5.4 — Newton’s third law — action and reaction

The third law

If object A exerts a force on object B, then B exerts an equal-magnitude, opposite-direction force on A.

\[ \boxed{\vec F_{A \to B} = -\vec F_{B \to A}} \]

Three notes:

• Always in pairs — a lone force doesn't exist
• Always on two different objects — one on A, the other on B
• Always simultaneous — no delay

Not obvious at first

Newton's third law is initially counterintuitive.

Earth pulls a ball: downward. Does the ball pull Earth? Yes! Equal magnitude, upward. Why doesn't Earth rise? Because Earth is 10²² times more massive — its acceleration is negligible.

Another example: a car crashes into a wall. The car is damaged, the wall isn't. But the force on the wall = force on the car. The difference is materials and structure, not force.

Common mistake — third law vs equilibrium

Crucial: the two forces in a third-law pair do not cancel each other — because they act on different objects.

Example — book on a table:

• Earth's gravity on book: \vec W = -mg\hat j (down)
• Third-law pair: book's gravity on Earth: +mg\hat j (up)
• Table's normal force on book: \vec N = +mg\hat j (up)
• Third-law pair: book's normal force on table: -mg\hat j (down)

Careful: \vec W (Earth on book) and \vec N (table on book) are not a third-law pair — both act on the same object (the book); they just happen to balance because the book is in equilibrium.

Example — walking

Why can you walk?

• Your foot pushes on the ground backward
• The ground pushes on your foot forward (third law)
• That forward force (friction) propels you

Without friction (on ice): you can push backward with your foot, but there's no reaction pair — so you don't accelerate forward.

Example — rocket propulsion

The rocket ejects gas backward (\vec F_{rocket \to gas}). The gas exerts an equal, opposite force on the rocket (\vec F_{gas \to rocket}). That thrust drives the rocket forward — even in vacuum, without pushing against anything.

Important: contrary to popular belief, a rocket doesn't push against the air. It's the interaction between rocket and exhaust.

Big consequence — momentum conservation

If \vec F_{A \to B} = -\vec F_{B \to A}, the effect on each object's momentum:

\[ \Delta \vec p_A = -\Delta \vec p_B \]

so total momentum is conserved:

\[ \vec p_A + \vec p_B = \text{const} \]

This is the foundation of momentum conservation — Chapter 9.

A few notes and common mistakes

1. Third-law pair: two objects, two forces. Never on the same object.

2. If two forces on one object balance, they aren't a third-law pair. Third-law pair sits on the paired object.

3. Third-law pairs are of the same kind. Gravity paired with gravity, not with normal force. Each interaction has its own pair.

What you should be able to do

• Write the third law in vector notation
• Identify third-law pairs in scenarios
• Distinguish third-law pair from balanced forces on one object
• Explain applications (walking, rockets, swimming) via the third law

Preview of §5.5

Mass and weight — two different things often confused. §5.5 sharpens the distinction.

📚 See also: Halliday Vol 1, Ch 5, §5.4.

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