We present the results from a study of the inelastic scattering of N-2 from water ice performed using time- and state-resolved molecular-beam and laser spectroscopic techniques. For low incident energies (90-300 meV/molecule), the scattering is entirely trapping-desorption. For incident energies above 300 meV/molecule, a direct scattering channel is observed in which the scattered molecules do not lose memory of their initial state. The final total energy (translational and rotational) of this direct channel is about 15% of the incident translational energy, remarkably low for a direct scattering process. We estimate that for an incident energy of 750 meV/molecule, 23% of the molecules scatter directly while the rest trap then desorb. By detailed analysis and modeling of the measured time-of-flight spectra, we place an upper limit of 2 mu s on the residence time of an N-2 molecule on a 100 K ice film. Experiments were performed off both amorphous and crystalline ice surfaces, with no measurable changes in scattering dynamics.