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It is shown that the angular acceleration and velocity of a rigid body can be determined from translational material acceleration measurements made by an observer on the body. An important contribution is the fact is that this can be done without integration. Therefore, ideally, the solutions are obtained at any instant simultaneously. Also shown is that whereas the angular acceleration solution is unique, there always exists two solutions for the angular velocity that only differ in sign, which cannot be resolved any further. The necessary and sufficient number of points is proven to be four, provided that they are not coplanar. Numerical simulations, in which sensor errors are also considered, are conducted for verification.