Conservation of Momentum

The principle of conservation of linear momentum is one of the most powerful tools in mechanics. It applies to collisions, explosions, and any interaction where no external resultant force acts on the system.

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1 Statement of the Principle

When no external resultant force acts on a system, the total momentum of the system is conserved.

For two particles:

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

This follows from Newton's Third Law: the internal forces between the particles are equal and opposite, so they produce equal and opposite impulses, and the total momentum change is zero.

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2 Types of Collision

Type	Momentum	Kinetic Energy
Perfectly elastic	Conserved	Conserved
Inelastic	Conserved	Not conserved (some lost)
Perfectly inelastic	Conserved	Maximum loss (particles coalesce)

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3 Worked Example 1 -- Coalescence

A 4 kg object moving at 6 m s^(-1) collides with a 2 kg object at rest. They coalesce. Find the common velocity.

Solution:

m_1 u_1 + m_2 u_2 = (m_1 + m_2) v

4(6) + 2(0) = (4 + 2)v