All subjects · Physics · WPH14 (Unit 4)

Mass-energy, particle energy units and relativistic lifetimes

In particle physics, mass and energy convert into each other. Energies are measured in MeV and GeV, and fast particles live longer than slow ones.

ΔE = c2Δm

Mass is a form of energy: ΔE = c2Δm.

Energy and mass units

Relativistic lifetimes

Unstable particles have a characteristic lifetime when at rest. When they move at speeds close to c, their lifetime measured in the laboratory is longer (time dilation, from special relativity). This matters when v is a significant fraction of c. Example: muons made by cosmic rays high in the atmosphere would mostly decay before reaching the ground if they lived only their rest lifetime, but many are detected at sea level because they move at about 0.99c.

Worked example

An electron and a positron at rest annihilate. Find the energy of each photon produced.

  1. Total mass = 2 x 9.11 x 10-31 = 1.822 x 10-30 kg
  2. ΔE = c2Δm = (3.00 x 108)2 x 1.822 x 10-30 = 1.64 x 10-13 J
  3. Each photon: 8.20 x 10-14 J

Answer: Each photon has 8.2 x 10-14 J (about 0.51 MeV)

Key idea: ΔE = c2Δm; pair production needs at least 2mc2; 1 MeV = 1.60 x 10-13 J; mass in MeV/c2; fast particles live longer in the lab frame.

Check yourself

  1. How much energy is released if 1.0 µg of mass is converted completely into energy? Answer in units of 107 J.
    Show answer

    9 x 107 J. ΔE = c2Δm = (3.00 x 108)2 x 1.0 x 10-9 = 9.0 x 107 J.

  2. Find the rest energy of a proton in MeV, to 3 s.f.
    Show answer

    939 MeV. E = mc2 = 1.67 x 10-27 x 9.00 x 1016 = 1.503 x 10-10 J; divide by 1.60 x 10-13 to get 939 MeV. (The accepted value is 938 MeV; the small difference comes from the rounded data values.)

  3. Convert 2.5 GeV into joules, in units of 10-10 J.
    Show answer

    4 x 10-10 J. 2.5 x 109 x 1.60 x 10-19 = 4.0 x 10-10 J.

  4. A particle has a mass of 125 GeV/c2. Find its mass in kg, in units of 10-25 kg, to 3 s.f.
    Show answer

    2.22 x 10-25 kg. E = 125 x 109 x 1.60 x 10-19 = 2.00 x 10-8 J; m = E / c2 = 2.00 x 10-8 / 9.00 x 1016 = 2.22 x 10-25 kg.

  5. Find the minimum photon energy needed to produce an electron-positron pair, in units of 10-13 J, to 3 s.f.
    Show answer

    1.64 x 10-13 J. E = 2m_ec2 = 2 x 9.11 x 10-31 x 9.00 x 1016 = 1.64 x 10-13 J.

  6. Why does an electron-positron pair at rest annihilate into two photons rather than one?
    • To conserve charge
    • Because photons have mass
    • To conserve momentum
    • To conserve lepton number
    Show answer

    To conserve momentum. The total momentum is zero, so the photons must move in opposite directions.

  7. When is the relativistic increase in a particle's lifetime significant?
    • When its speed is a significant fraction of the speed of light
    • When it is at rest
    • When it has a large charge
    • When it is in a magnetic field
    Show answer

    When its speed is a significant fraction of the speed of light. Time dilation is only noticeable at speeds close to c.

  8. Why do many muons made in the upper atmosphere reach the Earth's surface?
    • They are stable particles
    • They move so fast that their lifetime measured from Earth is much longer
    • They are slowed down by the atmosphere
    • They are made at ground level
    Show answer

    They move so fast that their lifetime measured from Earth is much longer. At about 0.99c, their lab-frame lifetime is several times their rest lifetime.

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