In the field of marine robotics, the problem of range based underwater target localization can be defined as that of localizing an unknown - fixed or moving - target from a surface vehicle called the tracker, using range information available about the target. In this set-up, the motion of the tracker (relative to the target) has a significant effect on its ability to localize the target. In general, without any prior knowledge about the target motion, this is a very challenging problem and one that requires, at the outset, a thorough observability analysis of the model adopted for localization system design. For a restricted class of target motions generated using constant linear speed and piecewise constant course rates throughout a given observation window, the target's initial position, constant speed, initial course angle, and course rate, completely characterize its motion. The conditions under which one can estimate any combinations of these four parameters using range information has not been fully studied, except for a simple isolated case where the initial target position is the only unknown parameter. In this paper, using range information, we address the observability properties of the target localization problem for a target moving along a straight line or an arc of a circumference with constant linear and angular speeds on a given observation window. In the first case there are three target parameters (initial position, linear speed, and course angle), while in the second case there are four (initial position, linear speed, angular speed, and course angle). To make the problem mathematically tractable, we consider simple kinematics models for the tracker (we assume a constant linear speed) and target with planar position and course angle as state variables. Under these conditions, for the aforementioned two cases, we investigate the observability properties of range-based target localization problem for various target parameter combinations. In particular, when the target is moving along a straight line, for most of the unknown target parameter combinations, we show that observability can be achieved by a nonzero constant tracker's course rate, which is a simple condition. In the case where the target moves along a circular path, with the knowledge of the angular speed, we derive sufficient conditions on the trackers input to achieve observability, which are more demanding and less straightforward, and do not lend themselves to a simple geometrical explanation.