1D Elastic Collision PhysicsMomentum and KE Both Conserved

Introduction

A one-dimensional elastic collision is a head-on impact in which the two bodies bounce apart with no kinetic energy lost, so momentum and kinetic energy are conserved at the same time. That double constraint is what makes the elastic case special. Momentum conservation alone leaves one free parameter and admits infinitely many outcomes; adding the kinetic-energy ledger closes the system and pins a single pair of post-impact velocities for any masses and any approach speeds. The simulator stages the event on a frictionless track: two carts start 16 m apart, close on each other, and separate with velocities the closed-form solution fixes in advance.

The mass ratio does all the work. Equal masses trade velocities exactly, a light cart striking a heavy one rebounds with most of its speed intact, and a heavy cart striking a light one barely notices the impact while flinging the light cart away at high speed. Billiards, neutron moderation in reactors, and gravitational slingshots are all the same two equations read at three different mass ratios.

The common expectation is that a collision must cost something, so the carts should come away slower than they arrived. The readouts refuse. With Mass 1 = 1.0 kg, Mass 2 = 3.0 kg, Velocity 1 = +4.0 m/s and Velocity 2 = −2.0 m/s, the v₁ readout flips from +4.00 m/s to −5.00 m/s while v₂ moves from −2.00 m/s to +1.00 m/s. Cart 1 leaves the impact faster than it entered, yet momentum holds at −2.00 kg·m/s and kinetic energy holds at 14.00 J.

The Physics Explained

Two statements do the whole job. Newton's third law makes the contact impulses equal and opposite, so the total momentum m₁v₁ + m₂v₂ is the same before and after the impact. The word "elastic" adds the second statement: the deformation during contact is fully reversible, nothing is dissipated as heat, sound, or permanent bending, so the total kinetic energy ½m₁v₁² + ½m₂v₂² is also the same before and after. At the default settings the individual momenta are p₁ = 1.0 · 4.0 = 4.00 kg·m/s and p₂ = 3.0 · (−2.0) = −6.00 kg·m/s, and their sum of −2.00 kg·m/s is the number the momentum readout holds all run.

Solving the two equations together gives the closed forms the simulator uses. Substituting the defaults of m₁ = 1.0 kg, m₂ = 3.0 kg, v₁ = 4.0 m/s and v₂ = −2.0 m/s returns v₁' = −5.00 m/s and v₂' = +1.00 m/s. Feeding those back through the ledgers confirms both: the momentum comes to 1.0 · (−5.00) + 3.0 · 1.00 = −2.00 kg·m/s and the kinetic energy to 0.5 · 1.0 · 25.0 + 0.5 · 3.0 · 1.0 = 14.00 J, matching the pre-impact 14.00 J exactly.

Equal masses collapse the algebra to a clean rule. When m₁ = m₂ the leading coefficient of each formula vanishes and the two carts simply exchange velocities. Setting Mass 1 = 2.0 kg, Mass 2 = 2.0 kg, Velocity 1 = 5.0 m/s and Velocity 2 = −3.0 m/s makes the v₁ readout land on −3.00 m/s and the v₂ readout on 5.00 m/s, each cart adopting the other's incoming velocity. The momentum readout stays at 4.00 kg·m/s and the kinetic-energy readout at 34.00 J, because a swap can never disturb either sum.

The stationary-target version of that swap is the most useful special case. With Mass 1 = 2.0 kg, Mass 2 = 2.0 kg, Velocity 1 = 6.0 m/s and Velocity 2 = 0 m/s, cart 1 arrives carrying the full 36.00 J of the system and leaves with the v₁ readout at 0.00 m/s while v₂ reads 6.00 m/s. Every joule transfers. That total handover happens only at a mass ratio of exactly one, and the transferred fraction falls away in both directions as the ratio moves off it, which is the single fact behind the reactor-moderator example below.

Key Equations

Momentum is the sum that survives every collision, elastic or not. At the defaults of m₁ = 1.0 kg, m₂ = 3.0 kg, v₁ = 4.0 m/s and v₂ = −2.0 m/s the left side is 1.0 · 4.0 + 3.0 · (−2.0) = −2.00 kg·m/s, and after the impact the right side is 1.0 · (−5.0) + 3.0 · 1.0 = −2.00 kg·m/s. The momentum readout never moves off that value, and the dashed reference line on the total-momentum bar marks it for comparison after impact.

This second ledger is what makes the collision elastic rather than merely momentum-conserving. Before impact at the defaults: 0.5 · 1.0 · 16.0 + 0.5 · 3.0 · 4.0 = 8.00 + 6.00 = 14.00 J. After impact: 0.5 · 1.0 · 25.0 + 0.5 · 3.0 · 1.0 = 12.50 + 1.50 = 14.00 J. The kinetic-energy readout holds at 14.00 J across the collision frame, which is the visual signature separating this sim from the inelastic case.

