Energy–Momentum Conservation in Relativistic Collisions

Source

Lecture: 16 — E = mc²
Simulation: Simulation 1 — Energy–momentum conservation in a collision (guided inquiry)
Concept: In relativistic collisions total four-momentum is conserved, so kinetic energy can convert into rest mass — the rest mass of products is generally not the sum of the incoming rest masses.
Serves: Derive the relativistic energy–momentum relation; explain rest energy; interpret mass–energy equivalence; apply energy–momentum conservation in relativistic processes.
Inquiry: Two identical particles of rest mass m collide head-on at equal & opposite β = 0.8 and merge. Predict the rest mass of the combined particle — 2m, more, or less?
Misconception: That rest masses add like ordinary masses (M = m₁ + m₂), ignoring the kinetic-energy contribution.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Every element serves: kinetic energy converts to rest mass via four-momentum conservation.
A2	Prune readouts	Speed selector hidden; only quantities tied to the conservation story are shown.
A3/A4	Formal layer optional	Equations + worked derivation live in “The Physics” revision sheet, opened from the top bar.
B1	Standard terms	"rest mass", "four-momentum", "Lorentz factor", "invariant mass" — no invented terms.
B3	Unambiguous symbols	Legend defines m, E=γm, p=γmv, γ, M; m₁,m₂ (inputs) vs M (product/system).
C1	Numbers earn place	Every table value is read by the conservation check or an inquiry prompt.
C2	Frame-labelled	Each E/p value carries its frame: "Frame S" or "Frame S′ (v_b = …)".
D1	Canvas primary	The collision canvas dominates the hero; tables sit in side panels.
D3	Particle identity	Coloured discs + adjacent mass/velocity labels; product a distinct colour.
D5	Many states, no clip	1-, 2-, and composite-particle states all draw cleanly; labels bounds-checked.
D6	Staged emphasis	Steps 1→2→3 add emphasis layers; never disable controls.
D7	Playable from load	All sliders, Play, mode toggle, boost, prediction work immediately.
D8	Labels vs narration	Canvas carries labels only; explanation lives in side panels.
E1	One-pass physics	Particles approach, collide once, products fly out and freeze; no loop.
E3	No decoration	No explosion effects — clean disappearance/appearance at the collision point.
F2	Qualitative predict	Asks ">2m, =2m, or <2m" — the number appears only after running.
F9–F14	Prompt wording	Open eliciting question, tension-holding choices, relational explore prompts, formative feedback.
G2	Frame invariance	Invariant mass² labelled "same in all frames"; visibly fixed under boost.
G3	Boost re-renders	Boost slider recomputes all E, p, v; particle speeds/directions update.
G4	Rest frame stationary	Boosting to a particle's rest frame visibly stops it; its E equals m.
G4b	One frame first	Default shows lab frame; Step 3 adds the side-by-side S′ comparison.
G4c	Identity preserved	Colours, labels, left/right arrangement persist across boosts.
G7	Operational quantities	Four-momenta are coordinate quantities; velocity = p/E in the shown frame.
H1	Misconception structural	The wrong idea is a prediction option, not a standing warning.
H2	Probe via option	Option (A) "M = 2m" is the probe; its rebuttal shows only when selected.
J1	Consistent notation	(E,p), m, M, γ, v used identically throughout.
J3	Correct model	Exact Lorentz boosts, COM-frame elastic kinematics, decay two-body kinematics.

Relativistic Collisions

Teach me

Speed

Guided Inquiry

1 · Two particles, one merger

Let’s set the scene. Two particles of rest masses m₁ and m₂ move toward each other along a line and merge into a single body of rest mass M. Each carries total energy E = γmc² and momentum p = γmv, with γ = 1/√(1 − v²/c²); the part of E above the rest energy mc², namely (γ − 1)mc², is its kinetic energy. Here both lab-frame speeds are β ≡ v/c = ±0.80.

2 · The merged body

Let’s start with the simplest case. Two identical particles, rest mass m each, approach head-on at β = 0.80 and merge into a single body. What do you think that body’s rest mass M will be?

Now press Play to run the merge, and read M off the four-momentum table.

3 · The energy books

So the merge has run, and the two bodies now travel as one, heavier than m₁ + m₂. Let’s check the energy books. How do you think the system’s total energy ΣE just after the merge compares with its value just before?

Take a look: the Before and After Σ rows of the four-momentum table now carry both totals.

4 · Counting the kinetic energy

Now let’s dig into the kinetic energy. Each incoming particle has β = 0.80. Newtonian mechanics assigns it kinetic energy ½mv² = 0.32 mc²; relativity assigns (γ − 1)mc². How do you think the two values compare?

Now glance at the mass budget: it shows ΔM = M − (m₁ + m₂).

5 · A second observer

Time to bring in a second observer. The same merge is described from a frame S′ gliding past the lab, and every individual E and p takes a new value there. Which of these do you think is the same number in both frames?

Try it: drag Boost to S′ and watch ΣE and Σp move, while the invariant mass holds.

