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We introduce the notion of kinematic controllability for second-order underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collision-free trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by time-optimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collision-free trajectory planning for a broad class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.