Relativistic Momentum Calculator

Relativistic momentum is the correct, frame-invariant definition of momentum under Einstein's special relativity: p = γmv, where γ is the Lorentz factor. Unlike the classical formula p = mv, this version remains conserved in collisions and interactions even when particles move at a significant fraction of the speed of light.

p = γmv, where γ = 1/√(1−β²) is the Lorentz factor, m is the particle's rest mass, and v is its velocity. As v approaches the speed of light c, γ grows without bound, so momentum increases far faster than the classical mv formula would predict.

Classical momentum p = mv assumes mass and velocity combine linearly with no upper speed limit. Relativistic momentum multiplies this by γ, which stays near 1 at low speeds (recovering the classical formula) but diverges as v → c, correctly reflecting that no massive particle can ever reach the speed of light no matter how much momentum it has.

Because γ diverges to infinity as β → 1, the relativistic momentum formula shows that accelerating a massive particle ever closer to the speed of light requires ever-increasing momentum (and correspondingly energy), which is why reaching or exceeding c is physically impossible for any object with nonzero rest mass.

Yes, relativistic momentum p = γmv is the quantity that is actually conserved in particle collisions and interactions across all inertial reference frames, exactly the role classical momentum plays in Newtonian mechanics. This is why particle physicists always use the relativistic definition, not the classical one, when analyzing high-energy collisions.

At low speeds (β well under about 0.1, or roughly 30,000 km/s), γ is extremely close to 1, so p ≈ mv to high precision. This is why classical mechanics works so well for everyday objects and low-speed engineering applications.

They are linked through the energy-momentum relation E² = (pc)² + (mc²)², which combines a particle's total relativistic energy, momentum, and rest mass energy into a single invariant equation valid in every reference frame.

Particle accelerators like the LHC push protons to enormous relativistic momenta (γ in the thousands), and cosmic rays observed on Earth can carry relativistic momenta corresponding to macroscopic kinetic energies concentrated in a single subatomic particle, both regimes where the classical mv formula would be wildly wrong.

No, like classical momentum, relativistic momentum is a vector whose magnitude is γmv and whose direction matches the particle's velocity direction; this calculator computes the magnitude for a given speed.

Results are given in kilogram-metres per second (kg·m/s), the SI unit for momentum, alongside the classical (non-relativistic) mv value for direct comparison.