The principle of conservation of momentum

Statement of the principle

For an isolated system (no net external force, or a pseudo-isolated system during the very brief duration of a collision), the total momentum is conserved:

p_total(before) = p_total(after)

This result follows from the principle of reciprocal actions (Newton’s third law): during a collision between two bodies A and B, the force that A exerts on B is equal and opposite to the force that B exerts on A, at every instant. The changes in momentum therefore exactly cancel each other out.

Equation for two bodies

For a collision between two masses m₁ and m₂, with initial velocities u₁ and u₂ and final velocities v₁ and v₂:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

This relationship is vectorial: it projects onto each axis independently. In one dimension, a single scalar equation suffices; in two dimensions, two are required.

Numerical example

Two minecarts weighing 5 kg and 3 kg are moving towards each other at 4 m/s and 2 m/s (the direction of the 5 kg minecart is positive). Before the collision:

p_before = 54 + 3(-2) = 20 - 6 = 14 kg·m/s

If they remain attached after the collision, their joint velocity v satisfies:

(5+3)*v = 14 -> v = 14/8 = 1.75 m/s

Pitfalls to avoid

• Forgetting the signs: you must choose a positive direction and ensure the signs of the opposing velocities are correct.
• Confusing the conservation of momentum (always true for an isolated system) with the conservation of kinetic energy (true only for elastic collisions).
• Applying the conservation principle to a subsystem that is still subject to a significant external force (friction, uncompensated gravity).