Abstract: This disclosure describes an improved electrical power apparatus for assisting a rider in powering a wheeled vehicle such as a bicycle. The apparatus controllably connects a rechargeable battery to a dc motor engageable with a wheel of the vehicle in a predetermined fashion, such that the wheel is driven with a torque that varies with the vehicle's speed according to a predetermined non-uniform profile. This control advantageously can be achieved by modulating on and off a switch at a duty cycle between 0% and 100%, which is selected to provide a motor torque that decreases monotonically with the vehicle's speed, from a maximum value at zero speed to a predetermined intermediate value at a predetermined intermediate speed, and that remains at that predetermined intermediate value for speeds between the intermediate speed and a predetermined maximum speed. The apparatus further includes a charger for charging the battery using a flyback-type dc-to-dc converter.