Since we use the same exact formula for perfectly inelastic collisions (in which 2 particles stick together after collision) and perfectly elastic ones. Why does one of them conserve energy and the other not.

There is no coefficient of restitution either to quantify the energy losses.

Edit: I know that all energy is conserved in the big picture,, I’m talking about the particles’ energy.

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Both elastic and inelastic collisions actually use 2 formulas. Only one of them are the same.

The first formula is the “conservation of momentum”, and it is indeed the same. However, this formula gives infinitely many solutions. We need an additional constraint. Elastic and inelastic collisions use different second constraint:

* Perfect elastic says “energy must be conserved”, so it obviously conserve energy. That limits to just 2 solutions, one of which is always the initial state. And because we know that bodies cannot stay in that state (because they are colliding) – the other solution must be the final one.

* Perfect inelastic says “final speeds are the same”. That limits to 1 solution.

That difference in the second constraint makes two collisions different.

Inelastic collisions don’t preserve KINETIC energy. Overall energy is conserved because the kinetic energy loss is converted to heat or sound.