Solving the two conservation equations simultaneously eliminates v₂' and leaves this closed form. At the defaults: v₁' = ((1.0 − 3.0) · 4.0 + 2 · 3.0 · (−2.0)) / (1.0 + 3.0) = (−8.0 − 12.0) / 4.0 = −5.00 m/s. The sign flip says cart 1 rebounds, and its speed rises from 4.00 m/s to 5.00 m/s because it borrows momentum from the heavier cart coming the other way.

The mirror-image solution gives the second cart. At the defaults: v₂' = ((3.0 − 1.0) · (−2.0) + 2 · 1.0 · 4.0) / (1.0 + 3.0) = (−4.0 + 8.0) / 4.0 = +1.00 m/s. Cart 2 reverses from −2.00 m/s to +1.00 m/s. Setting m₁ = m₂ makes the first term vanish in both formulas, which is exactly the velocity swap the equal-mass runs display.

Key Variables

Real World Examples

Why does the cue ball stop dead when it strikes another billiard ball head-on?

A clean head-on shot between two balls of the same mass is the closest everyday match to a perfectly elastic 1D collision. The two conservation laws leave exactly one outcome available: the moving ball hands over its entire velocity and the struck ball leaves with the speed the cue ball arrived at. Any other split would satisfy momentum but break the kinetic-energy ledger, so the physics simply does not allow it.

The simulator reproduces the shot directly. Setting Mass 1 = 2.0 kg, Mass 2 = 2.0 kg, Velocity 1 = 6.0 m/s and Velocity 2 = 0 m/s puts the momentum readout at 12.00 kg·m/s and the kinetic-energy readout at 36.00 J. After impact the v₁ readout falls to 0.00 m/s and the v₂ readout rises to 6.00 m/s, while both conserved quantities stay exactly where they started.

Real cue balls do not stop quite so perfectly because a struck ball is rolling rather than sliding, felt friction bleeds energy, and no impact is perfectly elastic. Players still rely on the idealisation constantly: the stop shot is taught as the reference stroke precisely because equal masses in a head-on elastic collision must trade velocities completely.

Why do nuclear reactors slow neutrons with light nuclei instead of heavy ones?

A fast neutron only becomes useful for sustaining fission once it has been slowed to thermal speeds, and the cheapest way to slow it is to let it bounce elastically off other nuclei. How much speed a single bounce removes depends entirely on the mass ratio, which is why moderator choice is a mass-ratio problem rather than a chemistry problem.

The simulator shows the losing case first. With Mass 1 = 0.5 kg as the light projectile, Mass 2 = 5.0 kg as a heavy target, Velocity 1 = 8.0 m/s and Velocity 2 = 0 m/s, the v₁ readout returns −6.55 m/s and the v₂ readout only 1.45 m/s. The light body rebounds with most of its speed intact and the kinetic-energy readout holds at 16.00 J, of which the heavy target has absorbed barely a third.

Swapping to equal masses transforms the transfer. At Mass 1 = 2.0 kg, Mass 2 = 2.0 kg, Velocity 1 = 6.0 m/s and Velocity 2 = 0 m/s the projectile stops dead and every joule of the 36.00 J moves to the target. Hydrogen in ordinary water is the closest common nucleus to a neutron in mass, which is why water and heavy water moderate so efficiently while a heavy nucleus such as lead barely slows a neutron at all.

How does a spacecraft gain speed from a gravitational slingshot?

A gravity assist is an elastic collision fought at a distance. No surfaces touch, but gravity exchanges momentum between the probe and the planet exactly as a spring would, and because gravity is conservative the encounter conserves kinetic energy in the two-body frame just as a perfectly elastic impact does. The planet is so much heavier that it absorbs the momentum change with an unmeasurable velocity shift while the probe leaves transformed.

The simulator scales the encounter down to sliders. Setting Mass 1 = 0.5 kg for the probe, Mass 2 = 5.0 kg for the planet, Velocity 1 = 4.0 m/s and Velocity 2 = −6.0 m/s produces a v₁ readout of −14.18 m/s after the encounter, against an incoming speed of 4.00 m/s. The momentum readout holds at −28.00 kg·m/s and the kinetic-energy readout at 94.00 J throughout, so the probe's gain is paid for entirely by the heavier body.

The exchange looks like free energy only because the planet's share is invisible. In the limit of a very heavy target the light body leaves with roughly its own approach speed plus twice the target's speed, which is why mission planners aim probes at the leading side of a moving planet. Voyager 2 climbed from Jupiter to Neptune on four such encounters, each one a wider version of the same two-line calculation.

Further Reading

• Inelastic collisions: the opposite limit, where the two bodies stick together, momentum still balances and a fixed fraction of the kinetic energy is lost to deformation.
• Elastic collisions: the same conservation pair explored through a second simulator, with the coefficient of restitution as the dial between the elastic and inelastic extremes.
• Two-dimensional collisions: what changes when the impact is off-centre and momentum has to balance along two axes at once.
• Momentum vector: why p = mv and KE = ½mv² respond so differently to the same change in mass or speed, on a single-cart track.