If Every Force Has an Equal Opposite, How Does Anything Move?
Learn Newton's third law and how to spot an action-reaction pair, why those forces never cancel, how to find the acceleration of connected blocks, and how momentum conservation gives recoil speeds.
If forces always come in equal opposite pairs, why is anything ever accelerated?
Because the two forces of a pair act on two different bodies, and forces only cancel when they act on the same body.
A horse pulls a cart forward. The cart pulls the horse backward with exactly the same force. It sounds as though nothing should move — and yet carts move.
The resolution is to look at one body at a time. To find out what the cart does, count only the forces on the cart: the horse pulls it forward, friction holds it back. If the pull exceeds the friction, the cart accelerates. The cart's backward pull on the horse never appears in that list at all, because it is not acting on the cart.
That single discipline — draw the forces on one object, and never put both halves of a pair in the same list — is what makes Newton's third law usable rather than paradoxical. This page covers the third part of the CBSE Class 9 Science chapter on how forces affect motion.
A horse pulls a cart forward. The cart pulls the horse backward with exactly the same force. It sounds as though nothing should move — and yet carts move.
The resolution is to look at one body at a time. To find out what the cart does, count only the forces on the cart: the horse pulls it forward, friction holds it back. If the pull exceeds the friction, the cart accelerates. The cart's backward pull on the horse never appears in that list at all, because it is not acting on the cart.
That single discipline — draw the forces on one object, and never put both halves of a pair in the same list — is what makes Newton's third law usable rather than paradoxical. This page covers the third part of the CBSE Class 9 Science chapter on how forces affect motion.
What does Newton's third law say, and how do you name the pair?
To every action there is an equal and opposite reaction, and the two act on different bodies.
The forces are equal in magnitude, opposite in direction, and they act simultaneously — neither comes first.
Action-reaction pairs, stated properly. The trick is to name both bodies in each force.
- You push the wall; the wall pushes you
- A book presses down on the table; the table pushes up on the book
- Your foot pushes the ground backwards; the ground pushes your foot forwards — which is how walking works
- A swimmer pushes the water backwards; the water pushes the swimmer forwards
- An oar pushes the water backwards; the water pushes the boat forwards
- A rocket pushes gases downwards; the gases push the rocket upwards
- A gun pushes the bullet forward; the bullet pushes the gun backward, which is the recoil
- The Earth pulls you down; you pull the Earth up
Everyday evidence. Step off a small boat onto a jetty and the boat slides backwards. Your foot pushed the boat back; the boat pushed you forward. On a heavy ship you notice nothing, because the same force on a far greater mass produces almost no acceleration.
**Name the pair as A on B and B on A.** That format makes mistakes almost impossible. The push of the ground on the foot and the push of the foot on the ground is a genuine pair. The weight of a book and the table's push on the book is not a pair, even though those two forces are equal and opposite here — both act on the book, and the partner of the book's weight is the book's pull on the Earth.
That last case is the one examiners test. Two forces being equal and opposite does not make them an action-reaction pair. They must also act on different bodies and arise from the same interaction. Checking the two bodies is the test.
The forces are equal in magnitude, opposite in direction, and they act simultaneously — neither comes first.
Action-reaction pairs, stated properly. The trick is to name both bodies in each force.
- You push the wall; the wall pushes you
- A book presses down on the table; the table pushes up on the book
- Your foot pushes the ground backwards; the ground pushes your foot forwards — which is how walking works
- A swimmer pushes the water backwards; the water pushes the swimmer forwards
- An oar pushes the water backwards; the water pushes the boat forwards
- A rocket pushes gases downwards; the gases push the rocket upwards
- A gun pushes the bullet forward; the bullet pushes the gun backward, which is the recoil
- The Earth pulls you down; you pull the Earth up
Everyday evidence. Step off a small boat onto a jetty and the boat slides backwards. Your foot pushed the boat back; the boat pushed you forward. On a heavy ship you notice nothing, because the same force on a far greater mass produces almost no acceleration.
