---
title: "Collisions and Invariant Mass Interactive Lab | Sandbox Physics"
description: "Why does every frame agree on a collision's invariant mass when energy and momentum change?"
canonical_url: "https://sandboxphysics.com/labs/collision-four-momentum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/collision-four-momentum/"
last_modified: "2026-10-29"
---

Special relativity · Energy and momentum

# Collisions and Invariant Mass

**Why does every frame agree on a collision's invariant mass when energy and momentum change?** Energy and momentum both change when you switch frames, but one combination of the totals does not: the invariant mass, equal to the total energy in the frame where the total momentum is zero. It decides what a collision can make. A kaon at 0.80 GeV/c hitting a resting proton gives 1.70 GeV, and making an antiproton needs a proton beam with six proton masses of kinetic energy.

Collide particles along one line, slide to other frames and watch energy and momentum change while the invariant mass stays fixed and decides what can be made.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Motion along one line only, with c = 1 and energies in GeV. Masses of the proton, charged kaon and neutral pion are PDG 2024 values; the two sticky lumps are an idealized 1 GeV example. Products are placed back to back in the centre-of-momentum frame (equal shares for the four-proton final state) and boosted to the chosen frame; real angular distributions, spin, cross-sections and detector effects are left out. A forbidden reaction is shown as a notice, not as invented products.

White and blue change with the frame; the green invariant mass does not.

Four-momentum table and the saved record

Particle $E\ (\mathrm{GeV})$ $pc\ (\mathrm{GeV})$ $\sqrt{E^2-(pc)^2}$

kaon · in $0.94$ $0.8$ $0.494$

proton · in $0.938$ $0$ $0.938$

kaon · out $0.524$ $-0.176$ $0.494$

proton · out $1.35$ $0.976$ $0.938$

The JSON record lists every particle's energy and momentum in the laboratory and in the viewing frame.

[Course: relativity as measurement, in eight chapters](https://sandboxphysics.com/stories/relativity-measurement/)

Physics tutorial

## One number every frame agrees on

**Background** In a collision, observers moving at different speeds record different energies and momenta for the same particles. Accelerator physicists still quote one number for the whole collision.

**Why it matters** That number decides what particles a collision can make, so it must not depend on who is watching.

Start with the essentials

- **Focus question**: Why does every frame agree on a collision's invariant mass when energy and momentum change?
- **One-sentence intuition**: Energy and momentum are components of one four-momentum. A boost mixes the components, but the total's invariant mass, its length in spacetime, stays fixed and sets the energy available for making particles.

### Core mathematical model

#### Mass shell of one particle

$$
E^2-(pc)^2=(mc^2)^2
$$

Every particle's energy and momentum sit on this hyperbola in every frame; photons sit on the light-like lines.

#### Invariant mass of the total

$$
M^2c^4=\Big(\sum_i E_i\Big)^2-\Big(\sum_i p_i c\Big)^2
$$

Add the four-momenta first, then take the length. Adding the masses instead gives the wrong answer.

#### Changing frame

$$
E'=\gamma\,(E-\beta\,pc),\qquad p'c=\gamma\,(pc-\beta E)
$$

In the frame moving at $\beta=pc/E$ of the total, the total momentum vanishes and the total energy equals $Mc^2$.

#### A beam on a resting target

$$
M^2c^4=m_b^2c^4+m_t^2c^4+2E_b\,m_tc^2
$$

For a proton beam on a resting proton, making three protons and an antiproton needs a total mass of four proton masses, so the beam energy must reach seven proton masses: a kinetic energy of six.

### Common difficulties

#### Mass is not additive

**Typical misconception** The mass of a system is the sum of its parts' masses.

**Better mental model** The invariant mass of the system includes the kinetic energy in its centre-of-momentum frame. Two photons with no mass can have a nonzero invariant mass together.

#### Energy is conserved, not invariant

**Typical misconception** If energy is conserved, every observer must agree on it.

**Better mental model** Each observer finds the same total before and after the collision, but different observers find different totals. The invariant mass is both conserved and the same for everyone.

#### What the model leaves out

**Typical misconception** The scene shows where real detector tracks would go.

**Better mental model** Motion along one line only, with c = 1 and energies in GeV. Masses of the proton, charged kaon and neutral pion are PDG 2024 values; the two sticky lumps are an idealized 1 GeV example. Products are placed back to back in the centre-of-momentum frame (equal shares for the four-proton final state) and boosted to the chosen frame; real angular distributions, spin, cross-sections and detector effects are left out. A forbidden reaction is shown as a notice, not as invented products.

### Run the experiment

1. 01

   #### Check a published number

   Start with the kaon preset: a kaon at 0.80 GeV/c hits a resting proton.

   **What to observe:** The invariant mass reads 1.70 GeV, the centre-of-momentum energy in the Particle Data Group's worked example.

2. 02

   #### Slide to other frames

   Drag the round tip of the total along its green curve, or move the frame slider.

   **What to observe:** Total energy and momentum change; the invariant mass does not. Where the total momentum crosses zero, the total energy equals the invariant mass.

3. 03

   #### Turn motion into mass

   Choose Motion turns into mass and raise the beam momentum.

   **What to observe:** The merged lump is heavier than the two lumps that made it. The extra mass is the kinetic energy of the centre-of-momentum frame.

4. 04

   #### Cross a threshold

   Choose the antiproton preset, then lower the beam momentum slightly.

   **What to observe:** The energy left over turns negative and the reaction is forbidden: below threshold no frame can supply four proton masses.

Physical conclusion

To judge what a collision can make, add the four-momenta and take the invariant mass; any frame gives the same answer.

## Related experiments and articles

Relativity as measurement · Reading path

### Relativity as measurement: clocks, light signals and four-momentum

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Four-momentum and the mass every frame agrees on**](https://sandboxphysics.com/stories/relativity-measurement/four-momentum-and-invariant-mass/)
2. [Lab **Particle Collider · Event Reconstruction Room** See how a detector reconstructs a collision](https://sandboxphysics.com/labs/particle-collider/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)