Why do forces always come in equal and opposite pairs acting on different objects, and how does this constrain interactions and underpin momentum conservation?
Topic 2.3 Newton's Third Law: state that forces arise in equal-and-opposite pairs on different objects, identify the members of a third-law pair, and use this to analyze interacting systems.
A focused answer to AP Physics C: Mechanics Topic 2.3, covering Newton's third law as equal-and-opposite force pairs on different objects, identifying the members of a third-law pair, why such pairs never cancel on a single object, and how the law underlies momentum conservation, with worked examples.
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What this topic is asking
The College Board (Topic 2.3) wants you to state Newton's third law, to identify the two members of a force pair, and to use the law to analyze interacting objects. The defining feature, and the source of most exam traps, is that the two forces in a third-law pair act on different objects, so they never cancel each other. This law is also the reason internal forces drop out of a system's center-of-mass motion and the basis of momentum conservation.
Stating the law
The law captures the idea that a force is an interaction between two objects, never a property of one alone. You cannot push without being pushed back: the swimmer pushes water backward and the water pushes the swimmer forward; the rocket expels gas downward and the gas pushes the rocket up. Every force you write on a diagram is one half of such a pair, and its partner acts on the other object involved.
Identifying the pair
To find the partner of a given force, name the two objects and the interaction, then reverse the roles. "The Earth pulls the book down (gravity)" pairs with "the book pulls the Earth up (gravity)." Crucially, the normal force the table exerts on the book is not the partner of the book's weight: the weight is a gravity interaction between book and Earth, while the normal force is a contact interaction between book and table. They happen to be equal here only because the book is in equilibrium (Newton's first law), a coincidence of magnitude, not a third-law pairing.
Why the pair does not cancel
Because the two forces act on different objects, they never appear together on a single free-body diagram, so they cannot cancel. A small car pushed by a large truck pushes back on the truck with an equal force, yet the car accelerates more because it has less mass (). The forces are equal; the accelerations are not. When you treat the car and truck as a single system, the push becomes internal and the pair does cancel in the system's center-of-mass equation, leaving only the external forces, the deep reason that momentum is conserved in isolated systems.
Try this
Q1. A rocket accelerates upward by expelling exhaust gas downward. Identify the third-law pair responsible. [2 points]
- Cue. The rocket pushes the gas down; the gas pushes the rocket up, equal and opposite, acting on different objects.
Q2. Explain why a horse can accelerate a cart even though the cart pulls back on the horse with an equal force. [2 points]
- Cue. The two forces act on different objects. The horse accelerates because the ground's forward friction on its hooves exceeds the cart's backward pull on the horse.
Exam-style practice questions
Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
AP 2023 (style)3 marksSection II (short FRQ). A loaded truck of mass kg pushes a stalled car of mass kg, and the two accelerate together at m/s squared on a level road (neglect friction). (a) Calculate the force the truck exerts on the car. (b) State the force the car exerts on the truck, with justification. (c) Explain why these two forces do not cancel even though they are equal and opposite.Show worked answer →
A 3-point FRQ testing the third law on an interacting pair.
(a) Force on the car (1 point): treat the car alone; the only horizontal force is the push from the truck, so N forward.
(b) Force on the truck (1 point): by Newton's third law the car pushes back on the truck with N, directed backward (opposing the truck's motion).
(c) No cancellation (1 point): the two forces act on different objects (one on the car, one on the truck), so they never appear on the same free-body diagram and cannot cancel. Each accelerates its own object.
Markers reward isolating the car to get the force and stating that third-law partners act on different bodies.
AP 2021 (style)1 marksSection I (multiple choice). A book rests on a table. The reaction force to the gravitational force the Earth exerts on the book is... (A) the normal force the table exerts on the book (B) the gravitational force the book exerts on the Earth (C) the weight of the table (D) the normal force the book exerts on the table. Justify your reasoning.Show worked answer →
A 1-point conceptual MCQ. The answer is (B).
A third-law pair involves the same two objects and the same type of interaction with the roles reversed. The Earth pulls the book down gravitationally; the partner is the book pulling the Earth up gravitationally, equal in magnitude and opposite in direction. The normal force (A, D) is a different interaction (book-table contact) and balances the weight only because the book is in equilibrium, not because it is the third-law partner. This is the classic trap.
Related dot points
- Topic 2.2 Forces and Free-Body Diagrams: identify the forces acting on a chosen object, represent them on a free-body diagram, and resolve them into components on chosen axes to find the net force.
A focused answer to AP Physics C: Mechanics Topic 2.2, covering contact and field forces, drawing a correct free-body diagram showing only the forces on the chosen object, choosing convenient axes (including tilted axes on an incline), and resolving forces into components to compute the net force, with worked examples.
- Topic 2.4 Newton's First Law: state the law of inertia, define translational equilibrium as zero net force, and apply the equilibrium conditions to find unknown forces.
A focused answer to AP Physics C: Mechanics Topic 2.4, covering Newton's first law and inertia, the meaning of translational equilibrium as zero net force, the difference between mass and weight, and applying the equilibrium conditions axis by axis to find unknown forces, with worked examples.
- Topic 2.5 Newton's Second Law: relate net force, mass and acceleration through the vector equation, apply it component by component, and extend it to connected systems and the general form with momentum.
A focused answer to AP Physics C: Mechanics Topic 2.5, covering Newton's second law as a vector equation applied axis by axis, the general form as the time rate of change of momentum, solving connected systems for the common acceleration and internal tension, and using it with variable forces, with calculus-based worked examples.
- Topic 2.1 Systems and Center of Mass: define a system, locate the center of mass by a mass-weighted average (including by integration for continuous bodies), and apply that only external forces accelerate the center of mass.
A focused answer to AP Physics C: Mechanics Topic 2.1, covering the idea of a system, the center of mass as a mass-weighted average for discrete particles and by integration for continuous bodies, the velocity and acceleration of the center of mass, and why only external forces change the center-of-mass motion, with calculus-based worked examples.
- Topic 4.3 Conservation of Linear Momentum: state that the total momentum of an isolated system is conserved, and apply it to recoil, explosions and interactions in one and two dimensions.
A focused answer to AP Physics C: Mechanics Topic 4.3, covering the condition for momentum conservation (zero net external force), why internal forces cannot change total momentum, and applying conservation to recoil, explosions and two-dimensional interactions by components, with worked examples.
Sources & how we know this
- AP Physics C: Mechanics Course and Exam Description — College Board (2024)