Skip to main content
Sign in Create a free account
Varsity Tutors
Sign in
Back to lessons

College Physics • Newton's Laws & Free-Body Modeling

Newton's Third Law

Every interaction produces equal and opposite forces, forming the foundation of all mechanical analysis.

Explore the lesson 19 min read
An atom with an orbital path and traveling waves.
Inside this lesson 10 sections 5 questions 9 figures

SECTION 1

Historical Context & Motivation

For most of human history, the study of motion was dominated by the Aristotelian view that a force was needed to sustain any movement—a seemingly intuitive notion that conflated the concepts of force and velocity. It was not until the scientific revolution of the seventeenth century that a rigorous, quantitative framework for dynamics began to emerge. The idea that forces always come in pairs was, in many ways, the most radical of Isaac Newton's contributions, because it shifted attention from individual objects to interactions between objects. Understanding the historical arc that led to Newton's Third Law illuminates why this principle remains central to every branch of physics and engineering.

1638 • Galileo's Dialogues

Galileo Galilei published Discourses and Mathematical Demonstrations Relating to Two New Sciences, establishing the concept of inertia and dismantling the Aristotelian requirement for sustained force to maintain motion. His thought experiments on idealized frictionless planes set the stage for Newton's later axioms.

1668 • Collision Experiments

John Wallis, Christopher Wren, and Christiaan Huygens independently presented studies on elastic and inelastic collisions to the Royal Society. Their experiments showed that momentum was conserved in two-body collisions—an empirical result that implicitly required the forces between colliding bodies to be equal and opposite.

1687 • Principia Mathematica

Isaac Newton published Philosophiæ Naturalis Principia Mathematica, codifying three laws of motion. The third law—Lex III—stated that 'to every action there is always opposed an equal reaction,' unifying contact forces, gravitational attraction, and collision dynamics under a single principle.

1743 • D'Alembert's Principle

Jean le Rond d'Alembert reformulated Newton's laws by introducing inertial (pseudo) forces, extending the third law's reach into non-inertial reference frames and paving the way for Lagrangian mechanics. His work demonstrated that the action–reaction structure was deeply embedded in the mathematical fabric of classical dynamics.

1918 • Noether's Theorem

Emmy Noether proved that every differentiable symmetry of a physical system's action corresponds to a conservation law. The translational symmetry of space implies conservation of momentum—providing the deepest theoretical justification for why Newton's Third Law holds: internal forces must cancel in pairs to preserve total momentum.

The historical thread reveals a persistent question: when two objects interact, how are the forces distributed between them? Newton's Third Law provides the definitive answer—the forces are always equal in magnitude and opposite in direction. This seemingly simple statement has profound consequences: it guarantees conservation of momentum, constrains every free-body diagram, and underpins the very definition of mass itself. The sections that follow develop each of these ideas rigorously.

SECTION 2

Core Principles & Definitions

Newton's Third Law is not a statement about a single body; it is a statement about an interaction between two bodies. Whenever object A exerts a force on object B, object B simultaneously exerts a force on A that is equal in magnitude and opposite in direction. These two forces are called an action–reaction pair (also known as a third-law pair or interaction pair). A critical point—often the source of student errors—is that the two forces in a third-law pair act on different objects and therefore never cancel each other in a free-body diagram of a single system.

1

Forces Come in Pairs

There is no such thing as a solitary force. Every force is part of a mutual interaction between two objects. If you push on a wall, the wall pushes back on you. The two forces are born simultaneously and cease simultaneously; one cannot exist without the other.
2

Equal Magnitude, Opposite Direction

The magnitudes of the action and reaction forces are always identical: |FA→B| = |FB→A|. Their directions are antiparallel. This equality holds regardless of the masses, velocities, or accelerations of the two objects.
3

Same Type of Force

Both forces in a third-law pair are of the same physical type. If A exerts a gravitational force on B, then B exerts a gravitational force on A—not a normal force or a friction force. This 'same-type' criterion is a reliable test for identifying genuine third-law pairs.
4

