In this paper the problem is addressed of modelling the encounter of two ships in a seaway from the dual points of view of collision-avoidance and pursuit-evasion maneuvers. The pronounced effect of speed-loss experienced by ships during a (fixed throttle) turn ins incorporated into the model. In the first part of the paper due attention is given to modelling issues germane to ship performance. During a fixed rudder turn the turning radius of a ship remains constant. This, in turn, allows us to employ the classical "game of two cars" and "homicidal chauffeur" kinematic models (which have been modified to include the bleeding off of speed during the turn). The methods of the theory of differential games are then employed to yield a solution to the qualitative (preliminary) problems of establishing the safe zone or the capture zone in collision avoidance or pursuit-evasion, respectively. We mainly analyze the simpler "homicidal chauffeur" model, but it also becomes evident how specific instances of the more difficult but also more realistic variable speed "game of two cars" formulation should be treated.