Momentum

Definition

Momentum is mass times velocity:

p = m·v

It's a vector quantity — direction matters. Units: kg·m/s.

For multi-particle systems, total momentum is the vector sum:

p_total = m1·v1 + m2·v2 + m3·v3 + ...

The conservation law

The total momentum of an isolated system is constant in time.

"Isolated" means no external forces (or net external force = 0). In such a system, regardless of what internal collisions, explosions, or interactions occur, total momentum before = total momentum after.

This is a direct consequence of Newton's third law. Internal forces in any system come in equal-opposite pairs, so the impulses (F·dt) cancel pairwise, leaving total momentum unchanged.

Conservation in action

System	Before	After	Why p conserved
Cars colliding (no friction)	p1 + p2 (equal velocities, opposite signs)	Same total p (one or both moving)	No external horizontal force during collision
Rifle firing a bullet	0 (rest)	p_bullet + p_rifle = 0	Internal force only
Rocket in space	0 (initially at rest)	p_rocket + p_exhaust = 0	No external forces in vacuum
Newton's cradle	p of one ball	p of one ball at far end	Steel-on-steel ≈ elastic; momentum passes through
Particle decay (e.g., π → μ + ν)	p of pion at rest = 0	p_muon + p_neutrino = 0	Forces are internal (weak)
Asteroid hitting Earth	p_asteroid + p_Earth ≈ p_asteroid	Same; Earth changes v slightly	External forces negligible during impact

Impulse-momentum theorem

From Newton's second law F = dp/dt, integrating both sides over time:

∫F·dt = Δp

The left side is called impulse (J). So impulse equals change in momentum:

J = Δp = m·Δv

This is hugely practical. To stop a moving object, you need to apply enough impulse. You can do this with:

• A large force over a short time (sudden braking, hard impact).
• A small force over a long time (gentle braking, gradual deceleration).

Same momentum change, very different forces felt. This is the principle behind airbags, crumple zones, and parkour rolls — extend the deceleration time to reduce peak force.

Elastic vs inelastic collisions

In elastic collisions, both momentum AND kinetic energy are conserved. Examples — billiard balls (approximately), atomic collisions, hard spheres.

In inelastic collisions, momentum is conserved but kinetic energy is NOT. Some KE goes into heat, sound, deformation. A perfectly inelastic collision is when the two objects stick together (max KE loss). Examples — clay balls colliding, bullet embedding in wood.

Collision type	Momentum	Kinetic energy	Example
Elastic	Conserved	Conserved	Billiard balls, idealized atomic
Inelastic	Conserved	Lost (some)	Most real collisions
Perfectly inelastic	Conserved	Lost (max)	Clay balls, train coupling
Explosive	Conserved	Increased (chemical → KE)	Bullet exits gun, fireworks

JavaScript — momentum-conserving collisions

// 1D elastic collision between two particles
function elasticCollision1D(m1, v1, m2, v2) {
  // Solve momentum and KE conservation simultaneously:
  // m1·v1 + m2·v2 = m1·v1' + m2·v2'
  // ½m1·v1² + ½m2·v2² = ½m1·v1'² + ½m2·v2'²
  const v1New = ((m1 - m2) * v1 + 2 * m2 * v2) / (m1 + m2);
  const v2New = ((m2 - m1) * v2 + 2 * m1 * v1) / (m1 + m2);
  return [v1New, v2New];
}

console.log(elasticCollision1D(1, 5, 1, 0));   // [0, 5] — masses swap velocities
console.log(elasticCollision1D(10, 5, 1, 0));  // ~[4.09, 9.09] — heavy keeps speed
console.log(elasticCollision1D(1, 5, 10, 0));  // ~[-4.09, 0.91] — light bounces back

// Verify momentum and KE conservation
function checkConservation(m1, v1Initial, m2, v2Initial) {
  const [v1Final, v2Final] = elasticCollision1D(m1, v1Initial, m2, v2Initial);
  const pInitial = m1 * v1Initial + m2 * v2Initial;
  const pFinal = m1 * v1Final + m2 * v2Final;
  const KEInitial = 0.5 * m1 * v1Initial**2 + 0.5 * m2 * v2Initial**2;
  const KEFinal = 0.5 * m1 * v1Final**2 + 0.5 * m2 * v2Final**2;
  console.log(`Momentum: ${pInitial.toFixed(4)} → ${pFinal.toFixed(4)} (conserved: ${Math.abs(pInitial - pFinal) < 1e-6})`);
  console.log(`KE:       ${KEInitial.toFixed(4)} → ${KEFinal.toFixed(4)} (conserved: ${Math.abs(KEInitial - KEFinal) < 1e-6})`);
}

checkConservation(1, 5, 2, -3);

// Perfectly inelastic — they stick
function inelasticCollision1D(m1, v1, m2, v2) {
  const vFinal = (m1 * v1 + m2 * v2) / (m1 + m2);
  const KEInitial = 0.5 * m1 * v1**2 + 0.5 * m2 * v2**2;
  const KEFinal = 0.5 * (m1 + m2) * vFinal**2;
  return { vFinal, KELost: KEInitial - KEFinal };
}

console.log(inelasticCollision1D(1, 10, 1, 0));  // {vFinal: 5, KELost: 25}

Rockets and momentum conservation

Rockets work entirely from momentum conservation. The rocket-plus-fuel system starts at rest. Burning fuel pushes exhaust gas out the back at high velocity. By conservation:

0 = p_rocket + p_exhaust
0 = m_rocket · v_rocket + m_exhaust · (-v_exhaust)
v_rocket = (m_exhaust / m_rocket) · v_exhaust

The rocket equation (Tsiolkovsky):

Δv = v_exhaust · ln(m_initial / m_final)

This is why rockets work in vacuum — no air to push against, just internal momentum exchange.

Where momentum shows up

• Collisions and impacts. Car crashes, billiards, sports physics, particle physics — momentum conservation solves them all.
• Rocket and propulsion physics. Tsiolkovsky's equation, ion drives, jet propulsion. All built on momentum conservation.
• Quantum mechanics. de Broglie wavelength λ = h/p. Heisenberg uncertainty Δx · Δp ≥ ℏ/2. Momentum is a fundamental quantum observable.
• Astrodynamics. Gravitational slingshots — spacecraft gain momentum from planetary flybys (planet loses an immeasurably tiny amount).
• Particle physics. Decay analysis, conservation in scattering, accelerator physics. Every collision experiment uses momentum conservation.
• Sports analytics. Football tackle physics, baseball pitch dynamics, tennis racket-ball impact.
• Engineering — recoil and propulsion. Gun design, rocket thrust, jet engines, drone hover physics.

Common mistakes

• Forgetting it's a vector. Momentum is mv with direction. Two equal-mass cars going opposite directions have zero total momentum, not 2mv. Always track vectors.
• Confusing momentum with kinetic energy. p = mv (linear); KE = ½mv² (quadratic). Conserved in different scenarios — both in elastic collisions, only momentum in inelastic.
• Ignoring external forces. "Isolated system" requires no external force (gravity, friction, applied). If external forces exist, total momentum changes; only momentum components perpendicular to external force are conserved.
• Using F = ma when F = dp/dt is needed. For variable-mass systems (rockets, ablating asteroids), use the dp/dt form. F = ma misses the dm/dt term.
• Treating relativistic regime as classical. At high speeds (v > 0.1c), use p = γmv, not p = mv. Particle accelerators, cosmic rays — relativistic momentum is essential.
• Computing p of light as zero. Photons have p = E/c = h/λ. Using p = mv with m=0 gives zero, but that's wrong — photons carry momentum.