6 · Invariant mass

Here’s the big idea to take away. The quantity M fixed by M²c⁴ = (ΣE)² − (Σpc)² is the system's invariant mass: one number for the whole system, the same in every frame, and never smaller than Σm. That surplus is what lets a collider build particles heavier than the beams it fires.

Listen

Controls

▾

m₁ [rest mass of particle 1]

1.00

m₂ [rest mass of particle 2]

1.00

v₁ [lab-frame speed, particle 1]

+0.80 c

v₂ [lab-frame speed, particle 2]

−0.80 c

Boost to S′, v_b [velocity of boosted frame S′]

0.00 c

M_parent [rest mass of decaying parent]

3.33

Hide output text

Four-momentum (E, p)

▾

m = rest mass · E = γm · p = γmv · γ = 1/√(1−v²) · M = invariant mass

Invariant mass²

▾

Rest energy & mass budget

▾

Lorentz factor

Energy & momentum (c = 1)

Energy–momentum relation

Invariant (system) mass

Lorentz boost to S′ (v_b ≡ boost velocity)

Worked: current values

The Physics

Energy–momentum conservation in a collision · what this sim shows, the physics behind it, and every control

At a glance

Energy and momentum are conserved in every collision, but relativity changes what counts as "the total". This sim runs a head-on collision as a merge, a bounce or a decay, and tabulates the totals in two frames, so you can check which quantities carry over and which depend on the observer.

What you'll learn

How to add up the energy and momentum of a system, using E = γmc² and p = γmv.
How to find the rest mass of the merged body, and how it compares with the rest masses that went in.
How relativity counts kinetic energy, compared with ½mv².
Which totals change when you change frame, and which do not.

Two bodies approach head-on with velocities v₁ and v₂ and merge into one body.

What's on screen

Cyan disc: particle 1. Its arrow points along its velocity and grows with speed.
Amber disc: particle 2.
Violet disc: the merged body, labelled with its rest mass M. In Decay mode it is the parent.
Stage panels: the collision in frame S (lab) and, below it, in frame S′, which moves at v_b.
Sidebar panels: the Four-momentum table, the Invariant mass² and the Rest energy & mass budget.

See itRun the default collision: two equal masses at ±0.80c. Watch for: the cyan and amber discs meeting and merging into one violet disc.

The controls

Press ⌖ on a row to see where that control sits in the sim. Speeds are fractions of c, and masses are in the sim's arbitrary units.

Control	What it changes	Range
Mode	Inelastic: the two bodies merge. Elastic: they bounce. Decay: one parent splits in two.	3 modes opens on Inelastic
m₁, m₂	The rest masses. In Decay mode they are the masses of the two products.	0.1–5 opens at 1.00 each
v₁, v₂	The speeds in the lab frame S. A minus sign means moving left. Locked in Decay mode, where the speeds follow from the masses.	−0.99c to +0.99c opens at +0.80c, −0.80c
Boost to S′, v_b	The speed of frame S′ relative to S. Changes the frame in which the table and the lower panel are read.	−0.99c to +0.99c opens at 0, or at +0.60c in free exploration
M_parent	The rest mass of the decaying parent. Active in Decay mode only.	0.2–10 opens at 3.33
Hide output text	Hides the numbers for the "after" state, so you can work them out first.	on / off
Four-momentum	Table of m, E and p for each body, with Σ rows before and after.	panel
Invariant mass²	(ΣE)² − (Σp)² and its square root M, in the frame you have chosen.	panel
Rest energy & mass budget	The rest energies in and out, and the difference ΔM (Decay mode: Q).	panel
▶ Play / ⏸ Pause	Runs the collision from the start, or freezes it. When it ends, Play runs it again.	top bar
Speed	Plays the collision faster or slower. No number changes.	0.25×–4×
↻ Reset	Restores the opening masses and speeds and starts again.	top bar

Fixed: c = 1, one space dimension, bodies treated as points.

What the collision conserves

A body of rest mass m moving at speed v has energy and momentum:

One body, one frame: v is measured in the frame you are standing in.

Together they form the four-momentum. Summed over all the bodies, it is the same before and after:

The only conservation law the sim enforces. The sim sets c = 1.

ΣE and Σp are each the same before and after, in any one frame.
Rest mass is not on that list. In the table, the Σ rows match across the collision; the m column need not.

Key idea. Add up E and p, not m. What the bodies become afterwards must carry exactly those two totals.

See itRun the default merge and read the Four-momentum table. Watch for: the Before and After Σ rows with the same E and p, while the m column changes.

Where the extra rest mass comes from

The default run: kinetic energy changes column but is not lost.

With Σp = 0 the merged body cannot move, so all of ΣE sits still. Energy of a body at rest is its rest energy, so M = ΣE.
The extra rest mass is exactly the kinetic energy the pair brought in:

Head-on with Σp = 0.

Kinetic energy is K = (γ − 1)mc², which starts as ½mv² and then grows faster:

At 0.80c, ½mv² gives 0.320 mc² and the true value is 0.667 mc².