**Name the pair as A on B and B on A.** That format makes mistakes almost impossible. The push of the ground on the foot and the push of the foot on the ground is a genuine pair. The weight of a book and the table's push on the book is not a pair, even though those two forces are equal and opposite here — both act on the book, and the partner of the book's weight is the book's pull on the Earth.
That last case is the one examiners test. Two forces being equal and opposite does not make them an action-reaction pair. They must also act on different bodies and arise from the same interaction. Checking the two bodies is the test.
Why do action and reaction not cancel each other out?
Because cancelling requires both forces to act on the same object, and an action-reaction pair never does.
When you add forces to find a net force, you are adding the forces on one body. An action-reaction pair puts one force on each of two bodies, so the two never appear in the same sum and can never cancel.
Worked example 1 — the horse and cart, resolved. Suppose the horse pulls the cart with N, and friction on the cart is N. Considering only the cart:
So the cart accelerates. The cart's N backward pull on the horse is balanced by the ground pushing the horse's hooves forward, which is a separate question about a separate body.
Worked example 2 — why the Earth does not visibly move. A kg student stands on the ground. The Earth pulls the student down with
and the student pulls the Earth up with exactly N. The forces are equal; the accelerations are not, because and the two masses are wildly different. The student's acceleration would be ; the Earth's is unmeasurably small because its mass is enormous.
Equal forces do not mean equal effects. That is the key idea, and it also explains the boat and the ship: the same push produces a large acceleration on a small mass and a negligible one on a large mass.
Worked example 3 — a rocket. Gases are pushed out downwards; the gases push the rocket up. Nothing outside the rocket is needed, which is why a rocket can accelerate in space where there is no air and nothing to push against. A student who thinks a rocket pushes against the air cannot explain space flight; the third law explains it without difficulty.
The two questions to keep apart. Are the forces equal? Always yes, by the third law. Are the effects equal? Only if the masses are equal. Almost every apparent paradox about the third law dissolves once those two questions are separated.
When you add forces to find a net force, you are adding the forces on one body. An action-reaction pair puts one force on each of two bodies, so the two never appear in the same sum and can never cancel.
Worked example 1 — the horse and cart, resolved. Suppose the horse pulls the cart with N, and friction on the cart is N. Considering only the cart:
So the cart accelerates. The cart's N backward pull on the horse is balanced by the ground pushing the horse's hooves forward, which is a separate question about a separate body.
Worked example 2 — why the Earth does not visibly move. A kg student stands on the ground. The Earth pulls the student down with
and the student pulls the Earth up with exactly N. The forces are equal; the accelerations are not, because and the two masses are wildly different. The student's acceleration would be ; the Earth's is unmeasurably small because its mass is enormous.
Equal forces do not mean equal effects. That is the key idea, and it also explains the boat and the ship: the same push produces a large acceleration on a small mass and a negligible one on a large mass.
Worked example 3 — a rocket. Gases are pushed out downwards; the gases push the rocket up. Nothing outside the rocket is needed, which is why a rocket can accelerate in space where there is no air and nothing to push against. A student who thinks a rocket pushes against the air cannot explain space flight; the third law explains it without difficulty.
The two questions to keep apart. Are the forces equal? Always yes, by the third law. Are the effects equal? Only if the masses are equal. Almost every apparent paradox about the third law dissolves once those two questions are separated.
How do you find the acceleration of two connected blocks?
Treat the whole system together to get the acceleration, then isolate one block to get the tension.
The reason the system method works is the third law itself. The string pulls the front block backwards and the rear block forwards with equal forces — an action-reaction pair. Considered as one system, those internal forces cancel, so only the external force is left.
Worked example 1. Two blocks of kg and kg are joined by a light string on a frictionless horizontal surface. A force of N is applied to the kg block, pulling the pair along.
Step 1 — the whole system.
**Step 2 — isolate the kg block.** The only horizontal force on it is the string's tension:
Step 3 — check with the other block. On the kg block, the applied force acts forward and the tension backward:
Consistent, so both answers are right.
Worked example 2. Blocks of kg and kg with a N force applied to the kg block.
Check: . Correct.
Worked example 3 — with friction. A kg box is pushed with N while friction opposes with N.
