Heavy-particle cross sections differential in the momentum transferred to the target are investigated using the classical trajectory Monte Carlo method. with the 3.6  MeV/u Au53++He system as a test case, it is shown that these cross sections are extremely sensitive to the initial target temperature. In particular, when thermal motion is varied for one of the target's initial momentum components between 0 and 25 K the absolute cross sections vary by orders of magnitude and, in addition, their relative shapes undergo major changes. We find that by setting one of the target's transverse momenta to a temperature of 16 K, previously reported major discrepancies between theory and experiment are removed.