Impedance Matching Analysis A Case of Simple 2-Particle Elastic Collision

This study investigates impedance matching in simple two-particle elastic collisions by analytically relating kinetic energy transfer to particle velocities, masses, and the restitution coefficient. Using Newton’s laws and work–energy principles, the authors derive an impedance matching efficiency model expressed in terms of velocity ratios (v₂/v₁, v₂′/v₁, v₁′/v₁), mass ratio (m₂/m₁), and restitution coefficient (e). They show that the efficiency of kinetic energy transfer is positively influenced by the ratios v₂/v₁ and v₂′/v₁ but negatively influenced by v₁′/v₁. For the special case where particle 2 is initially at rest, the model simplifies and reveals that only two velocity ratios control the impedance matching behavior. The results demonstrate that 100% kinetic energy transfer (maximum impedance matching) occurs only when the colliding and target particles have equal mass (m₂/m₁ = 1), the collision is perfectly elastic (e = 1), and the velocity of particle 1 is unchanged in magnitude before and after collision (v₁′/v₁ = 1). The analysis is supported by curve studies using Microsoft Excel and sensitivity analyses, clarifying how parameter variations modify energy transfer efficiency in mechanical collision systems.