(a) An open railroad coal car of mass \(M\) is rolling along a horizontal track at velocity \(v_{0}\) when a coal chute suddenly dumps a load of coal of mass \(m\) into the coal car, vertically in the frame of the ground. When the load of coal has come to rest relative to the coal car, how fast is the coal car moving?

(b) A similar coal car, of mass \(M\) and velocity \(v_{0}\), has a covered hole in its bottom; when the cover is suddenly opened a mass \(m\) of coal falls onto the tracks. How fast is the coal car then moving?