An acceleration control system controls acceleration of a vehicle to match the driver's feeling. In this control, the vehicle is controlled so that acceleration remains constant at the initial stage and then a differentiated value of a square power of a vehicle speed remains constant. This control is based on the finding of a normal acceleration operation attained by a driver. A linear relation exists between the acceleration in the initial stage of acceleration and the differentiated value of the square power of the speed. Switching of control from the constant acceleration to the constant differentiated value of the square power of the speed is made at a change-over speed determined by the linear relation.