Conservation of Momentum
The law of conservation of momentum states that the total momentum of a closed, isolated system remains constant if no external net force acts on it. This means the sum of momenta of all objects before an interaction equals the sum after the interaction. It is one of the most fundamental conservation laws in physics and applies equally to collisions, explosions, and all mechanical interactions.
Key Formula
m1×v1i + m2×v2i = m1×v1f + m2×v2f
LaTeX: m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}
| Symbol | Meaning | Unit |
|---|---|---|
| m₁, m₂ | Masses of objects 1 and 2 | Kilogram (kg) |
| v₁ᵢ, v₂ᵢ | Initial velocities of objects 1 and 2 | m/s |
| v₁f, v₂f | Final velocities of objects 1 and 2 | m/s |
Worked Example
Problem
A 5 kg ball moving at 6 m/s east collides with a stationary 3 kg ball. After the collision, the 5 kg ball moves at 2 m/s east. Find the velocity of the 3 kg ball after the collision.
Solution
Step 1: Write conservation of momentum equation. m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f Step 2: Substitute values. (5)(6) + (3)(0) = (5)(2) + (3)v₂f 30 = 10 + 3v₂f Step 3: Solve for v₂f. 3v₂f = 20 v₂f = 20/3 ≈ 6.67 m/s (east)
Answer
v₂f ≈ 6.67 m/s eastward
Conservation of Momentum in Different Scenarios
| Scenario | System Type | Momentum Conserved? | Kinetic Energy Conserved? |
|---|---|---|---|
| Elastic collision | Isolated | Yes | Yes |
| Perfectly inelastic collision | Isolated | Yes | No |
| Explosion (gun firing) | Isolated | Yes | No (energy added) |
| Object on rough surface | Non-isolated | No (friction) | No |
| Rocket propulsion | Isolated | Yes | No (chemical energy) |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Momentum
Momentum is the product of an object's mass and its velocity, representing the quantity of motion possessed by the object. It is a vector quantity, meaning it has both magnitude and direction aligned with the velocity. Momentum is fundamental to Newton's second and third laws and is the conserved quantity in isolated systems during collisions and interactions.
Elastic Collision
An elastic collision is one in which both the total kinetic energy and the total momentum of the system are conserved before and after the collision. No energy is lost to deformation, heat, or sound, making it an idealized model most closely approximated by atomic and subatomic particle interactions. Billiard ball collisions and gas molecule interactions are common approximations of elastic collisions.
Inelastic Collision
An inelastic collision is one in which the total kinetic energy of the system is not conserved, though the total momentum remains conserved. The lost kinetic energy is converted into other forms such as heat, sound, or deformation energy. When two objects collide and stick together, it is termed a perfectly inelastic collision, representing the maximum possible loss of kinetic energy.
The word "conservation" comes from Latin "conservare" meaning "to preserve". The law was first stated by René Descartes in 1644 and later rigorously derived from Newton's third law. Emmy Noether's theorem (1915) later showed it follows from spatial symmetry.