How does a falling cat change her orientation in midair without violating angular momentum constraint? This has become an interesting problem to both control engineers and roboticists. In this paper, we address this problem together with a constructive solution. First, we show that a falling cat problem is equivalent to the constructive nonlinear controllability problem. Thus, the same principle and algorithm used by a falling cat can be used for space robotic applications, such as reorientation of a satellite using rotors and attitude control of a space structure using internal motion, and other robotic tasks, such as dextrous manipulation with multifingered robotic hands and nonholonomic motion planning for mobile robots. Then, using ideas from Ritz Approximation Theory, we develop a simple algorithm for motion planning of a falling cat. Finally, we test the algorithm through simulation on two widely accepted models of a falling cat. It is interesting to note that one set of simulation results closely resembles the real trajectories employed by a falling cat.