Newton's Third Law — padho.wiki

In short

Newton's third law states that if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These two forces — the action and reaction — always act on different bodies, which is why they never cancel each other. Every force in the universe exists as part of such a pair: weight and the Earth's attraction upward, the normal force and the push of the object into the surface, the rocket exhaust pushing down and the rocket being pushed up.

Sit on a wheeled office chair and push against a wall. What happens? You roll backward. You pushed the wall, but the wall pushed you — and since you are on wheels, you moved. The wall, cemented into the building, barely budged. Yet the force the wall exerted on you was exactly as large as the force you exerted on the wall. Not roughly equal. Not approximately equal. Exactly equal.

This is Newton's third law in action. It is the most misunderstood of Newton's three laws — not because it is hard, but because the language is slippery. "Every action has an equal and opposite reaction" sounds like the two forces should cancel out and nothing should ever move. But they don't cancel, and things do move, and the reason is simple once you see it: the two forces act on different bodies.

The law — stated precisely

Here is the law in Newton's own framing, translated into modern language:

Newton's Third Law. If body A exerts a force \vec{F} on body B, then body B simultaneously exerts a force -\vec{F} on body A.

The minus sign means the reaction force points in the opposite direction. The magnitudes are identical:

|\vec{F}_{A \to B}| = |\vec{F}_{B \to A}|

Why: this is not a derived result — it is a fundamental law, an observation about how forces work in nature. Every experiment ever conducted confirms it. No isolated force has ever been found.

Three features make this law precise:

The forces are equal in magnitude. Not "roughly" or "on average" — exactly equal, at every instant.
The forces are opposite in direction. If A pushes B to the right, B pushes A to the left.
The forces act on different bodies. This is the critical point. \vec{F}_{A \to B} acts on B. \vec{F}_{B \to A} acts on A. They never appear in the same free body diagram.

The two forces in an action-reaction pair always act on different bodies. F(A on B) acts on body B, and F(B on A) acts on body A. They are never on the same object.

Action-reaction pairs — always on different bodies

The phrase "action and reaction" is an unfortunate one. It suggests that one force is the cause and the other is the effect, as if the reaction comes after the action. That is wrong. Both forces arise simultaneously, the instant the interaction begins. Neither is more "real" than the other.

A better way to think about it: forces always come in pairs, and the two forces in a pair live on different bodies.

Here is how to identify an action-reaction pair. Take any force and describe it as a sentence:

"Body A pushes body B."

The reaction is the sentence with A and B swapped:

"Body B pushes body A."

Same type of force. Same magnitude. Opposite direction. Different bodies.

The crucial test: are the forces on the same body?

If two forces act on the same body, they are not an action-reaction pair — no matter how equal and opposite they look. This is the single most common mistake in applying Newton's third law, and it trips up students from class 9 through JEE preparation.

Consider a book resting on a table. The book has two forces acting on it:

Its weight mg pulling it downward (the Earth pulls the book)
The normal force N pushing it upward (the table pushes the book)

These two forces are equal in magnitude and opposite in direction — N = mg for a book at rest. But they are not an action-reaction pair. Both forces act on the same body (the book). They happen to be equal because the book is in equilibrium (Newton's second law: \vec{F}_{\text{net}} = m\vec{a} = 0), not because of Newton's third law.

Why: the equality N = mg for a stationary book comes from Newton's second law (zero acceleration means zero net force). If you placed the book in a lift accelerating upward, N would be greater than mg. The third law pairs are different — they connect forces on different objects.

Why action-reaction pairs never cancel

This is the question that bothers every student who first hears the third law: if every force has an equal and opposite partner, how does anything ever accelerate?

The answer is in the phrase "different bodies." When you draw a free body diagram, you pick one body and draw only the forces acting on that body. The reaction forces act on other bodies — they don't appear in your diagram.

Take the office chair example again. You sit on the chair and push the wall with your hands at a force of 40 N.

Forces on you (person + chair system):

The wall pushes you backward: 40 N to the left
That is the only horizontal force on you

By Newton's second law, you accelerate to the left. You roll backward.

Forces on the wall:

You push the wall forward: 40 N to the right
The building's foundation holds the wall in place with a reaction force of 40 N to the left

The wall's net force is zero. The wall does not accelerate. Your net force is 40 N to the left. You accelerate.

The two 40 N forces (you-on-wall and wall-on-you) are an action-reaction pair. They live on different bodies. They never enter the same free body diagram. They cannot cancel.

