivial dynamic coupling make the exploration of aggressive motorcycle trajectories a rather challenging task. Previously (Hauser et al., 2004), we developed optimization techniques for constructing a suitable roll trajectory that (approximately) implements the desired plane trajectory. In that work, we found that the tracking error is usually quite small leading to the natural question: Given a smooth trajectory in the plane, does there exist a bounded roll trajectory that allows a simple motorcycle model to exactly track the plane trajectory? In this paper, we develop a technique for proving that such exact tracking is possible and apply it to a number of example cases. Our technique is based on the nonlinear system inversion work of Devasia and Paden (1998). Indeed, our algorithm is in the class that they propose. Unfortunately, we have been unable to directly use their results as the motorcycle system does not appear to satisfy the specific conditions required.