Relativistic Momentum Calculator

Relativistic momentum is the momentum of an object moving at speeds close to the speed of light, where classical Newtonian mechanics breaks down. It is defined as p = m₀ · γ · v, where m₀ is the rest mass, v is the velocity, and γ (the Lorentz factor) accounts for the relativistic effects. As v approaches c, γ increases without bound, making the momentum approach infinity.

Classical momentum is simply p = m₀ · v, which works well at everyday speeds. Relativistic momentum introduces the Lorentz factor γ = 1 / √(1 − v²/c²), effectively replacing the rest mass with the relativistic mass m₀γ. At low speeds, γ ≈ 1 and both formulas agree; at speeds near c, relativistic momentum grows much larger than classical momentum.

The Lorentz factor γ (gamma) describes how much time, length, and relativistic mass change for a moving object. It is calculated as γ = 1 / √(1 − v²/c²). At v = 0, γ = 1 (no relativistic effect). At v = 0.9c, γ ≈ 2.29. As v → c, γ → ∞.

As an object's velocity approaches the speed of light, the Lorentz factor γ increases toward infinity. This means relativistic momentum (and the energy required to accelerate the object) also approaches infinity. No finite amount of energy can accelerate a massive object to exactly c, making it physically impossible.

Relativistic momentum is expressed in kilogram-meters per second (kg·m/s), the same SI unit as classical momentum. In high-energy physics, momentum is often expressed in MeV/c or GeV/c, where c is the speed of light, because of the convenient relationship between mass, momentum, and energy via E² = (pc)² + (m₀c²)².

Relativistic effects are generally considered significant when an object's velocity exceeds about 10% of the speed of light (0.1c ≈ 29,979,246 m/s). At 0.1c, γ ≈ 1.005 — a 0.5% difference from classical. At 0.5c, γ ≈ 1.155, and at 0.9c, γ ≈ 2.294, representing much larger deviations from classical predictions.

This calculator uses the exact value c = 299,792,458 m/s (approximately 3 × 10⁸ m/s), as defined by the International System of Units (SI). Your velocity input is treated as a fraction of this value, so entering 0.5 means v = 149,896,229 m/s.

Yes — simply enter the rest mass in kilograms. The electron rest mass is approximately 9.109 × 10⁻³¹ kg, and the proton rest mass is approximately 1.673 × 10⁻²⁷ kg. This calculator works for any rest mass value, from subatomic particles to macroscopic objects.