My own field is electrical engineering and I often translate problems such as these into an electrical analogy when seeking a simple solution.
A moving mass is equivalent to an inductor having zero loss resistance, with its ends connected together and with a circulating current. Rather like a superconductor magnet. To represent a collision, we can connect two of these together and then suddenly cut a common shorting wire so the current now passes through both. The momentum is conserved, L1I1 L2I2 = L3I3. However, energy is lost. The question arises, where does it go? If there is no radiation of EM waves, it dissipates as heat. To maximise radiation, we would make the circuit physically large, so that radiation resistance appears and dissipates some of the energy. We might also add capacitance to the circuit to obtain a damped sine wave oscillation. The radiation resistance of structures can be calculated - that is antenna engineering. The same happens for colliding masses. The size and shape of the physical structure will dictate how much energy is radiated by coupling to the air. There is the possibility of a damped sine wave oscillation caused by springiness in the system. But I don't think that in general it will be practicable to calculate the acoustic radiation resistance for something like colliding balls and to predict the sound energy radiated.