  We study the linear response of a coherently driven polariton fluid in the
pump-only configuration scattering against a point-like defect and evaluate
analytically the drag force exerted by the fluid on the defect. When the system
is excited near the bottom of the lower polariton dispersion, the sign of the
interaction-renormalised pump detuning classifies the collective excitation
spectra in three different categories [C. Ciuti and I. Carusotto, physica
status solidi (b) 242, 2224 (2005)]: linear for zero, diffusive-like for
positive, and gapped for negative detuning. We show that both cases of zero and
positive detuning share a qualitatively similar crossover of the drag force
from the subsonic to the supersonic regime as a function of the fluid velocity,
with a critical velocity given by the speed of sound found for the linear
regime. In contrast, for gapped spectra, we find that the critical velocity
exceeds the speed of sound. In all cases, the residual drag force in the
subcritical regime depends on the polariton lifetime only. Also, well below the
critical velocity, the drag force varies linearly with the polariton lifetime,
in agreement with previous work [E. Cancellieri et al., Phys. Rev. B 82, 224512
(2010)], where the drag was determined numerically for a finite-size defect.
