Elastic vs Inelastic Collisions

Conservation laws

For an isolated system (no external forces during collision):

• Momentum is ALWAYS conserved — total p before = total p after. This holds regardless of collision type. Comes from Newton's third law (internal forces cancel).
• Kinetic energy may or may not be conserved — depends on collision type.

Types of collisions

Type	Momentum	KE	e (coefficient of restitution)	Examples
Elastic	Conserved	Conserved	e = 1	Atomic, hard-sphere, idealized billiards
Inelastic	Conserved	NOT conserved (some lost)	0 < e < 1	Most real collisions
Perfectly inelastic	Conserved	NOT conserved (max lost)	e = 0	Clay balls stick, train coupling, dart-board
Explosive (super-elastic)	Conserved	Increases (chemical/nuclear E added)	e > 1	Bullet from gun, fireworks, fission

Elastic 1D collisions

Conserve both momentum AND kinetic energy:

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²

Solving the system:

v₁' = ((m₁ - m₂)·v₁ + 2·m₂·v₂) / (m₁ + m₂)
v₂' = ((m₂ - m₁)·v₂ + 2·m₁·v₁) / (m₁ + m₂)

Special case	Result
Equal mass, one at rest (m₂ at rest)	v₁' = 0, v₂' = v₁ (velocities swap)
Heavy hits light (m₁ ≫ m₂, m₂ at rest)	v₁' ≈ v₁, v₂' ≈ 2v₁ (light one bounces away fast)
Light hits heavy (m₁ ≪ m₂, m₂ at rest)	v₁' ≈ -v₁, v₂' ≈ 0 (light one rebounds)
Equal masses, opposite velocities	v₁' = -v₁, v₂' = -v₂ (each reflects)

Perfectly inelastic collision

Objects stick together after collision. Use momentum conservation:

m₁v₁ + m₂v₂ = (m₁ + m₂)·v_f
v_f = (m₁v₁ + m₂v₂) / (m₁ + m₂)

Energy lost:

ΔKE = (½m₁v₁² + ½m₂v₂²) - ½(m₁ + m₂)·v_f²

This is the maximum possible KE loss (consistent with momentum conservation).

Coefficient of restitution

For 1D collisions, define:

e = (v₂' - v₁') / (v₁ - v₂)

i.e., the ratio of relative velocity after to relative velocity before.

e	Behavior	Examples
e = 1	Elastic — full bounce	Idealized billiards, atoms
e = 0.95	Nearly elastic	Real billiard balls, super balls
e = 0.75	Bouncy	Basketball, soccer ball
e = 0.5	Modest bounce	Baseball off bat, tennis ball
e = 0.2	Mostly inelastic	Glass marble on wood
e = 0	Perfectly inelastic	Clay balls, dart sticking

JavaScript — collision calculations

// Elastic 1D collision
function elastic1D(m1, v1, m2, v2) {
  const total_mass = m1 + m2;
  return [
    ((m1 - m2) * v1 + 2 * m2 * v2) / total_mass,
    ((m2 - m1) * v2 + 2 * m1 * v1) / total_mass
  ];
}

// Perfectly inelastic 1D
function inelastic1D(m1, v1, m2, v2) {
  const v_final = (m1 * v1 + m2 * v2) / (m1 + m2);
  const KE_initial = 0.5 * m1 * v1 * v1 + 0.5 * m2 * v2 * v2;
  const KE_final = 0.5 * (m1 + m2) * v_final * v_final;
  return { v_final, KE_lost: KE_initial - KE_final };
}

// Partial inelastic with restitution e
function partialInelastic(m1, v1, m2, v2, e) {
  // Use momentum conservation + e = (v2'-v1')/(v1-v2)
  const v_rel_initial = v1 - v2;
  const v_com = (m1 * v1 + m2 * v2) / (m1 + m2);  // CoM velocity
  // In COM frame, v_rel transforms to e*v_rel after
  const v1_prime = v_com - (m2 / (m1 + m2)) * e * v_rel_initial;
  const v2_prime = v_com + (m1 / (m1 + m2)) * e * v_rel_initial;
  return [v1_prime, v2_prime];
}

// Test
console.log('Elastic, equal masses, one at rest:');
console.log(elastic1D(1, 5, 1, 0));  // [0, 5] — velocities swap

console.log('Heavy hits light:');
console.log(elastic1D(10, 5, 1, 0));  // [4.09, 9.09] — light one shoots out

console.log('Inelastic clay collision:');
console.log(inelastic1D(1, 5, 1, -3));  // KE lost is significant

console.log('Partial bounce, e=0.5:');
console.log(partialInelastic(1, 5, 1, 0, 0.5));  // intermediate result

// Verify momentum conservation
function checkMomentum(m1, v1_init, m2, v2_init, v1_final, v2_final) {
  const p_init = m1 * v1_init + m2 * v2_init;
  const p_final = m1 * v1_final + m2 * v2_final;
  return { initial: p_init, final: p_final, conserved: Math.abs(p_init - p_final) < 1e-9 };
}

const [v1_p, v2_p] = elastic1D(2, 3, 5, -1);
console.log(checkMomentum(2, 3, 5, -1, v1_p, v2_p));

Where collision physics shows up

• Sports. Bat-ball impact, baseball/tennis/ping-pong physics, billiards, hockey collisions, football tackles. Coefficient of restitution determines bat performance.
• Auto safety. Crash analysis — kinetic energy must be absorbed. Crumple zones convert KE to deformation energy, reducing peak forces on passengers.
• Particle physics. All experiments are collisions — elastic scattering at low energies, inelastic at high (creating new particles). Detectors measure momentum and energy of products.
• Astronomy. Asteroid impacts, planetary formation, ring system collisions. Most planetary impacts are inelastic (heat, melting).
• Engineering. Hammers (elastic for max force on nail), pile drivers, presses (controlled inelastic for stamping), shock absorbers.
• Material science. Drop tests, impact tolerance, ballistic protection. Bulletproof vests use inelastic collision physics.
• Robotics. Walking robots, soccer-playing robots — collisions with ground and ball must be modeled.

Common mistakes

• Confusing momentum and KE conservation. Momentum ALWAYS conserved (in isolated systems). KE only sometimes (elastic). Don't conflate.
• Forgetting "isolated" requirement. If external forces act DURING the brief collision, momentum changes. For very short impacts, internal forces dominate, so external is usually ignorable.
• Misusing inelastic collision formulas. Elastic — 2 equations (p, KE), 2 unknowns. Perfectly inelastic — 1 equation (p), 1 unknown (since they stick). Partial — needs e to solve.
• Treating real collisions as elastic. Most real macroscopic collisions are inelastic to some degree. Sound and heat losses are usually small but nonzero.
• Ignoring 2D/3D nature. A glancing billiard collision is 2D — momentum conserved in both x AND y. Final speeds depend on impact angle.
• Treating e as material property only. Coefficient of restitution depends on BOTH materials, impact velocity, and even temperature. Tabulated values are approximate.