Two separate free body diagrams. The 40 N force from the wall appears on your FBD (left). The 40 N force from you appears on the wall's FBD (right). They are on different diagrams and can never cancel.

The normal force as a reaction

When you place a book on a table, the book pushes the table downward with a force equal to its weight. The table pushes the book upward with the normal force. This is an action-reaction pair:

Action: The book pushes the table downward (contact force, magnitude mg).
Reaction: The table pushes the book upward (normal force, magnitude N = mg for a stationary book).

These two forces are on different bodies — one on the table, one on the book. They are always equal and opposite, even if the book is accelerating. If you stack another book on top, the table's normal force increases to match the increased downward push.

Why: the normal force is not a mysterious new force. It is the electromagnetic repulsion between the atoms at the surfaces of the book and the table. When you push atoms closer together, they push back. The harder you push down, the harder the surface pushes up — automatically, and exactly enough to match. If the surface cannot generate enough force (say, you place a 500 kg safe on a flimsy cardboard box), the surface breaks.

But notice something subtle: the book's weight (mg downward, exerted by the Earth on the book) and the normal force (N upward, exerted by the table on the book) are not an action-reaction pair. They both act on the book. The actual third-law pairs are:

Force	On which body	Its third-law partner	On which body
Earth pulls book down (mg)	Book	Book pulls Earth up (mg)	Earth
Book pushes table down	Table	Table pushes book up (N)	Book

The Earth is being pulled upward by the book. Why doesn't the Earth accelerate toward the book? It does — but the Earth's mass is 6 \times 10^{24} kg, so the acceleration is immeasurably small: a = F/m = (1 \text{ N}) / (6 \times 10^{24} \text{ kg}) \approx 1.7 \times 10^{-25} m/s². That is why you never notice the Earth moving toward a falling object.

How you walk — the third law in every step

Walking is so automatic that you rarely think about the physics. But every step is a demonstration of Newton's third law.

When you take a step, your foot pushes the ground backward. By Newton's third law, the ground pushes your foot forward. That forward push — the friction force — is the only horizontal force on your body, and it is what accelerates you forward.

Why: your muscles generate force inside your body, but internal forces cannot accelerate your centre of mass (they cancel in pairs within the system). You need an external force, and that external force comes from the ground. No ground contact, no forward push — which is exactly why you cannot walk on perfectly frictionless ice.

Walking works because of Newton's third law. Your foot pushes the ground backward; the ground pushes your foot forward. The forward friction is the external force that accelerates you.

The same principle explains swimming. A swimmer pushes water backward with their arms. The water pushes the swimmer forward. The harder and faster you push water backward, the stronger the forward force on you.

And rowing a boat on a lake: the oars push water backward, and the water pushes the boat forward. The boat moves even though the water itself barely shifts — because the boat is much lighter than the lake.

Rocket propulsion — the third law without a surface

The most dramatic application of Newton's third law is rocket propulsion. A rocket works not by "pushing against the air" (it works in vacuum too) but by throwing mass backward at high speed.

The rocket engine burns fuel and expels hot gas downward at enormous velocity. By Newton's third law, the hot gas pushes the rocket upward with an equal force. This is why ISRO's GSLV rocket can lift off from Sriharikota and reach orbit — the exhaust gases rushing out of the nozzle at thousands of metres per second push the rocket upward with enough force to overcome gravity.

Why: if the rocket expels gas of mass \Delta m at velocity v_e in a time interval \Delta t, the force on the gas is F = \Delta m \cdot v_e / \Delta t downward. By Newton's third law, the gas exerts a force of the same magnitude on the rocket — upward. This is called thrust. The higher the exhaust velocity, the greater the thrust for a given rate of fuel consumption.

A rocket works by pushing exhaust gas downward. By Newton's third law, the exhaust gas pushes the rocket upward. No air or surface is needed — this works in the vacuum of space.

A common misconception is that rockets "push against the air." If that were true, rockets would be useless in space, where there is no air. In fact, rockets work better in vacuum, because there is no air resistance. The rocket pushes against its own exhaust — and by Newton's third law, the exhaust pushes the rocket.

Worked examples

Example 1: Pushing off a wall from a wheeled chair

Rahul sits on a wheeled office chair (total mass of Rahul + chair = 60 kg) and pushes against a concrete wall with a force of 90 N. The chair rolls on a smooth floor. Identify all action-reaction pairs and find Rahul's acceleration.

