Newton's third law

Newton's Third Law of Motion states: to every action there is an equal and opposite reaction, and the action and reaction act on two different bodies. If body A exerts a force on body B — call it the action, $F_{AB}$ — then body B simultaneously exerts a force back on body A, the reaction, $F_{BA}$. The two forces are always equal in size and opposite in direction.

where: $F_{AB}$ is the force exerted by A on B, $F_{BA}$ is the force exerted by B on A, and the minus sign records that the two forces point in opposite directions while having the same magnitude. Three conditions must all hold for a pair of forces to qualify as a genuine action-reaction pair under this law:

• They are equal in magnitude — neither force is ever bigger than the other, regardless of the size or mass of the two bodies involved.
• They are opposite in direction — if the action points one way, the reaction points exactly the other way, along the same line.
• They act on two different bodies, one force on each — never both forces on the same object.

That third condition is the one students most often get wrong, and it is also the reason the law works at all: because the action and its reaction land on different bodies, they never appear together when you draw the forces on a single object, so they never have a chance to cancel each other. The law is universal — it applies equally to contact forces, like a hand pushing a wall or a foot pressing the ground, and to non-contact forces, like the gravitational pull between Earth and the Moon or between any two masses.

The most common misunderstanding is thinking that if action and reaction are always equal and opposite, nothing should ever move — surely the two forces just cancel out. This reasoning fails because forces only cancel when they act on the same object; the action and reaction of Newton's third law never do. Take a horse pulling a cart. The horse pulls the cart forward (action, on the cart), and by the third law the cart pulls the horse backward with equal force (reaction, on the horse). These two forces do not fight each other, because one lands entirely on the cart and the other entirely on the horse — there is no single object experiencing both. Whether the cart actually starts moving depends only on the forces acting on the cart: the horse's forward pull versus friction and resistance opposing the cart's own motion. Similarly, whether the horse can walk forward depends only on the forces acting on the horse: its hooves pushing back on the ground (action) and the ground pushing the horse forward (reaction), compared with the cart's backward pull on it. Each body's motion is decided by Newton's second law applied to the forces on that body alone — and the third-law partner of any one of those forces always lives on a different object, so it plays no part in that decision. This is also why walking, swimming, and rocket propulsion work. A swimmer pushes water backward; the water pushes the swimmer forward — two different bodies, two different objects experiencing the two forces, so the swimmer accelerates even though her own push (on the water) is exactly cancelled, in strength, by the water's push (on her). A rocket expels hot gas downward (action, on the gas); the escaping gas pushes the rocket upward with equal force (reaction, on the rocket) — again on two separate bodies, so the rocket climbs.

A book of weight $20\ \text{N}$ rests on a table, at equilibrium, and a swimmer pushes water backward to move forward through a pool. For each situation, identify the true action-reaction pair and state which forces are wrongly mistaken for one.

Step 1 — the book on the table. The book presses down on the table with a force $F_{\text{book on table}}$; by the third law, the table presses back up on the book with an equal and opposite force $F_{\text{table on book}} = 20\ \text{N}$, upward. These two forces act on different bodies (table and book), are equal in magnitude, and are opposite in direction — a genuine action-reaction pair, and this upward push is the normal reaction.

Step 2 — the trap in the book example. It is tempting to pair the book's weight ($20\ \text{N}$ downward, the Earth's pull on the book) with the normal reaction ($20\ \text{N}$ upward, the table's push on the book). Both act on the same body — the book — so they cannot be a third-law pair; they merely happen to be equal because the book is in equilibrium, which is Newton's first law, a separate idea. The true third-law partner of the book's weight is the book's own gravitational pull on the Earth, acting on the Earth, not on the table.

Step 3 — the swimmer and the water. The swimmer's hands and feet push the water backward, $F_{\text{swimmer on water}}$ (action, on the water). By the third law, the water simultaneously pushes the swimmer forward with an equal and opposite force, $F_{\text{water on swimmer}}$ (reaction, on the swimmer).

Step 4 — why she moves forward. The reaction force lands on the swimmer herself, so it is the water's forward push on her — not her own backward push on the water — that propels her through the pool. The book-table pair is the normal reaction of $20\ \text{N}$, and the swimmer-water pair is what pushes her forward; in both cases the true partner force acts on the other body, never on the same one.

• Believing action and reaction cancel out and so nothing should move — they act on different bodies, so they never both appear in the force analysis of a single object.
• Pairing an object's weight with the normal reaction on it as a third-law pair — both act on the same body, so this is really a case of balanced forces (Newton's first law), not action-reaction.
• Assuming the law only applies to contact forces like pushes and collisions — it applies equally to non-contact forces, such as gravity between two masses.
• Thinking the reaction happens slightly after the action — action and reaction are exactly simultaneous, with no time lag between them.

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