Newton's Third Law

Forces never act alone — they come in pairs. Push a wall and it pushes back. Step off a boat and the boat slides away. Newton's third law captures this fundamental fact: every interaction has two sides.

Concept

Newton's Third Law: To every action there is an equal and opposite reaction. The forces always act on two different bodies.

If body A exerts a force $\vec{F}_{AB}$ on body B, then body B exerts a force $\vec{F}_{BA}$ on body A such that

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

The two forces are:

• equal in magnitude,
• opposite in direction,
• act along the line joining the two bodies,
• act on different bodies (so they never cancel each other on a free-body diagram of a single body),
• arise and disappear simultaneously.

Crucially, "action" and "reaction" are not first one and then the other — they appear together. There is no time delay.

Derivation

From conservation of momentum for an isolated two-body system,

$\vec{p}_1 + \vec{p}_2 = \text{constant}$

Differentiating,

$\frac{d\vec{p}_1}{dt} + \frac{d\vec{p}_2}{dt} = 0$

The first term is the force on body 1 (from body 2), and the second is the force on body 2 (from body 1):

$\vec{F}_{21} + \vec{F}_{12} = 0 \quad \Longrightarrow \quad \vec{F}_{12} = -\vec{F}_{21}$

So Newton's third law is equivalent to conservation of momentum for an isolated system.

Worked Example

A boy of mass 50 kg jumps off a stationary boat of mass 200 kg with a horizontal velocity of 4 m/s. The water is frictionless. Find the velocity of the boat just after the jump.

Solution:

The boy and the boat push each other (third-law pair). Take the boy's jump direction as positive, boat's recoil as negative.

By conservation of momentum (total initial momentum was zero),

$0 = m_{\text{boy}} v_{\text{boy}} + m_{\text{boat}} v_{\text{boat}}$

$0 = (50)(4) + (200) v_{\text{boat}}$

$v_{\text{boat}} = -\frac{200}{200} = -1 \, \text{m/s}$

The boat moves backward at 1 m/s.

The force the boy exerts on the boat (action) equals the force the boat exerts on the boy (reaction), but acts on the heavier boat for the same time, giving it less velocity.

Common Confusions

• "Action and reaction cancel." They do not cancel because they act on different bodies. To analyze a single body, only forces on that body count.
• Horse-and-cart paradox. "If horse pulls cart with force F, and cart pulls horse with force −F, how do they move?" Answer: the horse moves because the ground pushes the horse forward (the horse pushes the ground backward — another third-law pair). The cart moves because the horse pulls it forward.
• Weight is not the reaction to normal force. The reaction to a book's weight (Earth pulling book down) is the book pulling Earth up. The normal force is from the table, not Earth's gravity.
• Different magnitudes? Never. The pair is always equal in magnitude, no matter how different the bodies' masses or accelerations.

Key Takeaways

• Forces come in pairs; every action has an equal-and-opposite reaction.
• The two forces act on different bodies — they never cancel each other.
• The third law follows from conservation of momentum.
• Different masses experience the same force but different accelerations.
• The reaction to a gravitational pull is another gravitational pull, not a normal force.