The effect of elastomeric damping pads, softening the collision of hard objects, is investigated comparing the reference silicone elastomer and the polydomain nematic liquid crystalline elastomer, which has a far superior internal dissipation mechanism. We specifically focus not just on the energy dissipation, but also on the momentum conservation and transfer during the collision, because the latter determines the force exerted on the target and/or the impactor—and it is the force that does the damage during the short time of an impact, while the energy might be dissipated on a much longer time scale. To better assess the momentum transfer, we compare the collision with a very heavy object and the collision with a comparable mass, when some of the impact momentum is retained in the target receding away from the collision. We also propose a method to estimate the optimal thickness of an elastomer damping pad for minimising the energy in impactor rebound. It has been found that thicker pads introduce a large elastic rebound and the optimal thickness is therefore the thinnest possible pad that does not suffer from mechanical failure. We find good agreement between our estimate of the minimal thickness of the elastomer before the puncture through occurs and the experimental observations.