Separate FBDs for Rahul (left) and the wall (right). The 90 N pair (Rahul-on-wall and wall-on-Rahul) are on different diagrams. Rahul accelerates; the wall stays put.

Step 1. Identify the action-reaction pairs.

When Rahul pushes the wall to the right with 90 N, the wall pushes Rahul to the left with 90 N. This is the third-law pair.

Why: the sentence is "Rahul pushes wall." Swap the nouns: "Wall pushes Rahul." Same magnitude (90 N), opposite direction, different bodies.

Step 2. Draw the FBD for Rahul + chair (the system you care about).

The only horizontal force on Rahul is the wall's push: 90 N to the left. Vertically, weight and normal force balance (Rahul is not accelerating up or down).

Why: Rahul's push on the wall does not appear in Rahul's FBD — it acts on the wall, not on Rahul. Only forces on Rahul go into Rahul's diagram.

Step 3. Apply Newton's second law to Rahul.

F_{\text{net}} = ma

90 = 60 \times a

a = \frac{90}{60} = 1.5 \text{ m/s}^2

Why: the 90 N is the only horizontal force, so it is the net force. Dividing by Rahul's total mass (60 kg) gives the acceleration.

Result: Rahul accelerates at 1.5 m/s² away from the wall. The wall experiences zero net force and does not move.

What this shows: The wall pushes Rahul just as hard as Rahul pushes the wall. But Rahul is on wheels (low friction) while the wall is anchored to the Earth. Same force, very different accelerations — because the masses are vastly different. Newton's third law gives equal forces, but Newton's second law (a = F/m) gives unequal accelerations when the masses are unequal.

Example 2: A book on a table — the pairs that aren't pairs

A physics textbook of mass 1.2 kg rests on a table. Identify all the action-reaction pairs. (Hint: the weight of the book and the normal force from the table are NOT a third-law pair.)

The weight (Earth pulls book) and the normal force (table pushes book) are NOT a third-law pair — they both act on the book. The real pairs connect forces on different bodies, as shown in the box.

Step 1. List the forces on the book.

Two forces act on the book:

Weight W = mg = 1.2 \times 9.8 = 11.76 \approx 11.8 N downward (the Earth pulls the book)
Normal force N = 11.8 N upward (the table pushes the book)

Why: these are the only two forces on the book. They are equal because the book is in equilibrium (a = 0), so by Newton's second law, N - mg = 0, giving N = mg.

Step 2. Find the third-law partner of each force.

Force 1: Earth pulls book downward (gravitational, 11.8 N).

The third-law partner is: book pulls Earth upward (gravitational, 11.8 N).

Both are gravitational forces. They act on different bodies (book and Earth).

Force 2: Table pushes book upward (contact/normal, 11.8 N).

The third-law partner is: book pushes table downward (contact, 11.8 N).

Both are contact forces. They act on different bodies (book and table).

Why: notice that the two partners live on two different objects. The weight's partner is on the Earth. The normal force's partner is on the table. The weight and normal force are not partners of each other — they just happen to be equal because the book isn't accelerating.

Step 3. Verify the mistake.

If N and mg were truly a third-law pair, they would always be equal. But put the book in an accelerating lift:

Lift accelerating upward: N > mg (you feel heavier)
Lift in free fall: N = 0 while mg is unchanged (weightlessness)

The fact that N and mg can be unequal proves they are not a third-law pair. Third-law pairs are always equal — that's a law, not a coincidence.

Result: The two third-law pairs are (1) Earth-pulls-book / book-pulls-Earth, and (2) table-pushes-book / book-pushes-table. The weight and normal force are NOT a pair — they are two different forces that happen to balance on a stationary book.

What this shows: The "book on table" problem is the classic trap. Every force has its third-law partner on a different object. If you ever find both forces of a "pair" on the same FBD, you have made an error.

