All subjects · Physics · WPH14 (Unit 4)

Elastic and inelastic collisions; E_k = p2/2m

Momentum is always conserved when no external force acts, but kinetic energy is not. Checking the kinetic energy tells you what kind of collision happened.

Elastic or inelastic?

Total energy and momentum are conserved in every case. To decide, calculate the total E_k before and after.

Kinetic energy from momentum

Since p = mv, v = p / m. Substituting into E_k = (1/2)mv2:

E_k = (1/2)m(p/m)2 = p2 / 2m

This works for non-relativistic particles (speeds much less than c). It is useful in particle physics, where momentum is often what you know.

A useful consequence

Two objects with the same momentum do not have the same kinetic energy: the one with the smaller mass has the larger E_k. In an explosion from rest, the lighter fragment carries away more of the kinetic energy.

Worked example

A 2.0 kg trolley at 3.0 m/s hits a stationary 1.0 kg trolley and they stick together. Is the collision elastic?

  1. Momentum: 6.0 = 3.0v, so v = 2.0 m/s
  2. E_k before = 0.5 x 2.0 x 3.02 = 9.0 J
  3. E_k after = 0.5 x 3.0 x 2.02 = 6.0 J

Answer: Inelastic: 3.0 J of kinetic energy is transferred to other stores

Key idea: Elastic: E_k conserved. Inelastic: E_k falls. Momentum always conserved. E_k = p2/2m.

Check yourself

  1. An object of mass 4.0 kg has momentum 12 kg m/s. Find its kinetic energy in J.
    Show answer

    18 J. E_k = p2 / 2m = 144 / 8.0 = 18 J.

  2. An electron has momentum 2.0 x 10-24 kg m/s. Find its kinetic energy in eV, to 3 s.f.
    Show answer

    13.7 eV. E_k = (2.0 x 10-24)2 / (2 x 9.11 x 10-31) = 2.20 x 10-18 J; divide by 1.60 x 10-19 to get 13.7 eV.

  3. A proton has kinetic energy 3.2 x 10-13 J. Find its momentum, in units of 10-20 kg m/s, to 3 s.f.
    Show answer

    3.27 x 10-20 kg m/s. p = sqrt(2mE_k) = sqrt(2 x 1.67 x 10-27 x 3.2 x 10-13) = 3.27 x 10-20 kg m/s.

  4. A 0.50 kg trolley at 6.0 m/s hits a stationary 1.0 kg trolley and they stick together. How much kinetic energy is lost, in J?
    Show answer

    6 J. v = 3.0 / 1.5 = 2.0 m/s. E_k before = 9.0 J; after = 0.5 x 1.5 x 4.0 = 3.0 J; lost = 6.0 J.

  5. A 1.0 kg ball at 4.0 m/s hits a stationary 3.0 kg ball. Afterwards the 1.0 kg ball rebounds at 2.0 m/s and the 3.0 kg ball moves forward at 2.0 m/s. What type of collision is this?
    • Inelastic
    • Totally inelastic
    • Elastic
    • Impossible, because momentum is not conserved
    Show answer

    Elastic. Momentum: 4.0 = -2.0 + 6.0. E_k before = 8.0 J; after = 2.0 + 6.0 = 8.0 J. Kinetic energy is conserved.

  6. In a collision with no external forces, which quantity is always conserved?
    • Total momentum
    • Total kinetic energy
    • The speed of each object
    • The velocity of each object
    Show answer

    Total momentum. Momentum is conserved in every collision; kinetic energy only in elastic ones.

  7. A light particle and a heavy particle have the same momentum. Which has more kinetic energy?
    • The heavy particle
    • They have the same kinetic energy
    • The light particle
    • It depends on their directions
    Show answer

    The light particle. E_k = p2 / 2m, so with p fixed a smaller m gives a larger E_k.

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Last updated 7 November 2026. Spotted a mistake? Tell us.