Momentum
p = mv — the product of mass and velocity that is conserved in every collision.
Definition and Formula
The linear momentum p of an object is defined as the product of its mass m and velocity v:
p = mv
Momentum is a vector quantity — it has both magnitude and direction, pointing in the same direction as velocity. The SI unit of momentum is kilogram-metres per second (kg·m/s), equivalent to newton-seconds (N·s).
Newton's second law, in its most general form, is written as F = dp/dt — force equals the rate of change of momentum. For constant mass, this reduces to F = ma.
Variable Table
| Symbol | Quantity | SI Unit |
|---|---|---|
| p | Linear momentum (vector) | kg·m/s (= N·s) |
| m | Mass | kg |
| v | Velocity (vector) | m/s |
| J | Impulse | N·s (= kg·m/s) |
| F | Net force (vector) | N |
Conservation of Momentum
The law of conservation of momentum states: if no net external force acts on a system, the total momentum of the system remains constant.
For a two-object system: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′, where primes denote post-interaction velocities.
Conservation of momentum is a direct consequence of Newton's third law. In any interaction, the forces between objects are equal and opposite, so their momentum changes are equal and opposite — and the total is unchanged. Noether's theorem identifies momentum conservation as the consequence of spatial translation symmetry: the laws of physics are the same everywhere in space.
Impulse-Momentum Theorem
Impulse J = FΔt is the integral of force over time. The impulse-momentum theorem states: J = Δp = m(v′ − v). Impulse and momentum change have the same units (N·s = kg·m/s).
This theorem explains why safety features extend the time of collision. A car airbag increases the time Δt over which the occupant's momentum drops to zero, reducing the average force F = Δp/Δt on the body. The same principle underlies padded sports equipment, crumple zones, and falling technique in martial arts.
Collision Types
Elastic collisions conserve both momentum and kinetic energy. They are rare in macroscopic systems (billiard balls approximately, molecular collisions in ideal gases exactly). The equations give:
- v₁′ = [(m₁ − m₂)v₁ + 2m₂v₂] / (m₁ + m₂)
- v₂′ = [(m₂ − m₁)v₂ + 2m₁v₁] / (m₁ + m₂)
Inelastic collisions conserve momentum but not kinetic energy. The kinetic energy deficit goes into deformation, heat, or sound. A perfectly inelastic collision maximises energy loss: both objects stick together and move at v′ = (m₁v₁ + m₂v₂)/(m₁ + m₂).
Worked Examples
Example 1 — Finding momentum
A 0.15 kg cricket ball travels at 40 m/s. Find its momentum.
Solution: p = mv = 0.15 × 40 = 6 kg·m/s in the direction of travel.
Example 2 — Conservation in a collision
A 2 kg cart moving at 3 m/s collides with a stationary 1 kg cart. After impact they stick together. Find the final velocity.
Solution: Conservation of momentum: (2)(3) + (1)(0) = (2 + 1)v′. So 6 = 3v′ → v′ = 2 m/s in the original direction.
Kinetic energy check: KE_before = ½(2)(3²) = 9 J; KE_after = ½(3)(2²) = 6 J. Energy lost = 3 J (to deformation) — this is a perfectly inelastic collision.
Example 3 — Impulse and force
A 60 kg person's momentum changes from 0 to 300 kg·m/s in 0.5 s during a sprint start. Find the average net force.
Solution: J = Δp = 300 N·s. F = J/Δt = 300/0.5 = 600 N.
Common Mistakes
- Forgetting momentum is a vector. In 2D problems, treat x- and y-components separately. A ball bouncing off a wall at an angle changes direction — the momentum change is not zero even if speed is unchanged.
- Applying conservation when external forces act. Gravity, friction, and normal forces can all be external. Conservation only applies over time intervals short enough that external impulse is negligible, or in directions where no external force acts.
- Confusing momentum and kinetic energy. p = mv; KE = ½mv². A doubled speed doubles momentum but quadruples kinetic energy. They transform differently and are conserved under different conditions.
- Sign errors. Always define a positive direction before solving. Velocities in the opposite direction are negative.
Applications
- Rocket propulsion: Exhaust mass ejected backward gives the rocket forward momentum (conservation).
- Particle physics: Missing momentum in collision products signals an undetected particle (e.g., the neutrino was inferred this way).
- Sports biomechanics: Hitting technique, follow-through, and protective gear design all use impulse-momentum analysis.
- Vehicle crash safety: Crumple zones, airbags, and seatbelts extend collision time to reduce peak force.
- Stellar dynamics: Conservation of momentum governs binary star orbits, galaxy collisions, and gravitational slingshots.
Related Topics
Frequently Asked Questions
What is momentum?
p = mv — the product of mass and velocity. It measures how difficult it is to stop a moving object. A heavier or faster object has more momentum.
Is momentum always conserved?
In any system with no net external force, total linear momentum is conserved. This follows directly from Newton's third law.
Elastic vs inelastic collisions?
Elastic: both momentum and KE conserved. Inelastic: only momentum conserved (some KE lost to heat/deformation). Perfectly inelastic: objects stick together after impact.
What is impulse?
Impulse J = FΔt equals the change in momentum (Δp). A longer collision time means a smaller average force for the same momentum change — the principle behind airbags.
References
- Halliday, D., Resnick, R., & Krane, K. S. (2001). Physics (5th ed.). Wiley. Chapter 9.
- Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage. Chapter 9.
- Kleppner, D., & Kolenkow, R. J. (2014). An Introduction to Mechanics (2nd ed.). Cambridge University Press. Chapter 4.
- Taylor, J. R. (2005). Classical Mechanics. University Science Books. Chapter 3.