# Relativistic Momentum Calculator

p = γmv with p·c in eV-scale units, the Lorentz factor, and the classical mv comparison for electron, proton, or custom mass.

> Canonical page: https://elysiatools.com/en/tools/relativistic-momentum

- **Category:** Science & Education

- **Keywords:** relativistic momentum, gamma mv, lorentz factor, particle physics, pc in gev, special relativity

## Overview

The Relativistic Momentum Calculator computes the momentum of high-speed particles using the relativistic formula p = γmv, where γ is the Lorentz factor. It evaluates momentum in both standard SI units (kg·m/s) and particle physics units (p·c in eV, MeV, or GeV) while comparing the result directly against classical Newtonian momentum.

## Inputs

- **Particle** (select)
- **Custom mass (kg)** (number): e.g. 9.109e-31
- **Speed input** (select)
- **β (v/c)** (number): e.g. 0.8
- **Speed v (m/s)** (number): e.g. 2.4e8

## When to use

- Analyzing subatomic particle dynamics in accelerator physics or high-energy collisions.
- Solving modern physics and special relativity coursework problems involving relativistic motion.
- Evaluating the deviation between classical Newtonian mechanics (mv) and relativistic mechanics (γmv) as velocities approach the speed of light.

## How it works

- Select a particle preset (electron or proton) or choose custom mass to specify rest mass in kilograms.
- Choose your velocity input format as either a fraction of the speed of light (β = v/c) or absolute speed in meters per second (m/s).
- The tool calculates the Lorentz factor γ = 1/√(1 − v²/c²), computes relativistic momentum p = γmv alongside classical momentum mv, and converts the quantity to particle-physics energy units (p·c).

## Use cases

- Beamline instrumentation calculations determining electron or proton momentum in synchrotrons and linacs.
- Special relativity physics problem verification for university and advanced secondary education coursework.
- Astrophysical modeling of ultra-relativistic cosmic ray particle trajectories and kinetic distributions.

## Frequently asked questions

### What is the difference between relativistic momentum and classical momentum?

Classical momentum is defined as p = mv, whereas relativistic momentum incorporates the Lorentz factor p = γmv, accounting for spacetime curvature at velocities near the speed of light.

### Why is momentum frequently expressed as p·c in electron-volts?

In particle physics, multiplying momentum p by the speed of light c yields energy dimensions, making it straightforward to compare directly with particle rest mass and kinetic energy in MeV or GeV.

### What happens to the momentum as speed approaches the speed of light (c)?

As velocity approaches c, β approaches 1 and the Lorentz factor γ diverges toward infinity, requiring infinite momentum to accelerate a massive particle to the speed of light.

### Can I calculate relativistic momentum for arbitrary masses?

Yes, select the custom mass option and input the object's rest mass in kilograms.

### What is the relationship between the relativistic and classical momentum values?

The exact ratio of relativistic momentum to classical momentum (p / mv) equals the Lorentz factor γ.

## Related tools

- [Ampère Force Calculator](https://elysiatools.com/en/tools/ampere-force-calculator): F = B·I·L·sinθ for a current-carrying wire, with unit selectors, the force magnitude, the 90° reference, and the right-hand-rule direction.
- [Coulomb Force Calculator](https://elysiatools.com/en/tools/coulomb-force-calculator): Coulomb's law F = k·|q₁q₂|/r² with signed charges and unit selectors, reporting the force magnitude and whether the pair attracts or repels.
- [Electric Potential Calculator](https://elysiatools.com/en/tools/electric-potential-calculator): V = k·Q/r with the sign preserved, plus the electron potential energy U = −e·V in eV — at the Bohr radius this reproduces 27.2114 V, twice hydrogen's ionization energy.
- [Faraday Induction EMF Calculator](https://elysiatools.com/en/tools/faraday-induction-emf): Faraday's law |ε| = N·ΔΦ/Δt with turns, flux, and time selectors, plus the Lenz's-law sign explained — one turn and one weber per second induce exactly one volt.
- [Optical Magnification Calculator](https://elysiatools.com/en/tools/magnification-power): M = 250/f for magnifiers, (L/f_o)·(250/f_e) factorized for microscopes, f_o/f_e for telescopes.
- [Malus's Law Polarization Calculator](https://elysiatools.com/en/tools/polarization-malus-law): I = I₀·cos²θ: transmitted intensity and percentage for any polarizer angle, with the 0°/45°/90° landmarks.
- [Raoult's Law Calculator](https://elysiatools.com/en/tools/raoult-law-calculator): p_i = x_i·p_i* for ideal solutions, with total pressure and vapour-phase composition for binary mixtures.
- [Rest Mass Energy Calculator](https://elysiatools.com/en/tools/rest-mass-energy): E₀ = mc² for electron, proton, neutron, or custom mass: energy in J and eV-scale units, mass in kg and u — the collider mass table.

## Samples

- [Android Image Processing Java Samples](https://elysiatools.com/en/samples/android-image-processing-java): Android Java image processing examples including reading/saving images, scaling, and format conversion
- [Android Image Processing Kotlin Samples](https://elysiatools.com/en/samples/android-image-processing-kotlin): Android Kotlin image processing examples including reading/saving images, scaling, and format conversion
- [Web Image Processing Python Samples](https://elysiatools.com/en/samples/web-image-processing-python): Web Python image processing examples using PIL/Pillow including reading, saving, resizing, and format conversion
- [Web Image Processing Rust Samples](https://elysiatools.com/en/samples/web-image-processing-rust): Web Rust image processing examples including image read/save, scaling, and format conversion