Act on Different Objects

The action force acts on one object while the reaction force acts on the other. Because they target different bodies, they never appear on the same free-body diagram and cannot produce equilibrium of a single object by themselves.
5

Instantaneous & Universal

In classical mechanics the two forces arise and vanish together—there is no time delay. The law applies universally to all force types: gravitational, electromagnetic, normal, tension, friction, and spring forces, across all inertial reference frames.
✦

KEY TAKEAWAY

Think of a third-law pair like a telephone call: it requires exactly two participants, each experiencing the same conversation from opposite ends. You cannot have a caller without a receiver. Similarly, you cannot have a force without a reciprocal force on the other object. Just as the same call appears on both phone bills (equal 'magnitude'), the two forces in a third-law pair are recorded on different objects' free-body diagrams, never on the same one.

SECTION 3

Visual Explanation — Action–Reaction Pairs

Action–Reaction Pairs: Person Pushing a Box on a FloorPersonm₁ = 70 kgBoxm₂ = 30 kgFloor / EarthF₁→₂F₂→₁N_floor→personF_person→floorN_floor→boxF_box→floorThird-Law PairsPerson pushes box (F₁→₂)Box pushes person (F₂→₁)Floor on person (Normal)Person on floor (Reaction)Floor on box / Box on floor
A person pushes a box on a floor. The cyan arrow (F1→2) and pink arrow (F2→1) form one third-law pair: the contact force between person and box. The amber and emerald arrows form a second pair: the normal force between person and floor, paired with the force the person exerts downward on the floor. The violet and red arrows show the analogous pair for the box and floor. Notice that each pair involves two different objects.

The diagram above identifies three distinct third-law pairs in a single scenario. A common misconception is to pair the weight of the box with the normal force the floor exerts on the box; while these two forces are equal in magnitude when the box is in static equilibrium, they are not a third-law pair because they are both forces on the same object (the box) and they arise from different interactions—gravity involves the Earth, while the normal force involves the floor's surface. The true third-law partner of the gravitational force the Earth exerts on the box is the gravitational force the box exerts on the Earth, which is equal in magnitude but acts on the Earth rather than the box. The diagram reinforces the critical identification rule: a genuine third-law pair always involves two forces, of the same type, acting on two different objects.

⚠

Common Misconception

Students often ask, 'If every force has an equal and opposite reaction, how can anything accelerate?' The resolution lies in recognizing that the two forces act on different objects. When you draw the free-body diagram for a single object, only the forces on that object appear. The net force on that object is generally nonzero, producing acceleration via Newton's Second Law. The reaction force is on the other object and affects its motion separately.

SECTION 4

Mathematical Framework

Newton's Third Law can be expressed compactly in vector notation. The mathematical formulation makes explicit both the magnitude equality and the directional opposition, and it connects directly to the conservation of linear momentum for isolated systems.

NEWTON'S THIRD LAW

F⃗A→B=−F⃗B→AF⃗_A→B = −F⃗_B→A

F⃗A→B = force exerted by object A on object B (vector); F⃗B→A = force exerted by object B on object A (vector). The negative sign encodes that the two forces are antiparallel—equal in magnitude but opposite in direction.

MOMENTUM CONSERVATION FROM THE THIRD LAW

dp⃗total/dt=F⃗A→B+F⃗B→A=0⃗dp⃗_total/dt = F⃗_A→B + F⃗_B→A = 0⃗

For an isolated two-body system with no external forces, the total momentum p⃗total = m₁v⃗₁ + m₂v⃗₂ is constant. The internal forces cancel by the Third Law, so the rate of change of total momentum is zero. This is the direct derivation of conservation of linear momentum from Newton's axioms.