Common confusions

"Action and reaction cancel out." They do not, because they act on different bodies. You add forces when computing the net force on one body. The reaction force is on a different body and has no business in your free body diagram.
"The reaction comes after the action." No — both forces appear simultaneously, the instant the interaction exists. "Action" and "reaction" are just names. Neither comes first. You could swap the labels and the physics would be identical.
"Weight and normal force are a third-law pair." This is the most common textbook error. Weight (Earth pulls book) and normal (table pushes book) both act on the book. A true third-law pair always involves two different bodies. The proof: in an accelerating lift, N \neq mg, but a real third-law pair is always equal.
"The third law doesn't apply when objects have different masses." It always applies, regardless of the masses. A mosquito hitting a truck exerts the same force on the truck as the truck exerts on the mosquito. The forces are equal; the accelerations are vastly different because a = F/m and the masses are vastly different. The mosquito decelerates violently; the truck barely notices.
"A stronger person pushes harder, so the wall pushes back less." The wall always pushes back with exactly the same force you push it with, at every instant. If you push with 200 N, the wall pushes you with 200 N. The wall does not "decide" how much to push — the contact force adjusts automatically to match. If you exceed the wall's structural limit, the wall breaks.

If you understand that action-reaction pairs are always on different bodies, that they are always equal, and that they never cancel — you have the core of Newton's third law. What follows is for readers who want to see the formal connection to momentum conservation and explore subtle edge cases.

Newton's third law and conservation of momentum

Newton's third law is not just a statement about forces — it is the reason momentum is conserved.

Consider two bodies, A and B, interacting with each other and with no external forces. By Newton's third law:

\vec{F}_{A \to B} = -\vec{F}_{B \to A}

By Newton's second law, the force on B changes B's momentum:

\vec{F}_{A \to B} = \frac{d\vec{p}_B}{dt}

And the force on A changes A's momentum:

\vec{F}_{B \to A} = \frac{d\vec{p}_A}{dt}

Why: Newton's second law in its most general form says force equals the rate of change of momentum.

Since \vec{F}_{A \to B} = -\vec{F}_{B \to A}:

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

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

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

Why: the rate of change of the total momentum is zero. This means the total momentum \vec{p}_A + \vec{p}_B is constant — it does not change with time. This is the law of conservation of momentum, and it follows directly from Newton's third law.

This derivation shows that conservation of momentum is not an independent law — it is a consequence of the third law combined with the second law. Every time you use conservation of momentum to solve a collision problem, you are implicitly using Newton's third law.

The "horse and cart" paradox

A horse pulls a cart. By Newton's third law, the cart pulls the horse backward with an equal force. If the forces are equal, how does the system move?

The resolution: the horse-on-cart and cart-on-horse forces are internal to the horse+cart system. They cancel when you consider the system as a whole. But the horse also pushes the ground backward with its hooves. The ground pushes the horse forward (friction). This friction is an external force on the system, and it is what accelerates the horse+cart forward.

The forces:

Horse pulls cart forward: T
Cart pulls horse backward: T (third law pair)
Ground pushes horse forward: f (friction from hooves)
Ground pushes cart backward: rolling friction, f_r (small)

Net force on the system = f - f_r (forward), and the system accelerates as long as f > f_r.

Why: the confusion arises from mixing forces on different bodies. The horse-cart forces cancel inside the system. But the external friction force from the ground does not have its third-law partner on the system — the partner is on the Earth. So the net external force is nonzero, and the system accelerates.

Does the third law hold during radiation?

When the Sun exerts a gravitational force on the Earth, the Earth simultaneously exerts an equal gravitational force on the Sun. This is the third law in action. But what about light? When the Sun emits a photon toward the Earth, the photon carries momentum. When it arrives 8 minutes later and is absorbed by the Earth, the Earth gains momentum. Is the third law violated during those 8 minutes?

In Newtonian mechanics, forces are instantaneous (action at a distance), so the third law holds at every instant. In the framework of special relativity and electrodynamics, forces are mediated by fields that travel at the speed of light, and the simple form of the third law — that forces are always equal and opposite at every instant — does not hold for electromagnetic interactions between distant charges. What is conserved is the momentum of the system including the field. The field itself carries momentum, and when you account for that, total momentum is still conserved.

For all problems at the JEE level, you can treat Newton's third law as exact. The relativistic corrections matter only at speeds close to the speed of light or over astronomical distances.

Where this leads next

Free Body Diagrams — the systematic tool for identifying all forces on a single body, where the third law tells you which forces belong in the diagram and which do not.
Conservation of Momentum — the conservation law that follows directly from Newton's third law, and the key to solving collision and explosion problems.
Applications: Connected Bodies and Systems — pulleys, ropes, and multi-body problems where the third law determines the internal forces between connected objects.
Newton's Second Law — the law that converts the third law's equal forces into unequal accelerations, depending on mass.
Friction — the contact force that makes walking, driving, and braking possible, and which is itself a reaction force governed by the third law.