In a rack and pinion type motor-driven power steering apparatus, the motor-driven power steering apparatus is provided with a rack position detecting means for detecting a position of the rack, a rack speed arithmetically operating means for calculating a moving speed (s) of the rack, a memory means for storing a relation between a predetermined rack speed (s) near a stroke end of the rack and a motor damping control start rack position in which a distance from the stroke end is farther in accordance with the rack speed (s) as a relation curve (C) of a coordinate map A collating means is provided for collating with the relation curve (C) between the rack speed (s) stored in the memory means and the motor damping control start rack position so as to determine whether or not a point coordinate (s, p) of the combination between the rack speed (s) calculated by the rack speed arithmetically operating means and the rack position (p) detected by the rack position detecting means gets over the relation curve (C). A motor drive control means is provided for controlling the motor in a damping state on the basis of a predetermined motor assist decreasing rate (As) where the collating means determines that the point coordinate (s, p) of the combination gets over the relation curve (C).