Several embodiments of accelerating fuel systems and methods for delivering accelerating fuel to an internal combustion engine over an extended time period. In each embodiment, the accelerating pump delivers a portion of its discharge fuel to a storage chamber and the storage chamber redelivers the fuel to the engine after the completion of the pumping stroke of the accelerating pump. In some embodiments of the invention, the stored fuel is returned to the engine by gravity. In some embodiments, the stored fuel is delivered to the engine at a different rate than the rate at which it is stored. In some embodiments, the return rate is dependent upon the change of speed of the engine and in other embodiments, the return rate is responsive to the acceleration of the vehicle powered to the engine.