We investigate the velocity vs. position phase space of z ~ 1 cluster galaxies using a set of 424 spectroscopic redshifts in 9 clusters drawn from the GCLASS survey. Dividing the galaxy population into three categories: quiescent, star-forming, and poststarburst, we find that these populations have distinct distributions in phase space. Most striking are the poststarburst galaxies, which are commonly found at small clustercentric radii with high clustercentric velocities, and appear to trace a coherent ``ring in phase space. Using several zoom simulations of clusters we show that the coherent distribution of the poststarbursts can be reasonably well-reproduced using a simple quenching scenario. Specifically, the phase space is best reproduced if satellite quenching occurs on a rapid timescale (0.1 < tau_{Q} < 0.5 Gyr) after galaxies make their first passage of R ~ 0.5R_{200}, a process that takes a total time of ~ 1 Gyr after first infall. We compare this quenching timescale to the timescale implied by the stellar populations of the poststarburst galaxies and find that the poststarburst spectra are well-fit by a rapid quenching (tau_{Q} = 0.4^{+0.3}_{-0.4} Gyr) of a typical star-forming galaxy. The similarity between the quenching timescales derived from these independent indicators is a strong consistency check of the quenching model. Given that the model implies satellite quenching is rapid, and occurs well within R_{200}, this would suggest that ram-pressure stripping of either the hot or cold gas component of galaxies are the most plausible candidates for the physical mechanism. The high cold gas consumption rates at z ~ 1 make it difficult to determine if hot or cold gas stripping is dominant; however, measurements of the redshift evolution of the satellite quenching timescale and location may be capable of distinguishing between the two.
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