Inverse scattering methods based on exactly solvable models have been used to determine specific momentum dependent nucleon-nucleon potentials from nucleon-nucleon phase shifts in uncoupled partial waves. The novel method has been applied to the 1S 0 channel. The explicit momentum dependencies considered are restricted to those of the well known parametrisations of the Paris and the Nijmegen potentials which consist of a momentum independent term and one linear in the square of the momentum. The interplay between momentum dependent and independent parts is demonstrated, generating phase equivalent potentials to the Paris potential with different momentum dependencies. Furthermore the sensitivity of the triton binding energy on the momentum dependence has been studied.