Conservation of Momentum

The Core Idea: Why Momentum Stays Put

Newton’s second law tells us that a net external force is needed to change an object’s momentum. The natural flip side of this is: if no net external force acts on a system, its total momentum cannot change. This is the principle of conservation of momentum, and it is one of the most powerful and widely applicable laws in physics.

It is crucial to understand that this is not a new, independent law. It is a direct consequence of Newton’s second and third laws working together. The law applies to any system of particles or objects, from colliding billiard balls to a rocket blasting off.

Deriving Conservation from Newton's Laws

Consider a system of two particles, A and B, that interact with each other (they collide, push off, or attract each other). Let’s say they exert forces on each other. By Newton’s third law, the force on A due to B ($\vec{F}_{AB}$) is equal in magnitude and opposite in direction to the force on B due to A ($\vec{F}_{BA}$):

Now, apply Newton’s second law to each particle. The net force on a particle equals the rate of change of its momentum ($\vec{p}$).

For particle A: $\vec{F}_{AB} = \frac{d\vec{p}_A}{dt}$

For particle B: $\vec{F}_{BA} = \frac{d\vec{p}_B}{dt}$

If we add these two equations, we get the net force on the entire system of two particles:

But from Newton’s third law, $\vec{F}_{AB} + \vec{F}_{BA} = 0$. Therefore:

The derivative of the total momentum ($\vec{P} = \vec{p}_A + \vec{p}_B$) with respect to time is zero. This means the total momentum of the system is constant.

The Crucial Condition: An Isolated System

The derivation above worked because we only considered the forces between the two particles (internal forces). What if an external force, like friction or gravity, also acts on the system? Then the sum of forces on the system is no longer zero, and the total momentum will change.

The law of conservation of momentum is strictly valid only for an isolated system — a system on which no net external force acts.

Applying the Law: The Two-Body Collision

The most common application is a collision between two objects. Let’s say object 1 (mass $m_1$, initial velocity $\vec{u}_1$) and object 2 (mass $m_2$, initial velocity $\vec{u}_2$) collide. After the collision, their velocities are $\vec{v}_1$ and $\vec{v}_2$.

If the collision happens in an isolated system (no external forces like friction), the total momentum before the collision equals the total momentum after the collision.

This single vector equation is incredibly powerful. It gives us two or three scalar equations (one for each dimension) and is often enough to solve for unknown velocities, even when we don't know the details of the forces during the collision.

A Classic Example: The Recoil of a Gun

Consider a gun of mass $M$ firing a bullet of mass $m$ with a muzzle velocity $\vec{v}$ (relative to the ground). Before firing, both the gun and bullet are at rest, so the total initial momentum is zero.

$\vec{P}_i = 0$ …