IMPULSE–MOMENTUM RELATION FOR EACH BODY

Δp⃗1=F⃗B→A⋅Δt=−F⃗A→B⋅Δt=−Δp⃗2Δp⃗₁ = F⃗_B→A · Δt = −F⃗_A→B · Δt = −Δp⃗₂

During an interaction lasting time interval Δt, the impulse delivered to body 1 is equal and opposite to the impulse delivered to body 2. If body A has a larger mass, it experiences a smaller change in velocity than body B, but the magnitude of the momentum change |Δp⃗| is the same for both bodies.

An important corollary of the mathematical structure is the relationship between acceleration and mass for two interacting objects. Because F⃗A→B = −F⃗B→A, applying Newton's Second Law to each body yields mBa⃗B = −mAa⃗A. The object with the smaller mass experiences a larger acceleration, which is why a baseball accelerates enormously when struck by a bat, while the bat's deceleration is comparatively modest—even though the forces are identical in magnitude.

ACCELERATION RATIO

aB/aA=mA/mBa_B / a_A = m_A / m_B

The ratio of accelerations is the inverse ratio of the masses. This identity provides an operational definition of inertial mass: by measuring the accelerations of two interacting objects, one can determine their mass ratio without a balance or scale.

SECTION 5

Third-Law Pairs Across Force Types

Newton's Third Law applies to every type of force encountered in classical mechanics—contact forces (normal, friction, tension, applied) and field forces (gravitational, electrostatic, magnetic). However, identifying the correct third-law partner for each force requires careful attention to which two objects participate in the interaction. The following diagram and table catalog the most common force types and their third-law partners.

Free-Body Diagrams: Book on a TableIdentifying all third-law pairs for a static systemFBD of BookBookN (table→book)W = mg (Earth→book)ΣF = N − mg = 0 (equilibrium)Third-Law Partners (on other objects)TableN' (book→table)|N'| = |N| = mgEarthW' (book→Earth)|W'| = mg (negligible effect)Dashed lines connect each force on the book to its third-law partner on the other object.Pair 1: NormalPair 2: Gravity
Left: The free-body diagram of a book at rest on a table shows two forces—the normal force (upward, from the table) and weight (downward, from Earth's gravity). Right: Each force's third-law partner acts on a different object. The normal force's partner acts on the table; weight's partner acts on the Earth. The dashed lines connect each pair across the two panels.
Force on Object AThird-Law Partner on Object BForce Type
Normal force: table pushes book upwardBook pushes table downward (normal)Contact (normal)
Weight: Earth pulls book downwardBook pulls Earth upward (gravitational)Field (gravitational)
Friction: floor pushes box backwardBox pushes floor forward (friction)Contact (friction)
Tension: rope pulls block rightBlock pulls rope left (tension)Contact (tension)
Electrostatic: proton attracts electronElectron attracts proton (Coulomb)Field (electrostatic)

Common force types and their corresponding third-law partners

SECTION 6

Worked Example — Atwood Machine

An Atwood machine consists of two blocks of masses m₁ = 5.0 kg and m₂ = 3.0 kg connected by a massless, inextensible rope over a frictionless, massless pulley. Use Newton's Third Law and Second Law together to find the acceleration of the system and the tension in the rope.

Atwood Machine — Finding Acceleration and Tension

  1. 1

    Step 1 — Identify Third-Law Pairs

    The rope exerts an upward tension T on m₁ and an upward tension T on m₂. By Newton's Third Law, m₁ pulls the rope downward with force T at one end, and m₂ pulls the rope downward with force T at the other end. Because the rope is massless, these tensions are equal throughout the rope. Additional third-law pairs include the gravitational forces between each block and the Earth, but these do not constrain the calculation directly since we use the weight expression W = mg.
  2. 2

    Step 2 — Draw Free-Body Diagrams

    For m₁ (the heavier block, choosing downward as positive for m₁): net force = m₁g − T. For m₂ (the lighter block, choosing upward as positive for m₂): net force = T − m₂g. Both blocks share the same magnitude of acceleration a because the rope is inextensible.
  3. 3

