Principle

Conservation of momentum

(M4.2.1) M4.2 · Conservation of Momentum →

p_{total, i} = p_{total, f}

Statement

The total momentum of an isolated system is the same before and after any internal interaction.

Symbols and units

p_{total}

total momentum of the system

kg·m/s

When it applies

An isolated system — zero net external force during the interaction.

Derivation

By Newton's third law, every internal force between two parts of the system has an equal and opposite partner, so internal forces cancel in pairs when summed over the whole system:

\sum F_{internal} = 0

With no external force either, the total momentum has zero rate of change:

\frac{d p _{total}}{d t} = 0 \Rightarrow p_{total, i} = p_{total, f}

Worked examples

Recoil of a rifle

A 4.0 kg rifle fires a 0.010 kg bullet at 400 m/s:

0 = m_{bullet} v_{bullet} + m_{rifle} v_{recoil}

v_{recoil} = - \frac{( 0.010 ) ( 400 )}{4.0} = - 1.0 m/s

Watch out for

Momentum is conserved along each axis independently, even in collisions where kinetic energy is not conserved.

Connects to

Newton's third law