Why the tension needs one block, not the system. For the system as a whole the tension is internal and invisible — it cancels. To see it you must draw a boundary around one block, and then the tension becomes an external force acting on that block. That is the practical use of consider one body at a time.
The tension is the same throughout a light string. So the string pulls the kg block forward with N and the kg block backward with N — one magnitude, two directions, two bodies. The order of the blocks matters, though: applying the N to the kg block instead would give the same acceleration of but a tension of N, because the string now has to drag the heavier block.
The reason the system method works is the third law itself. The string pulls the front block backwards and the rear block forwards with equal forces — an action-reaction pair. Considered as one system, those internal forces cancel, so only the external force is left.
Worked example 1. Two blocks of kg and kg are joined by a light string on a frictionless horizontal surface. A force of N is applied to the kg block, pulling the pair along.
Step 1 — the whole system.
**Step 2 — isolate the kg block.** The only horizontal force on it is the string's tension:
Step 3 — check with the other block. On the kg block, the applied force acts forward and the tension backward:
Consistent, so both answers are right.
Worked example 2. Blocks of kg and kg with a N force applied to the kg block.
Check: . Correct.
Worked example 3 — with friction. A kg box is pushed with N while friction opposes with N.
Why the tension needs one block, not the system. For the system as a whole the tension is internal and invisible — it cancels. To see it you must draw a boundary around one block, and then the tension becomes an external force acting on that block. That is the practical use of consider one body at a time.
The tension is the same throughout a light string. So the string pulls the kg block forward with N and the kg block backward with N — one magnitude, two directions, two bodies. The order of the blocks matters, though: applying the N to the kg block instead would give the same acceleration of but a tension of N, because the string now has to drag the heavier block.
Formula
How does conservation of momentum give a recoil velocity?
In the absence of an external force, the total momentum of a system stays the same.
This follows from the third law combined with the second: the two bodies exert equal and opposite forces on each other for exactly the same time, so their momentum changes are equal and opposite and cancel in the total.
Worked example 1 — a collision where the bodies stick together. A kg ball moving at m/s strikes a stationary kg ball, and the two move off together.
Worked example 2 — the recoil of a gun. A gun of mass kg fires a bullet of mass kg at m/s. Find the recoil velocity.
Before firing, everything is at rest, so the total momentum is zero:
The gun recoils at m/s in the opposite direction to the bullet.
Worked example 3 — jumping off a boat. A boy of kg jumps from a stationary boat of kg at m/s.
The boat moves backwards at m/s.
Worked example 4 — two bodies moving towards each other. A kg object at m/s meets a kg object travelling at m/s in the opposite direction, and they stick together.
Taking the first direction as positive:
Positive, so the combined body moves in the direction of the heavier, faster object.
Worked example 5 — equal and opposite. Two objects of kg each approach at m/s and stick together.
They come to rest. The momentum was zero before and must be zero after, and that is the only way for a single combined body.
Assign a positive direction and use it throughout. Worked example 4 needed the m/s, and dropping that sign would have given instead of and a wrong answer of m/s. The signs carry the directions, and a momentum question with two bodies moving in opposite directions is testing exactly that.
Individual momenta change; only the total is conserved. The bullet gains forward momentum and the gun gains an equal backward momentum. Neither body's momentum is conserved on its own — the law is about the system, and a question asking whether the bullet's momentum is conserved is asking you to notice that distinction.
This follows from the third law combined with the second: the two bodies exert equal and opposite forces on each other for exactly the same time, so their momentum changes are equal and opposite and cancel in the total.
Worked example 1 — a collision where the bodies stick together. A kg ball moving at m/s strikes a stationary kg ball, and the two move off together.
Worked example 2 — the recoil of a gun. A gun of mass kg fires a bullet of mass kg at m/s. Find the recoil velocity.
Before firing, everything is at rest, so the total momentum is zero:
The gun recoils at m/s in the opposite direction to the bullet.
Worked example 3 — jumping off a boat. A boy of kg jumps from a stationary boat of kg at m/s.
The boat moves backwards at m/s.
Worked example 4 — two bodies moving towards each other. A kg object at m/s meets a kg object travelling at m/s in the opposite direction, and they stick together.