    Step 3 — Apply Newton's Second Law to Each Block

    For m₁: m₁g − T = m₁a …(1). For m₂: T − m₂g = m₂a …(2). These two equations contain two unknowns—a and T.
  4. 4

    Step 4 — Solve for Acceleration

    Add equations (1) and (2): m₁g − m₂g = (m₁ + m₂)a. Therefore a = (m₁ − m₂)g / (m₁ + m₂). Substituting: a = (5.0 − 3.0)(9.8) / (5.0 + 3.0) = (2.0)(9.8) / 8.0 = 19.6 / 8.0.
    a = 2.45 m/s²
  5. 5

    Step 5 — Solve for Tension

    Substitute a back into equation (2): T = m₂(g + a) = 3.0(9.8 + 2.45) = 3.0 × 12.25.
    T = 36.75 N
  6. 6

    Step 6 — Verify with Third-Law Consistency

    Check equation (1): m₁g − T = 5.0 × 9.8 − 36.75 = 49.0 − 36.75 = 12.25 N. This should equal m₁a = 5.0 × 2.45 = 12.25 N. ✓ The tension in the rope is the same on both sides, consistent with the massless-rope assumption and Newton's Third Law applied at each point of the rope.
💡

Why This Works

The Atwood machine elegantly illustrates how the Third Law constrains the internal forces (tension) so that only the net external force (the difference in weights) drives the system's acceleration. Without the Third Law ensuring that the tension is the same at both ends of the rope, we could not reduce the two-equation system to a single equation for a.

SECTION 7

Common Misconceptions & Clarifications

Newton's Third Law is among the most frequently misunderstood principles in introductory physics. Many conceptual errors arise from conflating the forces in a third-law pair with forces that produce equilibrium on a single body, or from failing to recognize that the law holds even for accelerating systems. The table below contrasts common misconceptions with the correct understanding.

MisconceptionCorrect Understanding
'Action and reaction cancel, so nothing can accelerate.'The two forces act on different objects. For acceleration, consider the net force on one object alone—its FBD may have an unbalanced force.
'The stronger or more massive object exerts a larger force.'The forces are always equal in magnitude regardless of mass. The larger mass simply accelerates less (a = F/m).
'The weight of a book and the normal force from the table are a third-law pair.'These are two different forces on the same object and arise from different interactions (gravity vs. contact). They happen to be equal only in equilibrium.
'Action comes first, then reaction follows.'Both forces arise simultaneously. The labels 'action' and 'reaction' are arbitrary—either force can be called the action.
'The Third Law doesn't apply to objects at rest.'The Third Law applies universally—static or dynamic, contact or field forces, inertial or non-inertial frames (in the Newtonian limit).

Common misconceptions about Newton's Third Law and their corrections

🔍

DIAGNOSTIC TEST

To verify you have found a genuine third-law pair, apply the three-check rule: (1) Are the two forces on different objects? (2) Are they the same type of force (both gravitational, both normal, both friction, etc.)? (3) Are they equal in magnitude and opposite in direction? If all three checks pass, you have a valid third-law pair. If any check fails, the forces are not third-law partners.

SECTION 8

Connection to Advanced Theory

Newton's Third Law, while beautifully effective in classical mechanics, has nuances that become apparent in more advanced theoretical frameworks. Understanding where the law fits—and where its naive form requires modification—prepares you for upper-division coursework in electrodynamics, analytical mechanics, and relativity.

FrameworkStatus of Third LawKey Insight
Newtonian Mechanics (this course)Exactly valid for all forcesForces are instantaneous and pairwise; momentum conservation follows directly.
Lagrangian / Hamiltonian MechanicsEncoded in symmetryNoether's theorem links translational symmetry to momentum conservation, which is the deeper origin of the Third Law.
Classical ElectrodynamicsWeak form only; strong form can failMoving charges can produce forces that are not collinear (violating the 'strong' form). Momentum is conserved only when field momentum is included.
Special RelativityRequires careful restatementSimultaneity is frame-dependent; 'at the same instant' is ambiguous. Conservation of four-momentum replaces the Third Law as the fundamental principle.
Quantum MechanicsReplaced by conservation lawsForces are not fundamental; interactions are described by exchange of virtual particles. Momentum conservation is enforced by symmetry at the Lagrangian level.

