In an elastic collision both momentum and kinetic energy are conserved.

\displaystyle m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'

\displaystyle \tfrac{1}{2}m_1v_1^2 + \tfrac{1}{2}m_2v_2^2 = \tfrac{1}{2}m_1v_1'^2 + \tfrac{1}{2}m_2v_2'^2

Elastic collision — internal kinetic energy (the sum of all the KEs) is the same before and after.
Perfectly elastic collisions happen mainly between atoms and subatomic particles. Air-track carts with spring bumpers and steel blocks on ice come close.
Shortcut when m₂ starts at rest (follows from the two equations above):

\displaystyle v_1' = \frac{m_1 - m_2}{m_1 + m_2}\,v_1

\displaystyle v_2' = \frac{2m_1}{m_1 + m_2}\,v_1

Case (target at rest)	What happens
Equal masses	The first object stops; the second leaves at the first one's speed (Newton's cradle)
Light hits heavy	Light one bounces back; heavy one moves forward slowly
Heavy hits light	Heavy one keeps going slightly slower; light one shoots forward at up to 2v

Worked example (Ex. 8.4): 0.500 kg at 4.00 m/s hits 3.50 kg at rest. v₁′ = (0.500 − 3.50)/4.00 × 4.00 = −3.00 m/s and v₂′ = 2(0.500)/4.00 × 4.00 = 1.00 m/s. Check: KE = 4.00 J and p = 2.00 kg·m/s both before and after.