An analytical stability analysis of the steady trajectory for a surface vessel with various damping models is presented in this work. The analysis is based on a control-oriented, three degrees-of-freedom model that considers vessel motion only in the horizontal plane. The goal of this study is to understand the vessel trajectories predicted by this reduced order model for model-based control design. Straight line and circular motion stability conditions for each trajectory are derived and presented for the various damping models. The results of this analysis show that either a straight line or a circular steady trajectory is possible, depending on the magnitude of the surge force and the form of the damping model used to represent viscous drag, vortex shedding, and losses due to the surface wake generated by the vessel motion. However, the straight line motion is much less likely for the vessel considered in this work.