What Is Momentum?
Momentum is, at its simplest, a measure of how much "motion" an object has — and how difficult it is to stop. Formally, the linear momentum of an object is defined as:
A 1,500 kg car travelling at 20 m/s has the same momentum as a 30 kg person running at 1,000 m/s — both have p = 30,000 kg·m/s. What's remarkable isn't the definition, but what happens when objects interact: the total momentum of an isolated system never changes.
This is the conservation of momentum — one of the most powerful and universally applicable principles in all of physics. It applies to billiard balls, subatomic particles, rocket engines, supernovae, and galaxy collisions. Wherever there's an interaction, momentum tells you what the outcome must be.
In an isolated system (no external forces), the total momentum before any interaction equals the total momentum after: p_total(before) = p_total(after).
Why Is Momentum Conserved? The Derivation from Newton's Third Law
Conservation of momentum isn't a separate postulate — it follows directly from Newton's Third Law: "for every action, there is an equal and opposite reaction." This derivation is one of the most elegant in classical physics, and it's worth going through carefully.
Consider two objects A and B that interact (collide, push, attract each other). By Newton's Third Law, the force that A exerts on B is equal and opposite to the force that B exerts on A:
By Newton's Second Law, force equals rate of change of momentum: F = dp/dt. So:
If the total rate of change of momentum is zero, then total momentum is constant. That's it. The conservation of momentum is Newton's Third Law in integral form. They're the same statement about physics expressed differently.
This connection also reveals when conservation of momentum breaks down: when there are external forces acting on the system. A ball rolling on the ground isn't isolated — friction is an external force, so its momentum decreases. The system you need to consider must include everything that exerts a force on anything inside it. Expand the boundary of your system until no external forces remain, and momentum is always conserved.
Impulse: The Bridge Between Force and Momentum Change
The impulse-momentum theorem connects the force acting on an object to the change in its momentum:
Impulse is why airbags save lives. A car crash changes your momentum by a fixed amount regardless of whether an airbag deploys. But Δp = FΔt: by increasing the time Δt over which the momentum change occurs (airbag extends the collision from ~10 ms to ~100 ms), the peak force F on your body decreases by roughly 10×. Same impulse, far smaller force, far less injury.
The same principle explains why martial artists break boards with bare hands: they aim to maximise force by minimising contact time (very hard, very fast strike), maximising F for a given impulse.
The impulse J = FΔt uses the average force over the collision interval. The peak force can be much higher. Engineers designing crumple zones, helmets, and packaging must consider the force-time profile, not just the impulse.
Types of Collisions: Elastic, Inelastic, and Perfectly Inelastic
Momentum is conserved in all collisions (for isolated systems). What distinguishes collision types is whether kinetic energy is also conserved:
| Collision Type | Momentum | Kinetic Energy | Real-World Examples |
|---|---|---|---|
| Elastic | Conserved ✓ | Conserved ✓ | Billiard balls, atomic/molecular collisions, Newton's cradle (approximately) |
| Inelastic | Conserved ✓ | Not conserved ✗ | Most macroscopic collisions — car crashes, dropped balls, bat hitting ball |
| Perfectly Inelastic | Conserved ✓ | Maximum loss ✗ | Car crash where vehicles stick together, clay catching a ball, bullet embedding in block |
In inelastic collisions, the "lost" kinetic energy doesn't disappear — it converts into heat, sound, and deformation (as required by energy conservation from the First Law of Thermodynamics). When a car crumples in a crash, the deformation absorbs kinetic energy deliberately — that's crash engineering working as designed.
Elastic Collision Equations (Two Objects)
For two objects with masses m₁, m₂ and initial velocities u₁, u₂, conserving both momentum and kinetic energy gives:
These look complicated, but check the special cases: if m₁ = m₂, then v₁ = u₂ and v₂ = u₁ — the objects swap velocities. This is exactly what billiard balls do when one strikes a stationary ball head-on.
Five Worked Examples
Perfectly inelastic collision — two carts
Conservation of momentum: m₁u₁ + m₂u₂ = (m₁+m₂)v
2(6) + 3(0) = (2+3)v → 12 = 5v → v = 2.4 m/s
KE before: ½(2)(36) = 36 J. KE after: ½(5)(5.76) = 14.4 J. Energy lost: 36 − 14.4 = 21.6 J (to heat/sound)
Elastic collision — verify kinetic energy is conserved
Total initial momentum: p_i = 4(5) + 2(−2) = 20 − 4 = 16 kg·m/s
Using elastic collision formula: v₁ = [(4−2)(5) + 2(2)(−2)]/(4+2) = [10 − 8]/6 = 2/6 = 0.33 m/s
v₂ = [(2−4)(−2) + 2(4)(5)]/(6) = [4 + 40]/6 = 44/6 = 7.33 m/s
Check momentum: 4(0.33) + 2(7.33) = 1.33 + 14.67 = 16 ✓. Check KE: ½(4)(25)+½(2)(4) = 54 J before; ½(4)(0.109)+½(2)(53.7) = 0.22+53.7 ≈ 54 J ✓
Explosion — gun recoil
Initial momentum = 0. By conservation: 0 = mv_b + MV_rifle
V_rifle = −mv_b / M = −(0.015 × 900)/3.5 = −13.5/3.5 = −3.86 m/s
The rifle recoils backward at 3.86 m/s — roughly 233 times slower than the bullet (as expected from the mass ratio).
