Newton’s Third Law of Motion in Physics

Newton’s third law of motion states that whenever one object exerts a force on another object, the second object exerts an equal force in the opposite direction on the first object. Often summarized as “For every action, there is an equal and opposite reaction,” the law explains why forces always occur in pairs. It governs interactions ranging from walking and swimming to rocket launches, collisions, and planetary motion, making it one of the fundamental principles of classical mechanics.

Key Points About Newton’s Third Law of Motion

• Forces always come in pairs: action and reaction.
• Action and reaction forces are equal in magnitude but opposite in direction.
• The forces act on different objects, so they do not cancel each other out.
• Newton’s third law explains the mutual interaction between two objects.
• The law applies universally, whether the objects are at rest or in motion.

Definition and Statement of Newton’s Third Law of Motion

Newton’s Third Law of Motion:

For every action, there is an equal and opposite reaction. In other words, when one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first object.

In mathematical form, if object A exerts a force F on object B, then object B exerts a force -F on object A:

F_A on B = −F_B on A​

The negative sign does not mean one force is weaker. It only indicates the two force vectors point in opposite directions.

What Does Newton’s Third Law Mean?

Newton’s third law describes the fundamental nature of forces, which always come in pairs. These paired forces are known as action-reaction force pairs. The law states that:

• Whenever a force is applied, another force of equal magnitude but in the opposite direction is simultaneously applied.
• These forces act on different objects, which is why they do not cancel each other out.

Forces in the universe do not act alone; every force has a partner force. This pairing is crucial to understanding interactions between objects, from the microscopic to the macroscopic scale.

Force Pairs

Force pairs consist of:

• The action force, which is the initial force applied by one object.
• The reaction force, which is the force exerted in response by the other object.

These forces are equal in magnitude but opposite in direction, and they act on separate objects.

The phrase “action-reaction pair” is traditional terminology, but many physics instructors prefer interaction force pair because neither force occurs first. Both forces arise simultaneously.

Examples Illustrating Newton’s Third Law

1. Walking:
   • When you walk, your foot pushes backward against the ground (action force).
   • The ground pushes forward with an equal force (reaction force), propelling you forward.
2. Recoil of a Gun:
   • When a bullet is fired, the gun exerts a force on the bullet (action force).
   • The bullet exerts an equal and opposite force on the gun (reaction force), causing the gun to recoil.
3. Rocket Launch:
   • The rocket’s engines expel exhaust gases downward (action force).
   • The gases push the rocket upward with an equal force (reaction force), allowing it to lift off.
4. Jumping Off a Boat:
   • When you jump off a boat, you push the boat backward (action force).
   • The boat pushes you forward with an equal force (reaction force), propelling you into the water.
5. Book on a Table:
   • A book exerts a downward force on the table due to gravity (action force).
   • The table exerts an equal and opposite upward force on the book (reaction force), preventing it from falling through.
6. Gravitational Attraction Between the Earth and the Moon:
   • Earth exerts a gravitational pull on the Moon, drawing it toward Earth’s center (action force).
   • The Moon exerts an equal and opposite gravitational pull on Earth, pulling Earth toward the Moon’s center (reaction force).

How to Identify an Action-Reaction Pair

When identifying Newton’s third-law pairs, ask three questions:

1. What two objects are interacting?
2. What force does object A exert on object B?
3. What equal and opposite force does object B exert on object A?

If both forces act on the same object, they are not an action-reaction pair.

Examples:

Newton’s Third Law vs. Other Laws of Motion

• Newton’s First Law states that an object remains at rest or in uniform motion unless acted upon by a net external force. It focuses on inertia and the condition of motion or rest.
• Newton’s Second Law quantifies the relationship between force, mass, and acceleration, expressed as F = ma. It explains how a net force affects the motion of an object.
• Newton’s Third Law, unlike the first and second laws, describes the interaction of forces between two objects. It focuses on the reciprocal nature of forces rather than the effects on a single object.

It’s important to distinguish between action-reaction force pairs (which are described by Newton’s third law) and balanced forces acting on a single object (often discussed in the context of Newton’s first and second laws). Objects only change their state of motion when there is an unbalanced net force acting on them. However, this concept is distinct from Newton’s third law of motion, which explains the reciprocal nature of forces between two interacting objects.