"Inelastic" does not mean energy is lost: ΣE is unchanged. Part of it changes from kinetic to rest energy.
If Σp ≠ 0 the product keeps moving and keeps some kinetic energy, so ΔM is smaller than K₁ + K₂.

See itMerge the default pair and open the mass budget. Watch for: ΔM equal to K₁ + K₂.

See itFire m₁ = 2 at +0.80c into m₂ = 1 at rest. Watch for: ΔM well below K₁ + K₂, and the merged body still moving.

The invariant mass

The rest mass of any system follows from its totals. Square the total energy, subtract the square of the total momentum, and take the root:

Here M² = (ΣE)² − (Σp)², with c = 1.

These numbers follow the sliders in the sim.

For two equal masses at equal and opposite speeds, M = 2γm. M grows with γ, not with v.
With Σp ≠ 0, M is smaller than ΣE. M is the energy in the frame where Σp = 0.

See itRaise both speeds from 0.80c to 0.95c. Watch for: M almost doubling, from 3.33 to 6.41.

Conserved is not invariant

The Boost slider reads the same collision from a frame S′ moving at +v_b relative to S. Every E and p transforms:

Conserved: the same before and after, within one frame. ΣE and Σp are.
Invariant: the same number in every frame. ΣE and Σp are not. At v_b = +0.60c the default run reads ΣE′ = 4.167 and Σp′ = −2.500, against 3.333 and 0.000 in S.
The combination survives: 4.167² − 2.500² = 11.11 = 3.333². M is the same in both frames.

See itOpen the two-frame view at v_b = +0.60c, then drag the Boost slider. Watch for: ΣE and Σp changing with the frame while M² stays at 11.11.

Elastic collisions and decays

Elastic. In the frame where Σp = 0, each body reverses its momentum. Rest masses do not change, ΔM = 0, and the total kinetic energy is the same before and after.
Decay. The merge run backwards: a parent of rest mass M at rest splits into A and B, with the masses set by m₁ and m₂. The products leave with equal and opposite momenta.
Q = M − (m_A + m_B) is the rest energy that becomes kinetic energy. If Q < 0 there is not enough energy, and the sim shows ✕ Decay forbidden.
The product energies are fixed by M alone: E_A = (M² + m_A² − m_B²)/2M.

See itSwitch to Elastic and run the default pair. Watch for: both discs bouncing back, and the budget panel showing no change in rest mass.

See itSwitch to Decay with M_parent = 3.33 and m_A = m_B = 1. Watch for: one violet disc at rest splitting into two discs leaving at ±0.80c.

See itLower M_parent to 1.50, below m_A + m_B = 2. Watch for: ✕ Decay forbidden on the stage.

Where it matters

The same ledger in three settings: ΣE and Σp hold, and rest mass can change.

Colliders. Two 6.8 TeV beams meet head-on with Σp = 0, so all of the 13.6 TeV is available as invariant mass, more than enough for a 125 GeV Higgs boson.
Particle decay. A π⁰ at rest has 135 MeV of rest energy. It decays into two photons of 67.5 MeV each, with opposite momenta.
The Sun. Four hydrogen atoms outweigh the helium-4 atom they become by about 0.7%. The difference, about 26.7 MeV per helium atom, is released.

Go deeper: why colliders aim beams at each other

A beam of energy E on a target of mass m at rest has ΣE = E + m and Σp = √(E² − m²). Then M² = (E + m)² − (E² − m²) = 2m² + 2Em.
A 6.8 TeV proton on a proton at rest gives M ≈ 113 GeV, less than the 125 GeV Higgs. Two such beams head-on give 13.6 TeV. The sim's "fire m₁ = 2 into m₂ at rest" case shows the same loss: most of the kinetic energy stays kinetic.

Summary

In one line. Energy and momentum are conserved together, and they fix the rest mass through M² = (ΣE)² − (Σp)², a number every frame agrees on.

Key points

ΣE and Σp are each conserved within a frame; rest mass is not on the list.
A merged body can outweigh its parts, because kinetic energy counts as rest energy once it sits inside the product.
ΣE and Σp change with the frame. M does not.
Decay is the merge run backwards, allowed only when Q ≥ 0.

Common mix-ups

"Rest masses add: M = m₁ + m₂." Only when the bodies are at rest relative to each other. Otherwise add the four-momenta first, then take M.
"An inelastic collision loses energy." ΣE is unchanged. Kinetic energy becomes rest energy.
"E = mc² is the general formula." It is the Σp = 0 case. In general E² = p²c² + m²c⁴.

Things to try on your own

With m₁ = m₂ = 1, set v₁ = +0.10c and v₂ = −0.10c, merge, and compare ΔM with K₁ + K₂ = 2(γ − 1).
Set m₁ = 2, v₁ = +0.80c, m₂ = 1, v₂ = 0. Work out M from ΣE and Σp, then check the Invariant mass² panel.
In Decay mode with m₁ = m₂ = 1, find the smallest M_parent that still allows a decay.