The temporal evolutions of the planetesimals scattered from the Jupiter zone for different masses of the proto-Jupiter [(a) 0.1 and (b) 1.0 of the present mass] are investigated. Due to the combined effects of the orbital evolution of the planetesimals and the elimination of these projectiles either via impact capture or injection into escape velocity by the outer planets, the whole scattering process lasts about 108 yr for case (a) and about 107 yr for case (b). The longer time scale may be a good estimate for the accretion time interval of Jupiter while the shorter one (107) gives the upper time limit of the late heavy-bombardment epoch of the terrestrial planets due to planetesimals scattered from the Jupiter zone. The limiting value of the encounter velocity U at the end of the scattering process is ≈0.6. Consideration of the collisional interaction of these projectiles with the asteroids indicates that the corresponding bombardment effect could be rather appreciable. Also, the asteroids on the inner edge of the main asteroid belt would have been bombarded more severely than those on the outer edge. From this point of view, the structure of the asteroidal belt could be affected significantly not only by Jupiter's gravitational perturbation effect but also by its early scattering process.