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I don't know how others may treat it, but I look at the collision in the center of mass frame, and define a totally inelastic collision to be one in which the post-collision kinetic energy is zero. A totally elastic collision is one in which the post-collision k.e. is the same as the pre-collision value. Anything in between, where the post-collision k.e. is somewhere between zero and the pre-collision value is called either partially elastic or partially inelastic (usually the former, since the latter isn't really great English).
I maybe haven't read this thread carefully enough, but could someone define
what a 'totally' inelastic collision actually is? In my mind, an elastic
collision conserves both momentum and kinetic energy while an inelastic
collision conserves momentum but not KE. It's the 'totally' that is
confusing me here.