Newton’s Third Law

Learning Goals

After this lesson, you should be able to:

• state Newton's third law in words and vector notation;
• identify the two forces associated with one interaction;
• use agent–receiver labels to test a proposed force pair;
• explain why a third-law pair does not cancel on one object's free-body diagram;
• compare the accelerations of unequal masses during an interaction; and
• analyze contact, gravitational, propulsion, and multi-object examples.

1. One Interaction, Two Forces

If A interacts with B, neither force occurs alone. The force of A on B and the force of B on A exist over the same interaction interval:

The minus sign means opposite vector directions. It does not mean that one force is weaker or physically less important.

2. The Four-Part Pair Test

Two forces form a Newton's-third-law pair only when all four statements are true:

1. They arise from the same interaction.
2. They have the same magnitude.
3. They point in opposite directions.
4. They act on different objects or systems.

A dependable notation test is to swap the two names: “A on B” pairs with “B on A.”

4. Why the Pair Does Not Cancel on One Object

Forces can cancel in the net force on an object only if they act on that same object. In the cart interaction, \(\vec F_{A\text{ on }B}\) acts on B, while \(\vec F_{B\text{ on }A}\) acts on A. They therefore belong to different single-cart free-body diagrams.

If the system boundary encloses both carts, both interaction forces are internal. Their vector sum is then zero within the combined system, leaving external forces to control the center-of-mass acceleration.

5. Balanced Forces Are Not Usually a Third-Law Pair

A book resting on a table has an upward force of the table on the book and a downward gravitational force of Earth on the book. These may be equal and opposite when the book has zero acceleration, but both act on the book and come from different interactions. They are a balanced-force pair, not a third-law pair.

6. Equal Forces Can Produce Unequal Accelerations

Newton's third law fixes the two interaction-force magnitudes, not the accelerations. Applying Newton's second law separately gives

The object with smaller mass has the larger acceleration magnitude. Equal accelerations occur only when the relevant masses are equal or when other forces make the net-force-to-mass ratios equal.

7. Worked Example: Two Skaters Push Apart

Question: A \(50\,\mathrm{kg}\) skater and a \(75\,\mathrm{kg}\) skater push on each other. During the push, each experiences a horizontal interaction force of magnitude \(150\,\mathrm N\). Ignore other horizontal forces. Find each acceleration magnitude.

Answer: The skaters accelerate in opposite directions. The \(50\,\mathrm{kg}\) skater accelerates at \(3.0\,\mathrm{m\,s^{-2}}\), while the \(75\,\mathrm{kg}\) skater accelerates at \(2.0\,\mathrm{m\,s^{-2}}\). The interaction forces remain equal.

8. Worked Example: Force Sensor Collision

Question: During a collision, cart A's force sensor reads a peak force of \(+240\,\mathrm N\) from cart B. What does B's sensor read at the same instant if both sensors use the same positive direction?

Answer: B's sensor reads \(-240\,\mathrm N\). Different cart masses, speeds, or amounts of damage do not change the equality of the paired forces.

9. Contact Time and Impulse

Because the paired forces act simultaneously over the same time interval, their impulses are also equal in magnitude and opposite in direction:

For approximately constant force, \(\vec J=\vec F\Delta t\). This connection explains why isolated interacting objects undergo opposite momentum changes.

10. Walking, Tires, and Propulsion

• Walking: a foot pushes the ground backward; the ground pushes the person forward.
• Driven car: a tire pushes the road backward; the road exerts forward static friction on the tire.
• Rocket: the rocket pushes exhaust backward; exhaust pushes the rocket forward. Air is not required.
• Swimming: the swimmer pushes water backward; water pushes the swimmer forward.

In every case, identify the material or object that receives the backward force and exerts the forward partner force.

11. Long-Range Interactions Also Come in Pairs

Contact is not required. Earth pulls an apple downward, and the apple pulls Earth upward with the same force magnitude. Their accelerations differ enormously because Earth's mass is much larger:

The paired gravitational forces are equal even though only the apple's acceleration is easily observed.

12. System Choice and Internal Forces

Suppose a person pushes a cart.

• Cart system: the person's force on the cart is external.
• Person system: the cart's force on the person is external.
• Person + cart system: those two forces are internal and cancel in the combined system's total.

The third-law relationship does not change; only the bookkeeping relative to the selected boundary changes.

13. Experimental Test with Two Force Sensors

Connect two force sensors with a light connector and have two students pull with changing strength.

1. Choose the same positive axis for both sensors.
2. Record both force–time graphs simultaneously.
3. Compare values at several shared times, including while the pull is changing.
4. Plot \(F_{A\text{ on }B}\) against \(-F_{B\text{ on }A}\); an ideal result has slope 1.
5. Repeat with one sensor attached to a cart of different mass.

Sensor uncertainty and timing offsets can explain small departures from perfect overlap; unequal cart accelerations do not disprove the third law.

14. Common Traps and Corrections

• Trap: pairing any equal and opposite arrows. Correction: require the same interaction and swapped agent–receiver names.
• Trap: saying the larger object exerts the larger force. Correction: mass affects acceleration, not third-law force equality.
• Trap: saying the reaction happens later. Correction: the paired forces are simultaneous.
• Trap: canceling both forces on one free-body diagram. Correction: each force acts on a different receiver.
• Trap: treating normal force and weight on a resting object as a third-law pair. Correction: find each force's actual partner by swapping names.
• Trap: claiming a rocket pushes on air. Correction: identify the rocket–exhaust interaction.

15. Key Takeaways

• Every interaction produces two simultaneous forces of equal magnitude and opposite direction.
• The two forces act on different objects and belong on different single-object free-body diagrams.
• Agent–receiver notation is the fastest reliable way to identify a pair.
• Equal interaction forces can cause different accelerations when masses differ.
• Third-law pairs become internal and cancel only when both interacting objects lie inside one system boundary.
• The law applies to contact and long-range interactions alike.