Question

In an elastic collision,
energy is conserved.

Answer 1

Mostly true — with a clarification.

In a perfectly elastic collision (and assuming an isolated system, no external forces), total kinetic energy is conserved. Momentum is also conserved. In one dimension for two masses m1, m2 with initial velocities v1, v2 and final velocities v1', v2':

- Momentum: m1 v1 + m2 v2 = m1 v1' + m2 v2'
- Kinetic energy: 1/2 m1 v1^2 + 1/2 m2 v2^2 = 1/2 m1 v1'^2 + 1/2 m2 v2'^2

Solving these gives the standard results
v1' = [(m1 - m2)/(m1 + m2)] v1 + [2 m2/(m1 + m2)] v2
v2' = [2 m1/(m1 + m2)] v1 + [(m2 - m1)/(m1 + m2)] v2

A compact physical statement: in the center-of-mass frame the velocities are simply reversed (relative velocity changes sign): v1' - v2' = −(v1 - v2).

Caveats:
- "Energy is conserved" is always true in the sense of total energy (including internal, thermal, potential, etc.). For an elastic collision specifically the kinetic energy is conserved; in inelastic collisions some kinetic energy is converted into other forms (heat, deformation, rotation).
- Conservation requires an isolated system with no external work or nonconservative forces doing net work.