Newton's Third Law across theoretical frameworks

The distinction between the weak form and the strong form of Newton's Third Law is worth noting. The weak form requires only that F⃗A→B = −F⃗B→A (equal and opposite). The strong form additionally requires that both forces act along the line connecting the two bodies—i.e., they are central forces. The strong form guarantees conservation of angular momentum in addition to linear momentum. Most forces in introductory physics (gravity, normal, tension, spring) satisfy the strong form. Magnetic forces between moving charges, however, can violate the strong form while still satisfying the weak form when field momentum is properly accounted for.

ℹ

Looking Ahead

In your upper-division mechanics course, you will see that Newton's Third Law is not an independent axiom but a consequence of translational symmetry of space via Noether's theorem. This perspective—that conservation laws arise from symmetries rather than from force laws—is one of the most profound unifying ideas in all of physics.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A horse pulls a cart forward. The cart pulls the horse backward with an equal and opposite force (Newton's Third Law). If these forces are truly equal, how is it possible for the horse–cart system to accelerate? Explain carefully, identifying all relevant forces and the objects they act on.
Show answer
The key is that the third-law pair (horse pulls cart forward, cart pulls horse backward) acts on different objects and therefore does not cancel on any single free-body diagram. To determine the motion of the horse, we draw its FBD: the ground exerts a forward static friction force on the horse's hooves, while the cart pulls the horse backward via the harness. If the friction force exceeds the backward pull from the cart, the net force on the horse is forward, so the horse accelerates. Similarly, the cart's FBD shows the horse pulling it forward and friction/drag pulling it backward; the net forward force accelerates the cart. The system accelerates because the external force (ground friction on the horse) exceeds the total resistive forces on the system.

Question 2: BASIC CALCULATION

A 60 kg ice skater pushes against a 90 kg ice skater on a frictionless surface. During the push, the lighter skater accelerates at 2.0 m/s². What is the acceleration of the heavier skater, and what force does each skater exert on the other?
Show answer
By Newton's Second Law, the force the heavier skater exerts on the lighter one is F = m₁a₁ = 60 × 2.0 = 120 N. By Newton's Third Law, the lighter skater exerts an equal force of 120 N on the heavier skater, but in the opposite direction. The acceleration of the heavier skater is a₂ = F / m₂ = 120 / 90 = 1.33 m/s². This confirms the acceleration ratio a₁/a₂ = m₂/m₁ = 90/60 = 1.5, and indeed 2.0/1.33 ≈ 1.5. Both skaters experience the same magnitude of force (120 N) but different accelerations because of their different masses.

Question 3: INTERMEDIATE

Two blocks are in contact on a frictionless horizontal surface. Block A (mass 4.0 kg) is pushed from the left by a horizontal applied force F = 24 N. Block B (mass 2.0 kg) is to the right of A. Find (a) the acceleration of the system, (b) the contact force between the blocks, and (c) identify the third-law pair associated with the contact force.
Show answer
(a) The system (A + B) has total mass 6.0 kg. Since the surface is frictionless, a = F_total / m_total = 24 / 6.0 = 4.0 m/s². (b) Consider the FBD of block B alone: the only horizontal force on B is the contact force from A, so F_contact = mBm_B × a = 2.0 × 4.0 = 8.0 N directed to the right. (c) The third-law partner is the force block B exerts on block A, which is 8.0 N directed to the left. We can verify: the net force on A is F_applied − F_contact(B on A) = 24 − 8.0 = 16 N to the right, and mAm_A × a = 4.0 × 4.0 = 16 N. ✓ The contact forces (A pushes B right at 8.0 N, B pushes A left at 8.0 N) form a valid third-law pair—same type (normal contact), same magnitude, opposite direction, acting on different objects.

