QUASARS can radiate up to a thousand times the energy of the entire Galaxy, yet this energy is generated in a small region approximately one light day across. (By comparison, the diameter of the Milky Way is about 100,000 light years.) Because of tbe high energy density in this region, it has often been suggested that radiation pressure might play an important dynamical role in quasars(1-3). Here we show that radiative acceleration can readily explain a prominent feature observed in the spectra of several broad-absorption-line (BAL) quasars. The broad absorption lines are themselves attributed to matter flowing towards the observer with velocities approaching one-tenth of the speed of light(4,5), and our results suggest that radiative acceleration is the dominant driving mechanism in these outflows. As most quasars are believed to have BAL outflows(6) (although they are seen in only about 10% of them because of viewing angle7,8), radiative acceleration is likely to be an important dynamical process in quasars in general.