---
title: "Relativistic Momentum and Energy"
description: "Relativistic momentum and energy preserve conservation laws in every inertial frame. A particle's rest mass determines its rest energy, while its total energy also depends on its motion."
canonical: https://lightmysky.com/learn/science/relativistic-momentum-and-energy-mt_zBq87F-j3B
source: https://lightmysky.com/learn/science/relativistic-momentum-and-energy-mt_zBq87F-j3B.md
retrieved: 2026-11-12
---

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# Relativistic Momentum and Energy

Relativistic momentum and energy preserve conservation laws in every inertial frame. A particle's rest mass determines its rest energy, while its total energy also depends on its motion.

Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20
Page: https://lightmysky.com/learn/science/relativistic-momentum-and-energy-mt_zBq87F-j3B

## Ready when they can

- Computes relativistic momentum, kinetic energy and total energy for a fast particle
- Relates energy, momentum and rest mass and applies the relation to a massless particle
- Explains rest energy and where it shows up in nuclear reactions

## Lesson: Energy hiding in mass itself

Let m be rest mass, u speed and c vacuum light speed. In classical mechanics, momentum is m u and kinetic energy is one half m u squared. Momentum p = gamma m u. Gamma = 1 / sqrt(1 - (u/c) squared) increases with speed and approaches 1 near rest.

Total energy E = gamma m c squared. At rest gamma = 1, giving rest energy m c squared even without motion.

Fission, fusion and annihilation can release rest energy. Decreased total particle rest mass supplies kinetic energy or radiation, conserving total energy.

E squared = (pc) squared + (m c squared) squared, where p is momentum magnitude. For photons m = 0, so E = pc and p = E/c. Photons travel at c in vacuum.

**Example.** Compute gamma = 1 / sqrt(1 - (u/c) squared), then p = gamma m u, E = gamma m c squared and K = (gamma minus 1) m c squared. At u = 0.6c, gamma = 1 / sqrt(0.64) = 1.25, giving K = 0.25mc squared.

Released rest energy equals the total rest-mass decrease times c squared. Square c before multiplying. The unit kg m squared / s squared equals J.

As u approaches c, gamma, momentum and energy grow without bound. No finite energy can accelerate a massive particle to c.

**Recap.** Stretch momentum and energy by gamma, keep rest energy in mass, and let massless light obey E equals p c.

*Original lesson and quiz text authored for this learning dataset.*

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Lorentz Transformation and Velocity Addition](https://lightmysky.com/learn/science/the-lorentz-transformation-and-velocity-addition-mt_uG3_gArw19)

## Opens up

- [Properties of Nuclei and the Nuclear Force](https://lightmysky.com/learn/science/properties-of-nuclei-and-the-nuclear-force-mt_HdMS_muGWa)