Action-Reaction Force Pairs (Newton’s Third Law)

When we talk about Newton’s third law, we refer to two forces that act on two different objects:

• For every action force, there is an equal and opposite reaction force.
• These two forces are equal in magnitude, opposite in direction, but act on different objects.

Example: Book on a Table

• The book exerts a downward force on the table due to its weight (action force).
• The table exerts an upward force on the book, equal in magnitude but opposite in direction (reaction force).

In this scenario:

• The downward force (weight of the book) acts on the table.
• The upward reaction force acts on the book.
• These forces do not cancel each other out because they act on different objects (the table and the book).

Balanced Forces Acting on a Single Object

This concept is different from balanced forces acting on a single object, which is described by Newton’s first law of motion. Balanced forces occur when multiple forces acting on the same object cancel each other out, resulting in no net force and no change in motion.

Example: The Book Itself

• The book experiences two forces: its weight (force of gravity) pulling it downward and the normal force from the table pushing it upward.
• These forces are equal in magnitude and opposite in direction, but they act on the same object (the book).
• Since these forces cancel each other out, the net force on the book is zero, and it remains at rest.

Summary of the Key Difference

Why This Distinction Matters

• In the context of Newton’s third law, action-reaction forces do not cancel each other out because they act on different objects.
• When we talk about an object being at rest or moving with constant velocity, we consider the net force acting on that object. In this case, we look at all the forces acting on the same object to determine if they are balanced.

Revisiting the Book on a Table Example

• Newton’s Third Law Pair: The book pushes on the table (action force), and the table pushes back on the book (reaction force). These forces act on different objects and do not cancel each other out.
• Balanced Forces on the Book: The weight of the book (gravity) acts downward, and the normal force from the table acts upward. These forces act on the book itself and cancel each other out, resulting in no net force and no movement.

Newton’s Third Law and Conservation of Momentum

Newton’s third law relates to the law of conservation of momentum. In a closed system where two objects interact:

1. The action-reaction force pairs exerted by each object on the other result in equal and opposite changes in momentum.
2. Since these changes in momentum are equal and opposite, they cancel each other out, leaving the total momentum of the system unchanged.

Example: Imagine two ice skaters pushing off from one another on frictionless ice:

• Each skater experiences an equal and opposite force from the other, pushing them apart.
• These forces act over the same time, imparting equal and opposite momentum changes to each skater.
• Although each skater’s momentum changes, the total momentum of the two-skater system remains zero (assuming it was initially zero).

This connection is why momentum is conserved in any interaction: Newton’s third law ensures that forces always occur in pairs, meaning any gain in momentum by one object is offset by an equal loss in momentum by another.

Newton’s Third Law in Collisions

Whenever two objects collide, each exerts an equal and opposite force on the other.

Examples include:

• billiard balls
• baseball bat and baseball
• bumper cars
• automobile crashes

Although the forces are equal, the resulting accelerations may differ because acceleration depends on mass according to Newton’s second law.

FAQs and Common Misconceptions

Q1: Do action and reaction forces cancel each other out?
A: No. The action and reaction forces act on different objects, so they do not cancel each other out.

Q2: Does Newton’s third law apply when objects are at rest?
A: Yes. Even at rest, objects exert equal and opposite forces, like a book resting on a table (normal force vs. gravitational force).

Q3: Can there be a force without an equal and opposite reaction force?
A: No. According to Newton’s third law, forces always come in pairs, with no exceptions.

Q4: Is the reaction force always of the same type as the action force?
A: Yes. For example, if the action force is a push, the reaction force is also a push in the opposite direction.

Q5: Why doesn’t a rocket push off the ground in space?
A: The rocket pushes against the expelled gases (action force), and the gases push back on the rocket (reaction force), propelling it forward even in the vacuum of space.

Q6: Does the Larger Object Exert a Larger Force?
A. No. During an interaction, both objects always exert equal-magnitude forces on each other.

For example, during a collision between a truck and a compact car, the truck pushes on the car with exactly the same force that the car pushes on the truck. The car usually experiences greater damage because its smaller mass results in greater acceleration (Newton’s second law), not because the truck exerts a larger force.