Taking the first direction as positive:
Positive, so the combined body moves in the direction of the heavier, faster object.
Worked example 5 — equal and opposite. Two objects of kg each approach at m/s and stick together.
They come to rest. The momentum was zero before and must be zero after, and that is the only way for a single combined body.
Assign a positive direction and use it throughout. Worked example 4 needed the m/s, and dropping that sign would have given instead of and a wrong answer of m/s. The signs carry the directions, and a momentum question with two bodies moving in opposite directions is testing exactly that.
Individual momenta change; only the total is conserved. The bullet gains forward momentum and the gun gains an equal backward momentum. Neither body's momentum is conserved on its own — the law is about the system, and a question asking whether the bullet's momentum is conserved is asking you to notice that distinction.
Exam tip
Exam tip: consider one body at a time, and fix a positive direction
Never put both halves of an action-reaction pair in the same force list. They act on different bodies and can never cancel.
Name each force as A on B: the push of the ground on the foot and the push of the foot on the ground. Two forces being equal and opposite is not enough — check that they act on different bodies.
Equal forces do not mean equal effects, because . That is why you accelerate and the Earth does not.
For connected bodies, find the acceleration from the total mass, then isolate one block to get the tension: , then N.
Check with the other block: .
For conservation of momentum, write and fix a positive direction before substituting.
If the bodies start at rest, the total momentum is zero — which is what makes recoil questions one-liners.
A body moving the other way takes a negative velocity: , not .
Only the total momentum is conserved, never an individual body's.
And give units and a direction on every answer — * m/s backwards*, not just m/s.
Name each force as A on B: the push of the ground on the foot and the push of the foot on the ground. Two forces being equal and opposite is not enough — check that they act on different bodies.
Equal forces do not mean equal effects, because . That is why you accelerate and the Earth does not.
For connected bodies, find the acceleration from the total mass, then isolate one block to get the tension: , then N.
Check with the other block: .
For conservation of momentum, write and fix a positive direction before substituting.
If the bodies start at rest, the total momentum is zero — which is what makes recoil questions one-liners.
A body moving the other way takes a negative velocity: , not .
Only the total momentum is conserved, never an individual body's.
And give units and a direction on every answer — * m/s backwards*, not just m/s.
Did you know
Why a rocket works where there is nothing to push against
It is a common assumption that a rocket flies by pushing against the air, in the way a swimmer pushes against water. If that were true, a rocket could not work in space.
Rockets work perfectly well in space, and the third law explains why. The rocket does not push against anything outside itself. It pushes against its own exhaust gases — burning fuel and throwing the products out of the nozzle at high speed. The rocket pushes the gases one way; the gases push the rocket the other.
Momentum conservation says the same thing from the other side. Before ignition the rocket and its fuel are at rest, so the total momentum is zero. Throw mass out of the back with some momentum, and the rest of the rocket must acquire an equal momentum forward so that the total stays zero — exactly the recoil calculation done above, repeated continuously instead of once.
The gun in that calculation is a rocket firing a single shot. A rocket is a gun firing continuously, which is why its speed builds up rather than arriving all at once.
Air turns out to be an obstacle rather than a help. It exerts drag on the rising rocket and contributes nothing to the thrust, which is why a launch gets more efficient as the rocket climbs out of the dense lower atmosphere.
The same principle appears in a garden sprinkler, which spins because water leaves its arms and pushes them the opposite way, and in a fire hose that a firefighter has to brace against. In each case something is thrown one way and something recoils the other — and nothing external is being pushed at all.
Rockets work perfectly well in space, and the third law explains why. The rocket does not push against anything outside itself. It pushes against its own exhaust gases — burning fuel and throwing the products out of the nozzle at high speed. The rocket pushes the gases one way; the gases push the rocket the other.
Momentum conservation says the same thing from the other side. Before ignition the rocket and its fuel are at rest, so the total momentum is zero. Throw mass out of the back with some momentum, and the rest of the rocket must acquire an equal momentum forward so that the total stays zero — exactly the recoil calculation done above, repeated continuously instead of once.
