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Relativistic Momentum Calculator

Find the relativistic momentum of a fast object.

How it is calculated

The speed is entered as a fraction of the speed of light, β = v ÷ c, and the mass in kilograms. The tool computes the Lorentz factor γ = 1 ÷ √(1 − β²) and then p = γ × m × β × c with c = 299,792,458 m/s. Momentum is shown in kg·m/s with two decimals, or in scientific notation (such as 1.5766 × 10⁻²²) when it is below 0.01, and γ with four decimals.

The mass must be greater than zero. β must be at least 0 and strictly below 1; exactly 1 or more gives no result, because γ would be infinite or undefined. Use the rest mass, not a speed-dependent mass. The tool reports momentum only, not energy. Very small masses are not rounded away: an electron (9.109 × 10⁻³¹ kg) at β = 0.5 shows 1.5766 × 10⁻²² kg·m/s.

Worked example

A 1 kg object moving at β = 0.6, which is 60% of the speed of light.

β² = 0.36, so 1 − β² = 0.64 and its square root is 0.8. Then γ = 1 ÷ 0.8 = 1.2500. Speed: 0.6 × 299,792,458 = 179,875,474.8 m/s. Momentum: 1.25 × 1 × 179,875,474.8 = 224,844,343.50 kg·m/s. The classical value m·v would be 179,875,474.80, so the relativistic one is 25% larger.

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Common mistakes

  • Typing the speed in m/s or km/h. The field takes the fraction of c, a number from 0 up to but not including 1, so 0.6 and not 180000000.
  • Entering a percentage such as 60. Values of 1 or more give no result; use 0.6.
  • Using relativistic mass instead of rest mass. The formula already includes γ, so the rest mass goes in.
  • Entering the mass of a subatomic particle in kilograms and overlooking the exponent. Momenta below 0.01 kg·m/s are printed in scientific notation, so 1.5766 × 10⁻²² is a tiny but non-zero value.
  • Expecting the classical m·v to be close at high speed. At β = 0.6 it is already 20% too low; at everyday speeds the two agree.

Sources

About this calculator

This calculator finds relativistic momentum using p = γ·m·v, where the classical formula p = m·v is scaled by the Lorentz factor γ = 1/√(1−v²/c²). At everyday speeds γ is essentially 1 and the result matches Newtonian mechanics, but as an object's velocity v approaches the speed of light c, γ grows sharply — momentum increases far faster than velocity does, which is part of why no massive object can actually reach light speed: it would take infinite momentum (and energy) to do so.

Particle physicists use this when designing accelerators and analyzing high-energy collisions, where particles routinely travel at a significant fraction of c; astrophysicists use it to model relativistic jets from black holes and the behavior of high-energy cosmic rays; and physics students use it as a core worked example when first studying special relativity.

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Comments (1)

  • Editorial team

    Compare against the everyday formula to see when relativity matters. For 1 kg at 0.1 c the factor gamma is 1.0050, only half a percent above classical momentum, while at 0.9 c gamma is 2.2942 and the classical value would be less than half the true one. Momenta below 0.01 kg·m/s print in scientific notation, so an electron at 0.5 c shows 1.5766 × 10⁻²² kg·m/s. What kind of particle or object are you modelling?

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