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Motivated by cooperative exploration missions, this paper considers constant velocity, level flight path planning for Unmanned Air Vehicles (UAVs) equipped with range limited, omni-directional sensors. These active energy-based sensors collect information about objects of interest at rates that depend on the range to the objects according to Shannon's channel capacity equation, where the signal-to-noise ratio is governed by the radar equation. The mission of the UAVs is to travel through a given area and collect a specified amount of information about each object of interest while minimizing the total mission time. This information can then be used to classify the objects of interest. An optimal path planning problem is formulated where the states are the Cartesian coordinates of the UAVs and the amounts of information collected about each object of interest, the control inputs are the UAV heading angles, the objective function is the total mission time, and the boundary conditions are subject to inequality constraints that reflect the requirements of information collection. Necessary conditions for optimality are given, whose solutions yield extremal paths, and whose utilization highlights analytical properties of these extremal paths. The problem exhibits several limiting regimes, including the so-called Watchtower and the Multi-Vehicle Traveling Salesman Problem. These results are illustrated on several time-optimal cooperative exploration scenarios.