Collisions
Collisions reveal why momentum and energy are complementary. Momentum conservation survives both elastic and inelastic collisions; kinetic-energy conservation does not.
What distinguishes an elastic collision from an inelastic one if momentum is conserved in both?
The difference is kinetic-energy accounting. In an inelastic collision, some kinetic energy becomes deformation, thermal, sound, or other internal energy.
Track both ledgers through the same collision.
Numerically zero apart from rounding in both collision models.
Zero for the elastic model.
For the sticking collision, this appears as deformation, thermal energy, sound, and other internal stores.
Momentum and kinetic energy are both conserved within the modeled system.
Momentum is conserved, but kinetic energy decreases as energy moves into other internal stores.
The objects leave with one shared velocity. This is the maximum kinetic-energy loss compatible with the initial momentum for two sticking bodies.