Momentum: p = m·v

Momentum (linear momentum, symbol p) is one of the most important quantities in classical mechanics. Defined as mass times velocity: p = m·v. It's a vector: magnitude, direction, and sense like velocity. SI unit: kg·m/s (equivalent to N·s). This calculator handles 4 modes: single object momentum, 1D elastic collisions, perfectly inelastic collisions, and impulse J = F·Δt.

Fundamental formulas

• p = m · v — momentum of an object
• KE = p² / (2m) = ½·m·v² — link to kinetic energy
• p_tot = Σ p_i — vector sum for particle system
• J = F·Δt = Δp — impulse equals momentum change
• F = dp/dt — Newton's second law general form

Momentum conservation

The law of momentum conservation: in an isolated system (no external forces), total momentum is constant:

m₁·v₁ + m₂·v₂ = m₁·v₁' + m₂·v₂'

Holds for any internal interaction: collisions, explosions, separations. Follows from space homogeneity (Noether's theorem, 1915).

1D elastic collisions

In elastic collision, both momentum and kinetic energy are conserved. Typical:

• Billiard balls (approximately)
• Atom/molecule collisions (in ideal gases)
• Subatomic particles
• Steel balls on rigid surfaces

Final velocities in 1D:

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

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

Special cases

• m₁ = m₂: velocities swap. Classic billiard pattern.
• m₁ ≫ m₂: v₁' ≈ v₁, v₂' ≈ 2v₁
• m₁ ≪ m₂ (ball on wall): v₁' ≈ -v₁, v₂' ≈ 0
• v₂ = 0: v₁' = ((m₁-m₂)/(m₁+m₂))·v₁, v₂' = (2m₁/(m₁+m₂))·v₁

Perfectly inelastic collisions

In perfectly inelastic collisions, objects stick together. Momentum conserved but kinetic energy not (some becomes heat, deformation, sound):

v_f = (m₁·v₁ + m₂·v₂) / (m₁+m₂)

Energy lost:

ΔKE = (½·m₁·m₂)/(m₁+m₂) · (v₁-v₂)²

Inelastic examples

• Car crashes with deformation
• Bullet embedding in target
• Meteor impacting planet
• Neutron capture by nucleus

Partially inelastic collisions

Most real collisions are between extremes. Characterized by coefficient of restitution:

e = |v₂' - v₁'| / |v₁ - v₂|

• e = 1: perfectly elastic
• 0 < e < 1: partially inelastic
• e = 0: perfectly inelastic

Real examples:

• Basketball on hardwood: e ≈ 0.75-0.85
• Tennis ball on concrete: e ≈ 0.75
• Golf ball: e ≈ 0.78
• Ping-pong: e ≈ 0.9
• Superball: e ≈ 0.95

Practical numerical examples

Example 1: moving car

1,500 kg car at 20 m/s (72 km/h).

• p = 1,500 × 20 = 30,000 kg·m/s
• KE = ½ × 1,500 × 400 = 300,000 J = 300 kJ
• Check: KE = p²/(2m) = 300,000 J ✓

Example 2: car-motorcycle elastic

Car 1,500 kg at 20 m/s hits stationary 200 kg motorcycle:

• v₁' = ((1500-200)·20)/1700 = 15.3 m/s (car slows)
• v₂' = (2·1500·20)/1700 = 35.3 m/s (bike launches)
• Momentum before: 30,000, after: 22,950 + 7,060 ≈ 30,000 ✓
• Ideal elastic (not realistic for cars/bikes)

Example 3: inelastic car crash

Car A (1,500 kg, 20 m/s) rear-ends car B (1,200 kg, stationary). Stuck together.

• v_f = (1500·20)/2700 = 11.1 m/s
• KE before = 300 kJ, after = ½·2700·11.1² = 166 kJ
• Energy lost = 134 kJ (45% dissipated in deformation, heat, sound)
• Basis of "crumple zone" design — deforming absorbs energy protecting occupants

Example 4: rifle firing

4 kg rifle fires 20 g bullet at 800 m/s.

• Before: p_tot = 0
• Bullet momentum: 0.020 × 800 = 16 kg·m/s
• Rifle momentum (recoil): -16 kg·m/s
• v_recoil = -16/4 = -4 m/s backward
• Significant recoil: why butt is pressed against shoulder

Example 5: airbag impulse

5 kg head at 20 m/s must stop in crash.

• Δp = 100 kg·m/s to reduce to zero
• No airbag (Δt = 0.01 s against wheel): F = 10,000 N → fatal
• With airbag (Δt = 0.1 s): F = 1,000 N → survivable
• Same Δp but F reduced 10×: airbag saves life by distributing deceleration over time

Impulse J = F·Δt

Impulse:

J = ∫F·dt = Δp

For constant force: J = F·Δt. Equals momentum change (impulse theorem).

