Conservation of Linear Momentum

The law

When no net external force acts on a system, the total linear momentum of the system remains constant:

$\vec{p}_{total} = m_1\vec{v}_1 + m_2\vec{v}_2 + \cdots = \text{constant}$

For a two-body system:

$m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}$

Total momentum before = total momentum after.

Derivation from Newton's Third Law

Consider two bodies A and B interacting — no external forces.

By Newton's Third Law, the force A exerts on B is equal and opposite to the force B exerts on A:

$\vec{F}_{BA} = -\vec{F}_{AB}$

By Newton's Second Law for each body:

$\vec{F}_{BA} = \frac{d\vec{p}_B}{dt}, \qquad \vec{F}_{AB} = \frac{d\vec{p}_A}{dt}$

Adding:

$\frac{d\vec{p}_A}{dt} + \frac{d\vec{p}_B}{dt} = \vec{F}_{AB} + \vec{F}_{BA} = 0$

$\frac{d}{dt}(\vec{p}_A + \vec{p}_B) = 0$

$\vec{p}_A + \vec{p}_B = \text{constant}$

Total momentum is conserved. This derivation shows conservation of momentum is a direct consequence of Newton's Third Law.

Internal vs external forces

Internal forces are forces between bodies within the system. They always come in action-reaction pairs and cancel when summed over the whole system — they cannot change total momentum.

External forces are forces from outside the system. They change total momentum.

The law holds when net external force is zero — not when all forces are zero. Internal forces can be large; they just don't affect total momentum.

Example: Two ice skaters pushing off each other. The push forces are internal to the skater system. Total momentum before (both at rest) = 0. Total momentum after = 0, so they move in opposite directions with momenta that cancel:

$m_1\vec{v}_1 + m_2\vec{v}_2 = 0 \implies \vec{v}_1 = -\frac{m_2}{m_1}\vec{v}_2$

Applications

Recoil of a gun

Before firing: gun + bullet at rest. Total momentum = 0.

After firing: bullet moves forward with momentum $m_b v_b$, gun recoils backward:

$m_b v_b + m_g v_g = 0 \implies v_g = -\frac{m_b v_b}{m_g}$

The gun recoils with speed $\frac{m_b v_b}{m_g}$. Since $m_g \gg m_b$, the recoil speed is small.

Rocket propulsion

The rocket expels exhaust gas backward. The system (rocket + gas) has no external force in space. As gas momentum increases backward, rocket momentum increases forward — total stays constant.

Collisions

Conservation of momentum applies to all collisions — elastic, inelastic, and perfectly inelastic — as long as the collision time is short enough that external forces (like friction) do negligible impulse during the collision.

Conservation in components

Since momentum is a vector, conservation holds independently in each direction:

$p_{x,total} = \text{constant}, \quad p_{y,total} = \text{constant}, \quad p_{z,total} = \text{constant}$

Even if momentum is not conserved in one direction (due to an external force), it may still be conserved in other directions. For example, during a horizontal collision on a surface with friction: momentum is not conserved horizontally (friction acts), but if there are no vertical external forces other than normal and gravity (which cancel), vertical momentum is conserved.