What does β represent?

β (beta) is the ratio v/c — the particle speed divided by the speed of light in vacuum (c ≈ 2.998 × 10⁸ m/s). A value of 0.99 means the particle moves at 99% of c.

Why does energy go to infinity as β → 1?

The Lorentz factor γ = 1/√(1−β²) diverges as β approaches 1. Since total energy is γmc², it also diverges. This means an infinite amount of energy would be needed to accelerate a massive particle to exactly c.

What is the Lorentz factor?

The Lorentz factor γ (gamma) is a dimensionless number ≥ 1 that quantifies time dilation, length contraction, and energy increase in special relativity. At rest γ = 1; at β = 0.99 it is about 7.09.

Rest energy E₀ = mc² is the energy equivalent of a particle's mass when it is stationary. For a proton (~1.673 × 10⁻²⁷ kg), E₀ ≈ 938.3 MeV. It does not change with velocity.

How does relativistic kinetic energy differ from classical KE?

Classical kinetic energy is ½mv². The relativistic formula KE = (γ−1)mc² reduces to ½mv² at low speeds (β ≪ 1) but grows much faster as β → 1, matching the infinite-energy barrier.

What units are used for the results?

Results are given in joules (J) for SI consistency, and in MeV and GeV (mega/gigaelectronvolts), which are standard in particle physics. 1 MeV = 1.602 × 10⁻¹³ J.

Relativistic Energy Particle Calculator

Enter a particle rest mass and speed (as a fraction of c) to get total relativistic energy, rest energy, kinetic energy, and Lorentz factor.

Particle Parameters

Quick Presets

Rest Mass (kg)

Proton default. Use scientific notation (e.g. 1.67e-27).

Velocity β = v/c

Must be between 0 (exclusive) and 1 (exclusive).

Results

Results will appear here after calculation.

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Summary

This calculator applies Einstein's special relativity to find the total energy, rest energy, and kinetic energy of a moving particle. Given a rest mass and a velocity expressed as a fraction of the speed of light (β = v/c), it computes the Lorentz factor γ = 1/√(1−β²) and then derives E_total = γmc², E_rest = mc², and KE = (γ−1)mc². Results are shown in joules, MeV, and GeV — the units most common in particle physics and astrophysics.

How it works

Enter the particle's rest mass in kg, or select a preset (electron, proton, neutron, muon).
Set the velocity β = v/c as a decimal between 0 and 1 (exclusive), or use the slider.
The calculator computes the Lorentz factor γ = 1/√(1−β²).
Rest energy E₀ = mc² is derived from the mass alone.
Total energy E = γmc² and kinetic energy KE = (γ−1)mc² are calculated.
Results appear instantly in joules, MeV, and GeV for easy cross-reference.

Use cases

Estimate the kinetic energy of a cosmic ray proton at near-light speed.
Compare rest energy and total energy for electrons in a particle accelerator.
Verify textbook special relativity problems.
Explore how the Lorentz factor grows as β approaches 1.
Convert particle energies between joules and electron-volts.
Understand why rockets cannot reach c — energy diverges as β → 1.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu