  The collapse of spherical neutron stars is studied in General Relativity. The
initial state is a stable neutron star to which an inward radial kinetic energy
has been added through some velocity profile. For two different equations of
state and two different shapes of velocity profiles, it is found that neutron
stars can collapse to black holes for high enough inward velocities, provided
that their masses are higher than some minimal value, depending on the equation
of state. For a polytropic equation of state of the form $p=K\rho^\gamma $,
with $\gamma = 2$ it is found to be $1.16 (\frac{K}{0.1})^{0.5} \msol$, whereas
for a more realistic one, it reads $0.36 \msol $. In some cases of collapse
forming a black hole, part of the matter composing the initial neutron star can
be ejected through a shock, leaving only a fraction of the initial mass to form
a black hole. Therefore, black holes of very small masses can be formed and, in
particular, the mass scaling relation, as a function of initial velocity, takes
the form discovered by Choptuik for critical collapses.
