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I still think most people are missing the reason for the question about
exactly 1/2 of the KE being transferred in a perfectly inelastic collision
between same mass objects, one at rest.
The question, as I read it, is HOW is exactly 1/2 transferred? Restating
the math of momentum conservation or stating that no real-life situation
actually behaves this way, is not, if I interpret the question properly, an
answer. Here is what I imagine a student thinking:
"A ball is moving along at speed v, has momentum mv, and KE .5mv^2. Now
there is this collision, and after the collision, the math says that only
half of the energy is KE. But where did the rest of the energy go? Heat,
sound, deformation--OK, but HOW does exactly 1/2 of the original energy find
its way into these other channels, none of which are very well defined nor
necessarily exactly the same from collision to collision."
Rick