We have measured the polarizations of J/psi and psi(2S) mesons as functions of their transverse momentum p_{T} when they are produced promptly in the rapidity range |y|<0.6 with p_{T}>/=5 GeV/c. The analysis is performed using a data sample with an integrated luminosity of about 800 pb;{-1} collected by the CDF II detector. For both vector mesons, we find that the polarizations become increasingly longitudinal as p_{T} increases from 5 to 30 GeV/c. These results are compared to the predictions of nonrelativistic quantum chromodynamics and other contemporary models. The effective polarizations of J/psi and psi(2S) mesons from B-hadron decays are also reported.
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