Elastic collision of a system demands conservation of both momentum and kinetic energy. To solve problems involving one-dimensional elastic collisions between two objects, the equations for conservation of momentum and conservation of internal kinetic energy can be used. For the two objects, the sum of momentum before the collision equals the total momentum after the collision. An elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals the sum after the collision. We can understand this better with the help of a problem.

Let’s try to calculate the velocities of two objects following an elastic collision, given that m_A = 0.50 kg, m_B = 3.5 kg, v_A1x = 4.0 m/s, and v_B1x = 0 m/s.

The initial conditions imply that a small object strikes a larger object that is initially at rest. There are two unknowns (the final velocities v_A2x and v_B2x), which should be found. Because this collision is elastic, the equations for conservation of momentum and conservation of energy can be used. They are given by,

Both can be simplified because object B is initially at rest, and thus v_B1x = 0 m/s.

Substituting the known values into these equations, we obtain v_A2x = -3 m/s and v_B2x= 4 m/s. The negative sign indicates that the first object bounces backward. The result of this example is reasonably intuitive: a small object strikes a larger one at rest and bounces backward. The larger one is knocked forward but with a low speed. This is like a compact car bouncing backward off a full-size SUV that is initially at rest.

This text is adapted from Openstax, University Physics Volume 1, Section 9.4: Types of Collisions.