The gun in that calculation is a rocket firing a single shot. A rocket is a gun firing continuously, which is why its speed builds up rather than arriving all at once.
Air turns out to be an obstacle rather than a help. It exerts drag on the rising rocket and contributes nothing to the thrust, which is why a launch gets more efficient as the rocket climbs out of the dense lower atmosphere.
The same principle appears in a garden sprinkler, which spins because water leaves its arms and pushes them the opposite way, and in a fire hose that a firefighter has to brace against. In each case something is thrown one way and something recoils the other — and nothing external is being pushed at all.
Exam relevance
How is conservation of momentum tested in JEE Main and NEET?
Conservation of momentum is one of the small number of ideas used across the whole of physics, and this page is where it is first applied to numbers.
This is the foundation for the Class 11 Physics chapter Laws of Motion, examined in JEE Main and in NEET Physics. The third law is stated there in the same words, and the connected-block method on this page becomes the standard free-body diagram technique, extended to blocks on inclines and over pulleys.
Where momentum conservation is reused. Class 11 Work, Energy and Power uses it for collisions, dividing them into elastic and inelastic. The cases worked above — two bodies sticking together — are perfectly inelastic collisions, and the Class 11 addition is that kinetic energy is not conserved in them while momentum still is. That contrast is a favourite JEE Main question. Class 11 System of Particles and Rotational Motion extends it to the centre of mass, and Class 12 uses it for nuclear and particle reactions.
The third law also underlies tension in every string problem, the normal reaction in every surface problem, and the recoil calculations in Class 12 Nuclei, where an unstable nucleus emitting a particle recoils exactly as the gun does here.
What the questions look like. Numericals dominate, and the recoil and sticking-collision shapes worked above are the two standard forms. Assertion-reason items favour two statements from this page: that action and reaction do not cancel, and that momentum is conserved while kinetic energy need not be. Match-the-column questions pair a situation with its action-reaction pair. And diagram-based connected-block problems asking for the tension are common, because they require both the system view and the isolated view in one answer.
How board and competitive emphasis differ. A board paper asks you to state the third law and explain the recoil of a gun, then sets a one-step conservation sum. A competitive paper gives two bodies moving in opposite directions, or asks for the tension rather than the acceleration, so the sign convention and the isolation step are what is being tested. Board papers also accept a verbal explanation of why forces do not cancel; competitive papers test it by giving a pair that is not an action-reaction pair and asking you to say so.
The single trap that costs the most marks. Dropping the negative sign on a body moving the other way. In worked example 4, gives and gives — two completely different answers, and only the first is right. Write the positive direction down before the first substitution.
A second trap worth naming. Calling the weight of a book and the table's push on it an action-reaction pair. They are equal and opposite in that particular situation, but both act on the book, so they are a balanced pair rather than an action-reaction pair. The partner of the book's weight is the book's pull on the Earth — and being able to say that is what the question is checking.
This is the foundation for the Class 11 Physics chapter Laws of Motion, examined in JEE Main and in NEET Physics. The third law is stated there in the same words, and the connected-block method on this page becomes the standard free-body diagram technique, extended to blocks on inclines and over pulleys.
Where momentum conservation is reused. Class 11 Work, Energy and Power uses it for collisions, dividing them into elastic and inelastic. The cases worked above — two bodies sticking together — are perfectly inelastic collisions, and the Class 11 addition is that kinetic energy is not conserved in them while momentum still is. That contrast is a favourite JEE Main question. Class 11 System of Particles and Rotational Motion extends it to the centre of mass, and Class 12 uses it for nuclear and particle reactions.
The third law also underlies tension in every string problem, the normal reaction in every surface problem, and the recoil calculations in Class 12 Nuclei, where an unstable nucleus emitting a particle recoils exactly as the gun does here.
What the questions look like. Numericals dominate, and the recoil and sticking-collision shapes worked above are the two standard forms. Assertion-reason items favour two statements from this page: that action and reaction do not cancel, and that momentum is conserved while kinetic energy need not be. Match-the-column questions pair a situation with its action-reaction pair. And diagram-based connected-block problems asking for the tension are common, because they require both the system view and the isolated view in one answer.
