It is typically assumed that radiation pressure driven winds are accelerated to an asymptotic velocity of V ~ v_esc, where v_esc is the escape velocity from the central source. We note that this is not the case for dusty shells and clouds. Instead, if the shell or cloud is initially optically-thick to the UV emission from the source of luminosity L, then there is a significant boost in V that reflects the integral of the momentum absorbed as it is accelerated. For shells reaching a generalized Eddington limit, we show that V ~ (4R_UV L/M_sh c)^1/2, in both point-mass and isothermal-sphere potentials, where R_UV is the radius where the shell becomes optically-thin to UV photons, and M_sh is the mass of the shell. The asymptotic velocity significantly exceeds v_esc for typical parameters, and can explain the ~1000-2000km/s outflows observed from rapidly star-forming galaxies and active galactic nuclei if the surrounding halo has low gas density. Similarly fast outflows from massive stars can be accelerated on few - 10^3 yr timescales. These results carry over to clouds that subtend only a small fraction of the solid angle from the source of radiation and that expand as a consequence of their internal sound speed. We further consider the dynamics of shells that sweep up a dense circumstellar or circumgalactic medium. We calculate the momentum ratio Mdot v/(L/c) in the shell limit and show that it can only significantly exceed ~2 if the effective optical depth of the shell to re-radiated FIR photons is much larger than unity. We discuss simple prescriptions for the properties of galactic outflows for use in large-scale cosmological simulations. We also briefly discuss applications to the dusty ejection episodes of massive stars, the disruption of giant molecular clouds, and AGN.
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