Last updated: 2026-06-16

Relativistic Momentum

Relativistic momentum is the corrected form of momentum that remains valid at high speeds. In Newtonian mechanics momentum is simply mass times velocity, $p = mv$, but that expression quietly fails to be conserved when motion is observed from frames moving at an appreciable fraction of the speed of light $c \approx 3 \times 10^8\ \text{m/s}$. Special relativity repairs this by multiplying by the Lorentz factor.

This calculator takes a mass $m$ and a velocity $v$, and returns the Lorentz factor $\gamma$, the relativistic momentum $p = \gamma m v$, and the classical momentum $mv$ for comparison.

How it works

The defining relation is $p = \gamma m v$, where $\gamma = 1/\sqrt{1 - v^2/c^2}$. The factor $\gamma$ is the same one that appears in time dilation and length contraction, and it ties the three relativistic effects together. At low speeds $\gamma \approx 1$ and the momentum is indistinguishable from the Newtonian $mv$. As $v$ rises toward $c$, $\gamma$ grows without bound, so the momentum can become arbitrarily large even though the speed itself can never reach $c$.

This unbounded growth of momentum is the mechanical counterpart of the energy result: it takes ever more push to gain ever less speed, which is why massive objects cannot be accelerated to the speed of light.

Formula

As $v \to 0$, $\gamma \to 1$ and $p \to mv$. As $v \to c$, $\gamma \to \infty$ and $p \to \infty$.

Worked example

A 1 kg object moves at v = 0.8c.

Step 1 — Lorentz factor:

Step 2 — Classical momentum:

Step 3 — Relativistic momentum:

The relativistic momentum is about two-thirds larger than the classical estimate. Enter these values in the calculator to reproduce the result.

Momentum at different speeds

Real-world relevance

Relativistic momentum is essential in particle physics. The deflection of a charged particle in a magnetic field depends on its momentum, so detectors at colliders measure the relativistic $p = \gamma m v$ to identify particles and reconstruct collisions. Cosmic rays, synchrotron light sources and medical particle beams all involve particles whose momentum is dominated by the Lorentz factor rather than by speed, which is already pinned just below $c$.

Limitation: this calculator covers special relativity only

The formula here applies to a free object in flat spacetime, with $m$ being the invariant rest mass. It does not account for forces, fields, or the curvature effects of general relativity near very massive bodies. For kinematics and particle physics in inertial frames, the special-relativistic result is sufficient.