Conservation of Momentum & Collisions

You’ll be able to

State conservation of momentum for an isolated system and justify it from Newton's third law
Distinguish elastic from inelastic collisions by whether kinetic energy is conserved
Solve one-dimensional collision problems, including the perfectly inelastic case

Total momentum is conserved without external forces

When no net external force acts, the total momentum of a system is constant. The reason is Newton's third law: the internal forces objects exert on each other are equal and opposite, so they cancel in the total. During a brief collision the internal impact forces dwarf outside forces like gravity and friction, so momentum is conserved through the collision even when energy is not.

Conservation of momentum

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

Total momentum before equals total momentum after. It is a vector equation — apply it component by component, keeping track of signs.

Elastic versus inelastic collisions

Momentum is conserved in every collision, but kinetic energy is not. In an elastic collision, kinetic energy is also conserved (ideal, like billiard balls). In an inelastic collision, some kinetic energy is lost to heat and deformation. The extreme case is a perfectly inelastic collision, in which the objects stick together and move off with one common velocity — this loses the most kinetic energy possible while still conserving momentum.

Perfectly inelastic collision

m₁v₁ + m₂v₂ = (m₁ + m₂) v_f

The objects stick and share one final velocity v_f. Momentum is conserved; kinetic energy is not.

Worked example

A 2 kg cart moving at 3 m/s collides head-on with a 1 kg cart at rest and they stick together. Find their common velocity and the kinetic energy lost.

1.Momentum is conserved: m₁v₁ + m₂v₂ = (m₁ + m₂)v_f → (2)(3) + (1)(0) = (3)v_f.
2.Solve: 6 = 3 v_f → v_f = 2 m/s.
3.Kinetic energy before: ½(2)(3²) = 9 J. After: ½(3)(2²) = 6 J.
4.Energy lost = 9 − 6 = 3 J, converted to heat and deformation — momentum stayed the same but KE did not.

Answer: v_f = 2 m/s; kinetic energy drops from 9 J to 6 J, so 3 J is lost

Checkpoint

A 3 kg cart moving at 4 m/s collides with and sticks to a 1 kg cart at rest. What is their common velocity afterward?

Checkpoint

In a perfectly inelastic collision between two objects, which of the following is conserved?

Watch out

Never assume kinetic energy is conserved in a collision unless you are explicitly told it is elastic. Momentum conservation always holds (absent external forces); kinetic-energy conservation is the special elastic case. Mixing the two is the most common collision mistake.

Answer the 2 checkpoints as you read.