How board and competitive emphasis differ. A board paper asks you to state the third law and explain the recoil of a gun, then sets a one-step conservation sum. A competitive paper gives two bodies moving in opposite directions, or asks for the tension rather than the acceleration, so the sign convention and the isolation step are what is being tested. Board papers also accept a verbal explanation of why forces do not cancel; competitive papers test it by giving a pair that is not an action-reaction pair and asking you to say so.
The single trap that costs the most marks. Dropping the negative sign on a body moving the other way. In worked example 4, gives and gives — two completely different answers, and only the first is right. Write the positive direction down before the first substitution.
A second trap worth naming. Calling the weight of a book and the table's push on it an action-reaction pair. They are equal and opposite in that particular situation, but both act on the book, so they are a balanced pair rather than an action-reaction pair. The partner of the book's weight is the book's pull on the Earth — and being able to say that is what the question is checking.
Key takeaways
Newton's third law and momentum conservation: quick revision
- Newton's third law: to every action there is an equal and opposite reaction, acting on two different bodies, and the two act simultaneously.
- Name a pair as A on B and B on A: foot on ground and ground on foot; swimmer on water and water on swimmer; gun on bullet and bullet on gun.
- Two forces being equal and opposite is not enough — they must act on different bodies and come from the same interaction. A book's weight and the table's push both act on the book.
- Action and reaction never cancel, because cancelling needs both forces on the same body.
- The horse and cart: with a N pull and N of friction on the cart, the net force on the cart is N forward.
- A kg student is pulled down with N and pulls the Earth up with N. Equal forces, unequal effects, because .
- A rocket pushes its own exhaust gases, not the air, which is why it works in space.
- Connected blocks: find from the total mass, then isolate one block for the tension.
- N on a kg and kg pair gives and N, checked by .
- N on a kg and kg pair gives and N.
- A kg box pushed with N against N friction gives .
- Tension is internal to the system and external to one block — which is why the tension needs the isolated view.
- Conservation of momentum: , when no external force acts.
- A kg ball at m/s hitting a stationary kg ball and sticking gives m/s.
- A kg gun firing a kg bullet at m/s **recoils at m/s** backwards.
- A kg boy jumping at m/s from a kg boat sends the boat back at m/s.
- kg at m/s meeting kg at m/s gives , so m/s.
- Two kg bodies meeting at m/s each come to rest.
- Fix a positive direction first, and remember that only the total momentum is conserved.
Work out the recoil of a kg gun firing a g bullet at m/s, then check that the two momenta really do cancel — if they do, you have used the law rather than the formula.
- Name a pair as A on B and B on A: foot on ground and ground on foot; swimmer on water and water on swimmer; gun on bullet and bullet on gun.
- Two forces being equal and opposite is not enough — they must act on different bodies and come from the same interaction. A book's weight and the table's push both act on the book.
- Action and reaction never cancel, because cancelling needs both forces on the same body.
- The horse and cart: with a N pull and N of friction on the cart, the net force on the cart is N forward.
- A kg student is pulled down with N and pulls the Earth up with N. Equal forces, unequal effects, because .
- A rocket pushes its own exhaust gases, not the air, which is why it works in space.
- Connected blocks: find from the total mass, then isolate one block for the tension.
- N on a kg and kg pair gives and N, checked by .
- N on a kg and kg pair gives and N.
- A kg box pushed with N against N friction gives .
- Tension is internal to the system and external to one block — which is why the tension needs the isolated view.
- Conservation of momentum: , when no external force acts.
- A kg ball at m/s hitting a stationary kg ball and sticking gives m/s.
- A kg gun firing a kg bullet at m/s **recoils at m/s** backwards.
- A kg boy jumping at m/s from a kg boat sends the boat back at m/s.
- kg at m/s meeting kg at m/s gives , so m/s.
- Two kg bodies meeting at m/s each come to rest.
- Fix a positive direction first, and remember that only the total momentum is conserved.
Work out the recoil of a kg gun firing a g bullet at m/s, then check that the two momenta really do cancel — if they do, you have used the law rather than the formula.