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Lets say two bodies experience head on collision with each other. It is said in my textbooks that it is found that the ratio of relative velocities of the two bodies before and after collision is fixed for a certain combination objects. But no reasoning or proof is provided for this quantitative relation in the textbook.

Is there any proof or reasoning behind this relation?

CuriousOne
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    Such as? An example might clarify what you are asking. – Jon Custer Jan 27 '24 at 20:11
  • Lets say two tennis balls collided with each other and it is observed the ratio of the total initial kinetic energy and the total final kinetic energy of the balls are nearly fixed for all scenarios. My question is that is there any proof or reasoning to justify this relation instead of just empirical observation. – CuriousOne Jan 27 '24 at 20:42
  • In an elastic collision between two balls, kinetic energy is conserved. (In a real collision some is lost to, e.g., sound or heat.) – Jon Custer Jan 27 '24 at 20:46
  • yeah but can you show through reasoning that the initial and final kinetic energy of the objects in a collision would always bear nearly a fixed ratio for a certain combination of materials. – CuriousOne Jan 27 '24 at 21:20
  • Newton's cradle? One ball comes to an almost complete stop while the other ball picks up all of the kinetic energy and momentum. Total energy and momentum are conserved, but the ratios between energies and momenta of the two balls can be an almost arbitrary factor... which is also observer dependent. Not sure what you are asking. – FlatterMann Jan 27 '24 at 21:37
  • Related: https://physics.stackexchange.com/questions/398476/why-dont-all-objects-bounce-like-rubber-balls – Claudio Saspinski Jan 27 '24 at 21:43
  • My bad. The ratio of relative velocity of the objects before and after the collision is what is fixed in the event of collision of certain combination of objects which id also known as the coefficient of restitution. Could anyone explain the reason behind this phenomenon. – CuriousOne Jan 28 '24 at 08:22
  • This not a physical law. The difference between kinetic energy before and after an inelastic collision is the energy lost from inelastic deformation of the materials during the collision. The higher the energy of the collision the more deformation the more energy lost, so to a first approximation the energy loss is proportional to the initial energy, making the reduction a constant ratio. – Malcolm Jan 30 '24 at 15:13

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