The conservation laws for linear and angular momenta in a Schrödinger-Chern-Simons field theory modelling vortex dynamics in planar superconductors are studied. In analogy with fluid vortices it is possible to express the linear and angular momenta as low moments of vorticity. The conservation laws are shown to be consistent with those obtained in the moduli space approximation for vortex dynamics, valid close to the Bogomol'nyi limit. For Bogomol'nyi vortices, the relevant moments of vorticity can be evaluated fairly explicitly, as can the integral of log||², where is the scalar field. Conservation of angular momentum prevents a single vortex from escaping to infinity.