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Relativistic Momentum Calculator

Work out relativistic momentum instantly with clear inputs, formula shown and shareable results.

About Relativistic Momentum Calculator

Relativistic momentum is gamma.m0.v, which diverges from the classical value as speed rises. A proton at 0.95c has gamma = 3.203, so its momentum is more than three times the Newtonian estimate. Accelerator design depends on this: beyond a few percent of light speed the classical formula badly underestimates the magnetic field needed to bend a beam.

Formulas

Relativistic momentum

p = gamma m0 v

Energy-momentum relation

E^2 = (pc)^2 + (m0 c^2)^2

Worked example

With the default values below, Relativistic Momentum Calculator returns Relativistic Momentum of 1.52560 × 10⁻¹⁸. Change any input above and every figure here is recalculated in your browser.

Inputs used

Rest massminimum 0 kg
0 kg
Speed as a fraction of light speedaccepts 0–1
0.95

Results

Relativistic Momentum
1.52560 × 10⁻¹⁸ kg m/s
Classical Momentum For Comparison
4.7637 × 10⁻¹⁹ kg m/s
Lorentz Factor
3.2026
Total Energy
3.0049 × 10³ MeV

Step by step

  1. 1Values used
  2. 2Relativistic momentum
  3. 3Energy-momentum relation
  4. 4Relativistic Momentum= 0.00000 kg m/s
  5. 5Classical Momentum For Comparison= 0.0000 kg m/s
  6. 6Lorentz Factor= 3.2026
  7. 7Total Energy= 3,004.8800 MeV

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

When does the classical formula become inadequate?

Above about 0.1c the error passes 0.5%, and by 0.5c momentum is understated by 15%. Below 0.01c the difference is negligible.

Do photons have momentum without mass?

Yes - p = E/c. The energy-momentum relation still holds with the rest mass term set to zero.

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