Question 4: APPLIED

A 1200 kg car collides head-on with a 3600 kg truck. During the collision, the truck decelerates at 5.0 m/s². (a) What is the magnitude of the force the car exerts on the truck? (b) What is the magnitude of the force the truck exerts on the car? (c) What is the car's acceleration during the collision? (d) Why is the car damaged more severely than the truck, even though the forces are equal?
Show answer
(a) The force the car exerts on the truck: F = m_truck × a_truck = 3600 × 5.0 = 18,000 N. (b) By Newton's Third Law, the truck exerts an equal force of 18,000 N on the car. (c) The car's acceleration: a_car = F / m_car = 18,000 / 1200 = 15.0 m/s², which is three times the truck's deceleration. (d) Although the forces are equal, the car experiences three times the acceleration (and therefore three times the change in velocity) due to its smaller mass. Greater acceleration means more rapid deformation of the car's structure, higher g-forces on occupants, and more severe damage. The asymmetry in outcomes is due to the mass difference, not a force difference—a direct consequence of Newton's Second Law applied in conjunction with the Third Law.

Question 5: CRITICAL THINKING

Consider two charged particles: a proton (mass mpm_p = 1.67 × 10⁻²⁷ kg, charge +e) and an electron (mass mem_e = 9.11 × 10⁻³¹ kg, charge −e) separated by distance r. (a) Show that Newton's Third Law requires the electrostatic forces between them to be equal in magnitude. (b) Compute the ratio of their accelerations. (c) A student claims that the Third Law must fail for the electromagnetic interaction between two charges moving at relativistic speeds because the forces are not instantaneous. Critically evaluate this claim—does the Third Law fail, or does the concept of 'force' need to be generalized?
Show answer
(a) By Coulomb's law, F_proton→electron = ke²/r² directed toward the proton (attractive), and F_electron→proton = ke²/r² directed toward the electron (attractive). The magnitudes are identical because the same charges (e) and the same separation (r) appear in both expressions; the directions are opposite (each force points toward the other particle). This is fully consistent with Newton's Third Law. (b) The acceleration ratio is aea_e/apa_p = mpm_p/mem_e = (1.67 × 10⁻²⁷)/(9.11 × 10⁻³¹) ≈ 1836. The electron accelerates about 1836 times faster than the proton. (c) The student's claim has a kernel of truth but is imprecise. In classical electrodynamics, when charges move at high speeds, the forces between them are mediated by electromagnetic fields that propagate at the speed of light. The instantaneous force between two moving charges can appear to violate Newton's Third Law in the naive (particle-only) formulation. However, the resolution is not that the Third Law 'fails' but rather that the electromagnetic field itself carries momentum. When both the particle momenta and the field momentum are included, total momentum is conserved—which is the deeper principle underlying the Third Law. In Newtonian mechanics (the domain of this course), where speeds are much less than c, the Third Law holds exactly.

SUMMARY

Lesson Summary

Summary

Newton's Third Law states that whenever object A exerts a force on object B, object B simultaneously exerts a force on A that is equal in magnitude and opposite in direction: F⃗A→B = −F⃗B→A. These action–reaction pairs always involve the same type of force acting on two different objects, which is why they never cancel on a single free-body diagram. The law holds universally for all force types—gravitational, normal, friction, tension, electrostatic—and is valid regardless of whether the system is at rest or accelerating.

From the Third Law, one can derive the conservation of linear momentum for isolated systems, since internal forces cancel pairwise. The acceleration ratio of two interacting bodies equals the inverse of their mass ratio, explaining why lighter objects experience greater acceleration despite equal forces. To identify valid third-law pairs, apply the three-check rule: (1) different objects, (2) same force type, (3) equal magnitude and opposite direction. Looking ahead, Newton's Third Law is ultimately a consequence of the translational symmetry of space, as formalized by Noether's theorem—one of the deepest organizing principles in theoretical physics.

Varsity Tutors • College Physics • Newton's Third Law