Newton’s third law of motion

What is Newton’s third law?

In plain words: forces always come in pairs. If body A pushes or pulls body B, then body B pushes or pulls body A at the same instant, with a force of the same size in the opposite direction. Either force can be called the action; the other is then the reaction.

The third law as an equation

Here $\vec F_{AB}$ is the force on A due to B, and $\vec F_{BA}$ is the force on B due to A. The minus sign means the two forces point in opposite directions, and their magnitudes are equal: $|\vec F_{AB}| = |\vec F_{BA}|$⁠.

Properties of action-reaction pairs

• They act on different bodies. One force acts on A and the other on B, so they never cancel in the motion of either body. Two forces can cancel only when they act on the same body. If you treat A and B together as one system, the pair become internal forces that add up to zero, which is why only external forces count in the second law for a system.
• They are the same type of force. If the action is a gravitational pull, so is the reaction; if the action is a contact push, so is the reaction.
• They act at the same instant. The reaction does not come after the action, and neither force causes the other.
• They hold whether the bodies are at rest or moving, and whether they touch (a push, friction) or not (gravity, the pull between two magnets). A pseudo force, used only in an accelerating frame, has no partner, because no body exerts it.

The weight of a book on a table and the normal force from the table are not a third-law pair, even though they are equal and opposite: both act on the book, and they balance because the book is at rest (Newton’s first law of motion). The partner of the book’s weight is the book’s gravitational pull on the Earth, and the partner of the normal force is the book’s push on the table.

Examples

• Walking. Your foot pushes the ground backward through friction, and the ground pushes your foot forward. That forward force from the ground moves you. On smooth ice the friction is small, so walking is hard.
• A rocket. The engine pushes hot gases out backward at high speed, and the gases push the rocket forward. A rocket needs no air to push against, so it works in space.
• Recoil of a gun. The gun pushes the bullet forward, and the bullet pushes the gun backward.
• Swimming. A swimmer pushes the water backward with the hands and feet, and the water pushes the swimmer forward.

Common mistakes to avoid

Quick revision summary

• Forces always come in pairs: when A pushes or pulls B, B pushes or pulls A.
• The two forces are equal in size, opposite in direction, of the same type and simultaneous.
• They act on different bodies, so they never cancel in the motion of either body; for the system A + B they are internal forces that add to zero.
• Equal forces can give very different accelerations: the lighter body accelerates more.
• Walking, swimming, rockets and the recoil of a gun are all third-law effects.

Questions students ask

How is Newton’s third law related to conservation of momentum?

Newton’s third law, together with the second law, gives the law of conservation of momentum. When two bodies interact and no external force acts, they exert equal and opposite forces on each other for the same time, so they receive equal and opposite impulses: Δp_A = −Δp_B. The momentum one body gains, the other loses, so their total momentum stays constant; NCERT Section 4.7 (pages 57–58) extends this to any number of bodies.

What is the difference between balanced forces and action-reaction forces?

Balanced forces act on the same body and add up to zero, so they cancel; an action-reaction pair acts on two different bodies, so its two forces never cancel in the motion of either body. For a bucket hanging at rest from a rope, the rope’s pull and the bucket’s weight are balanced forces, while the partner of the rope’s pull on the bucket is the bucket’s pull on the rope. Balanced forces can be of different types and need not exist together (cut the rope and the weight remains), but the two forces of a third-law pair are always of the same type and exist together.

How does Newton’s third law apply to a person in a lift?

A person standing in a lift pushes down on the floor and the floor pushes up on the person with a force of the same size N at every instant: these two forces are a third-law pair. When the lift accelerates upward at a, N grows to m(g + a) for a person of mass m, as the second law gives, but the push of the person on the floor grows with it, so the pair stays equal and opposite. The person’s weight mg and N are not a pair, because both act on the person; that is why they can differ while the lift accelerates.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A person of mass 60 kg stands on frictionless ice and throws a 2 kg ball horizontally at 6 m s⁻¹ relative to the ground. With what speed does the person move back?

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   The person and the ball push on each other with equal and opposite forces for the same time, so their momenta are equal and opposite: 60 V = 2 × 6, which gives V = 12/60 = 0.2 m s⁻¹, opposite to the ball.

2. Q2. A book rests on a table. Are its weight and the normal force from the table an action-reaction pair?

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   No. Both forces act on the book, so they cannot be a third-law pair; they are equal because the book is in equilibrium. The reaction to the weight is the book’s gravitational pull on the Earth, and the reaction to the normal force is the book’s push on the table.

3. Q3. In a collision, a car of mass 1000 kg and a truck of mass 5000 kg push on each other with a force of 5 × 10⁴ N. Ignoring all other forces during the collision, find the acceleration of each.

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   By the third law, each feels a force of 5 × 10⁴ N. Car: a = 5 × 10⁴/1000 = 50 m s⁻². Truck: a = 5 × 10⁴/5000 = 10 m s⁻². The forces are equal, but the lighter car gets five times the acceleration.