Superoperator methods for calculating cross sections. IV

A decoupled motions approximation to the time evolution of binary inelastic collisions is presented. The translational motion of the particles is described by a reference Hamiltonian while the internal motion is parameterized by the average trajectory associated with the translational motion. This description is applicable to both classical and quantum mechanics and any combination thereof. Thus it includes the standard impact parameter approximation. Cross sections for this decoupled evolution are derived from first principles. Explicit results for the impact parameter approximation are given