For example, lets assume there is an elastic collision between two balls. The first ball has a mass of 2 kg and an initial velocity of 4 m/s. The second ball has a mass of 4 kg and is at rest so it has an initial velocity of 0. Let's assume that after the collision, the first ball stops, and the second ball starts moving.
Then according to the law of conservation of momentum, the second ball should have a velocity of 2 m/s. But in order for the collision to be elastic, the second ball should have a velocity of √8 m/s.
How can both laws be true if they give different values for an elastic collision?