Unit: N·s (= kg·m/s). Explains protective phenomena:

• Airbags: increase Δt → reduce F_avg
• Shock absorbers: longer deceleration time
• Helmets: foam compresses
• Parachute landing: rolling increases Δt
• Karate: short contact maximizes F
• Rocket thrust: J = ∫F·dt determines Δv

Momentum vs kinetic energy

Doubling v: momentum doubles, KE quadruples. Same p, larger mass = less KE.

Conservation applications

Galileo (ships)

Showed inside a uniformly moving ship nothing reveals motion (Galilean relativity). Inside isolated inertial system momentum conserved.

Rocket propulsion (Tsiolkovsky)

Rocket equation (1903) from momentum conservation: ejecting gas at v_e, remaining mass gains Δv = v_e · ln(m₀/m).

Particle colliders

LHC: protons at 99.999999% c collide. Momentum conservation lets reconstruct invisible particle tracks (neutrinos) by measuring "missing momentum".

Galaxy physics

Angular momentum (rotational analog) conserved during galaxy formation, explaining flat rotation.

2D and 3D collisions

Momentum conservation per component independently:

Σ p_x_before = Σ p_x_after

Σ p_y_before = Σ p_y_after

Non-head-on collisions require separate x, y component analysis.

Quantum momentum

De Broglie (1924) wavelength:

λ = h / p

h = 6.626×10⁻³⁴ J·s. Electron at 10⁶ m/s: λ ≈ 0.73 nm. 70 kg person at 1 m/s: λ ≈ 10⁻³⁵ m (unobservable).

Heisenberg uncertainty:

Δx · Δp ≥ ℏ/2

Relativistic momentum

Near c, p = mv fails. Einstein:

p = γ·m·v, γ = 1/√(1-v²/c²)

At low v γ ≈ 1, recovers classical. At v = 0.9c, γ ≈ 2.3. v → c requires infinite energy.

History

Galileo (1638): inertia intuition.

Descartes (1644): proposed m·|v| conserved — wrong (no vector).

Huygens (1669): discovers conservation of m·v (with sign) in collisions. Correct.

Newton (1687): F = dp/dt. Third law → momentum conservation for pairs.

Emmy Noether (1915): theorem. Momentum ↔ space homogeneity.

Applications

Car safety

• Belts, airbags, crumple zones
• Crash tests: deceleration and peak F
• Design minimizes peak F via deceleration time

Sports

• Billiards: ball-ball collisions
• Soccer: foot-ball momentum transfer (~0.4 s contact)
• Karate/boxing: contact time calibration
• Baseball: bat transfers momentum
• Tennis: racket strings

Ballistics

• Firearm recoil
• Projectile penetration
• Crime scene reconstruction

Aerospace

• Rocket thrust (momentum conservation)
• Gravitational slingshots (momentum exchange)
• Orbital rendezvous

Nuclear physics

• Particle scattering
• Nuclear reactions
• Gamma spectroscopy

Astronomy

• Solar system formation (angular momentum)
• Binary stars
• Galaxy mergers

FAQ

Momentum formula?

p = m·v, m in kg, v in m/s. Result in kg·m/s. Vector with direction of v.

Is momentum always conserved?

Yes in isolated system (no external forces). Most robust law in physics. External forces change total momentum via F_ext = dp/dt.

Elastic vs inelastic collision difference?

Elastic: both momentum and KE conserved. Inelastic: only momentum, some energy to heat/deformation. Perfectly inelastic: objects stick.

How does an airbag work?

Increases deceleration time Δt (from 0.01 s to 0.1 s), reducing average force F = Δp/Δt by factor 10. Same final Δp (body must stop), but max F much lower → less trauma.

Is momentum conservation absolute?

Yes, holds in relativity (p = γmv) and quantum mechanics. Linked to spatial translation symmetry (Noether 1915). Most solid law in physics.

Why rifle recoil equals bullet momentum?

By momentum conservation: before shot p_tot = 0. After: bullet has +p_p, so rifle must have -p_p. Rifle velocity v_r = -m_p·v_p/m_f is small because m_f ≫ m_p.

Momentum scalar or vector?

Vector. Magnitude m·|v|, direction of v. 1D uses signed magnitude. 2D/3D uses components (p_x, p_y, p_z), each conserved independently.

Angular momentum

Angular momentum: rotational analog:

L = r × p = I·ω

r radius, I moment of inertia, ω angular velocity. Conserved without external torques. Explains:

• Spinning skater pulling arms in (I decreases → ω increases)
• Gyroscope stability
• Planetary orbits (Kepler's 2nd law)
• Subatomic particle spin (quantized in ℏ)

Common errors

• Forgetting vector nature: velocity sign matters
• Confusing p with KE: p linear, KE quadratic
• Using elastic for car crashes: real crashes are inelastic
• Ignoring external forces: friction breaks isolation
• Forgetting impulse sign: Δp = p_final - p_initial
• Classical formula near c: need p = γmv

Updated July 2026