  We employ the Gross--Pitaevskii theory to model a quantized vortex depinning
from a small obstacle in a two-dimensional superfluid due to an imposed
background superfluid flow. We find that, when the flow's velocity exceeds a
critical value, the vortex drifts orthogonally to the flow before subsequently
moving parallel to it away from the pinning site. The motion of the vortex
around the pinning site is also accompanied by an emission of a spiral-shaped
sound pulse. Through simulations, we present a phase diagram of the critical
flow velocity for vortex depinning together with an empirical formula that
illustrates how the critical velocity increases with the height and width of
the pinning site. By employing a variety of choices of initial and boundary
conditions, we are able to obtain lower and upper bounds on the critical
velocity and demonstrate the robustness of these results.