Bullet embeds in block — find initial speed
Perfectly inelastic collision: mu = (m+M)v_f
u = (m+M)v_f / m = (2.01 × 1.2)/0.01 = 2.412/0.01 = 241.2 m/s
Energy check — KE before: ½(0.01)(241.2²) = 291 J. KE after: ½(2.01)(1.44) = 1.45 J. Energy converted to heat/deformation: 289.5 J — nearly all of it, as expected for a perfectly inelastic collision with a huge mass ratio.
Two-dimensional glancing collision
Equal masses in elastic collision: by the elastic result, total KE is conserved, so v_A² + v_B² = u_A² = 16 m²/s²
x-momentum: 0.3(4) = 0.3 v_A cos30° + 0.3 v_B cosθ → 4 = v_A(0.866) + v_B cosθ
y-momentum: 0 = 0.3 v_A sin30° − 0.3 v_B sinθ → v_A(0.5) = v_B sinθ
For equal-mass elastic collisions, the two post-collision velocities are perpendicular: θ = 90° − 30° = 60°. Then: v_A = 4 cos30° = 3.46 m/s; v_B = 4 sin30° = 2.0 m/s. Check: 3.46² + 2.0² = 12 + 4 = 16 ✓
Centre of Mass
The centre of mass (CoM) of a system is the single point where, for the purposes of overall motion, all the mass can be treated as concentrated. For a two-body system:
The key theorem: the centre of mass of an isolated system moves at constant velocity (including at rest, if total momentum is zero). No internal forces — no matter how violent the explosion or collision — can shift the CoM's trajectory.
This is why an astronaut floating in space can't propel themselves by waving their arms wildly. Their CoM stays fixed; arms moving one way means the rest of the body moves the other. To actually move, you need to throw something (a wrench, exhale gas) — expelling mass in one direction to recoil in the other.
Angular Momentum: Momentum's Rotational Cousin
Just as linear momentum p = mv describes translational motion, angular momentum L describes rotational motion:
Angular momentum is conserved when no net external torque acts on a system. This explains the figure skater spin: when she pulls her arms in, her moment of inertia I decreases, so her angular velocity ω must increase to keep L = Iω constant. The spin-up isn't powered by her muscles — it's conservation of angular momentum. For the full picture, see our Classical Mechanics guide on rotational dynamics.
Momentum Conservation in the Real World
- Rocket propulsion: A rocket accelerates by ejecting mass (exhaust) at high speed backward. The reaction (Newton's Third Law) pushes the rocket forward. Momentum is conserved: the backward momentum of exhaust = forward momentum gain of rocket.
- Particle physics: At CERN, when two protons collide, physicists measure the momenta of all outgoing particles. If the total doesn't match the input, they know a particle has escaped undetected — this is how neutrinos and dark matter candidates are inferred.
- Jet engines and helicopters: Both work by throwing air backward/downward, gaining an equal and opposite forward/upward momentum. There's no magic — just Newton's Third Law.
- Car safety: Crumple zones extend the collision time to reduce peak force (same impulse, smaller F). See the energy article for the energy side of crash physics.
Frequently Asked Questions
Next: Energy and the Bigger Picture
Momentum tells you velocities. Energy tells you what's possible and what's lost. Read the companion article on kinetic and potential energy.
What Is Energy? → Full Mechanics GuideSources & Further Reading
- Halliday, D., Resnick, R., & Krane, K. S. (2002). Physics (5th ed., Vol. 1). Wiley. Chapter 9: Systems of Particles and Chapter 10: Collisions.
- Goldstein, H., Poole, C., & Safko, J. (2002). Classical Mechanics (3rd ed.). Addison-Wesley. §3.1: Noether's theorem and conservation laws.
- Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage. Chapter 9: Linear Momentum and Collisions.
- Tipler, P. A., & Llewellyn, R. A. (2012). Modern Physics (6th ed.). W. H. Freeman. Momentum conservation in particle physics.
- Knight, R. D. (2016). Physics for Scientists and Engineers (4th ed.). Pearson. Chapter 11: Impulse and Momentum.