  We put forward a new fiber optic sensor for measuring linear velocity with
picometer/second sensitivity with Weak-value amplification based on generalized
Sagnac effect [Phys. Rev. Lett.\textbf{93}, 143901(2004)].The generalized
Sagnac effect was first introduced by Yao et al, which included the Sagnac
effect of rotation as a special case and suggested a new fiber optic sensor for
measuring linear motion with nanoscale sensitivity. By using a different scheme
to perform the Sagnac interferometer with the probe in momentum space, we have
demonstrated the new weak measure protocol to detect the linear velocity by
amplifying the phase shift of the generalized Sagnac effect. Given the maximum
incident intensity of the initial spectrum, the detection limit of the
intensity of the spectrometer, we can theoretically give the appropriate
pre-selection, post-selection, and other optical structures before the
experiment. Our numerical results show our scheme with Weak-value amplification
is effective and feasible to detect linear velocity with picometer/second
sensitivity which is three orders of magnitude smaller than the result
$\nu$=4.8 $\times$ $10^{-9}$ m/s obtained by generalized Sagnac effect with
same fiber length.
