Time-of-flight spectra (TOF) of supersonic He and D-2 beams in the energy range 100 less than or equal to E-i less than or equal to 250 meV have been measured after scattering from a clean Cu(001) surface at surface temperatures between 100 and 950 K. The TOF spectra of both: He and D-2 exhibit broad featureless distributions over the whole range of incident beam energies and surface temperatures. The intensities of the He TOF spectra are a factor of 5 to 7 higher than those of D-2 when the incident beam energies are the same and below 200 meV. For the highest incident beam energies E-i greater than or equal to 200 meV and surface temperatures T-s > 700 K the difference between the He and D-2 TOF spectra reduces to about a factor of 3. A theoretical model is employed which reproduces the TOF spectra to a very good approximation. The comparison of the best-fit parameters for He and D-2 provides valuable information on the interaction parameters and their dependence on incident energy. The analysis of the energy and temperature dependence of the peak intensities of the D-2 TOF spectra allows for the separation of the contribution of rotational excitations in the collision mechanism.