Common Student Challenges and Tips for Mastery

Sometimes Newton’s third law is challenging for students to grasp fully, especially when it comes to correctly identifying action-reaction pairs. Here are some common challenges and tips to overcome them:

1. Confusing Action-Reaction Pairs with Balanced Forces
   • Challenge: Students often confuse action-reaction pairs with balanced forces on a single object.
   • Tip: Remember that action-reaction pairs act on different objects and do not cancel out. Balanced forces, on the other hand, act on the same object and result in no net force.
2. Difficulty Identifying Force Pairs
   • Challenge: Identifying which forces form action-reaction pairs is not always intuitive.
   • Tip: Focus on the two interacting objects. Ask, “If object A exerts a force on object B, what force does object B exert back on object A?”
3. Forgetting that Newton’s Third Law Applies to All Forces
   • Challenge: Some students mistakenly believe that Newton’s third law only applies to contact forces.
   • Tip: Newton’s third law applies to all types of forces, including gravitational, magnetic, and electrostatic forces. Always look for an equal and opposite force in any interaction, whether contact or non-contact.
4. Neglecting the Vector Nature of Forces
   • Challenge: Students sometimes overlook the direction of forces, especially in complex scenarios.
   • Tip: Forces are vectors, meaning they have both magnitude and direction. Always consider the direction to correctly apply Newton’s third law.
5. Overthinking Symmetrical Scenarios
   • Challenge: In symmetrical scenarios (like a book on a table), students often overthink the forces involved, assuming that forces cancel each other on different objects.
   • Tip: In such cases, revisit the principle that action and reaction forces act on different objects, even if they are equal in magnitude. Double-check which objects each force acts upon.

Common Mistakes

Worked Example Problems

Here are some example physics problems that rely on Newton’s third law of motion for their solution:

Problem 1: Recoil of a Gun

A 5-kg rifle fires a 0.01-kg bullet at a velocity of 800 m/s. What is the recoil velocity of the rifle?

Solution:

Using conservation of momentum:

m_bulletv_bullet = m_riflev_rifle

(0.01 kg)(800 m/s) = (5 kg)(vrifle)

v_rifle = −1.6 m/s

The negative sign indicates the rifle moves in the opposite direction of the bullet.

Problem 2: Skater Pushes Off a Wall

A 60-kg skater pushes against a wall with a force of 120 N. What is the acceleration of the skater?

Solution:

By Newton’s third law, the wall exerts an equal and opposite force of 120 N on the skater.

Using Newton’s second law:

F = ma

120 N = (60 kg)(a)

a = 2 m/s²

The skater accelerates at 2 m/s² away from the wall.

Problem 3: Identifying the Action-Reaction Pair

A person leans against a wall.

Which two forces form Newton’s third-law pair?

Solution:

The person pushes on the wall.

The wall pushes back on the person with an equal and opposite force.

These are the action-reaction pair.

The person’s weight and the floor’s normal force are not the third-law pair because they involve different interactions.

Historical Context

Newton introduced the third law in Philosophiæ Naturalis Principia Mathematica (1687). Together with his first and second laws, it formed the foundation of classical mechanics and remained the standard description of motion until the development of relativity and quantum mechanics. Even today, Newton’s third law accurately describes most everyday interactions.

References

• Eisenbud, Leonard (1958). “On the Classical Laws of Motion”. American Journal of Physics. 26 (3): 144–159. doi:10.1119/1.1934608
• Halliday, David; Krane, Kenneth S.; Resnick, Robert (2001). Physics Volume 1 (5th ed.). Wiley. ISBN 978-0471320579.
• Knight, Randall D. (2008). Physics for Scientists and Engineers: A Strategic Approach (2nd ed.). Addison-Wesley. ISBN 978-0805327366.
• Newton, Isaac; Chittenden, N. W.; Motte, Andrew; Hill, Theodore Preston (1846). Newton’s Principia: The Mathematical Principles of Natural Philosophy. University of California Libraries.
• Rosengrant, David; Van Heuvelen, Alan; Etkina, Eugenia (2009). “Do students use and understand free-body diagrams?”. Physical Review Special Topics – Physics Education Research. 5 (1): 010108. doi:10.1103/PhysRevSTPER.5.010108