  Motivated by recent observations that show starless molecular cloud cores
exhibit subsonic inward velocities, we revisit the collapse problem for
polytropic gaseous spheres. In particular, we provide a generalized treatment
of protostellar collapse in the spherical limit and find semi-analytic
(self-similar) solutions, corresponding numerical solutions, and purely
analytic calculations of the mass infall rates (the three approaches are in
good agreement). This study focuses on collapse solutions that exhibit nonzero
inward velocities at large radii, as observed in molecular cloud cores, and
extends previous work in four ways: (1) The initial conditions allow nonzero
initial inward velocity. (2) The starting states can exceed the density of
hydrostatic equilibrium so that the collapse itself can provide the observed
inward motions. (3) We consider different equations of state, especially those
that are softer than isothermal. (4) We consider dynamic equations of state
that are different from the effective equation of state that produces the
initial density distribution. This work determines the infall rates over a wide
range of parameter space, as characterized by four variables: the initial
inward velocity $\vin$, the overdensity $\overdense$ of the initial state, the
index $\Gamma$ of the static equation of state, and the index $\gamma$ of the
dynamic equation of state. For the range of parameter space applicable to
observed cores, the resulting infall rate is about a factor of two larger than
found in previous theoretical studies (those with hydrostatic initial
conditions and $\vin = 0$).
