Newton's Third Law

The statement

For every force that body A exerts on body B, body B exerts an equal and opposite force on body A:

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

$\vec{F}_{AB}$ means "force on A due to B". $\vec{F}_{BA}$ means "force on B due to A".

They are equal in magnitude, opposite in direction, and act on different bodies.

Forces always come in pairs

This is the central insight: forces never exist alone. Every force is one half of an action-reaction pair.

• You push a wall → the wall pushes you back
• Earth pulls you down (gravity) → you pull Earth up with equal force
• A rocket expels gas backward → the gas pushes the rocket forward
• A bat hits a ball → the ball hits the bat with the same force

The two forces in a pair are always:

• Equal in magnitude
• Opposite in direction
• Of the same type (both gravitational, both normal, both friction, etc.)
• Acting on different bodies — never on the same body

The most important point: they act on different bodies

This is where students most commonly go wrong.

Consider a book on a table. Two pairs of action-reaction forces:

Pair 1:

• Earth pulls book downward (gravity on book)
• Book pulls Earth upward (gravity on Earth)

Pair 2:

• Table pushes book upward (normal force on book)
• Book pushes table downward (normal force on table)

The book is in equilibrium because the gravity on the book and the normal force on the book are equal and opposite — but these are not an action-reaction pair. They are two different forces both acting on the book, and they happen to be equal because the book is in equilibrium.

The action-reaction pairs each have their two forces acting on different objects.

Why doesn't the reaction cancel the action?

A common confusion: "If every action has an equal reaction, why does anything accelerate?"

Answer: because the two forces act on different bodies. They cannot cancel each other.

Example: You kick a football.

• You exert force $F$ on the ball → ball accelerates
• Ball exerts force $F$ back on your foot → your foot (and you) feel the impact

The force on the ball accelerates the ball. The force on your foot acts on you — a completely different body. These two forces never act on the same object, so they never cancel.

Applications

Rocket propulsion: The rocket expels hot gas backward at high speed. By the Third Law, the gas pushes the rocket forward. No external surface to push against is needed — the rocket carries its own reaction mass.

Walking: Your foot pushes backward on the ground (action). The ground pushes your foot forward (reaction). That forward reaction is what propels you forward. Without friction, the ground cannot push back — you slip.

Swimming: Your hands push water backward. Water pushes you forward.

Recoil of a gun: The gun exerts force on the bullet forward. The bullet exerts equal force on the gun backward — recoil.

Newton's Third Law and conservation of momentum

The Third Law is the reason momentum is conserved.

Consider two bodies A and B interacting. By the Third Law:

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

By the Second Law, $\vec{F} = \frac{d\vec{p}}{dt}$:

$\frac{d\vec{p}_A}{dt} = -\frac{d\vec{p}_B}{dt}$

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

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

Total momentum is constant. Conservation of momentum is a direct consequence of Newton's Third Law.