Newton's Third Law
When object A exerts a force on object B, object B simultaneously exerts an equal-magnitude, opposite-direction force on object A. The two forces belong to one interaction but act on different objects, so they never cancel on a single-object free-body diagram.
Topic sequence · 3 of 10
Learning objectives
Objective 1
Identify the two objects in an interaction and name both forces in the pair.
Objective 2
Distinguish third-law pairs from balanced forces acting on one object.
Objective 3
Use system boundaries to decide when an interaction pair is internal.
- Force as an interaction and free-body diagrams from Topic 2.2.
- System boundaries from Topic 2.1.
Definitions first
Representations you must read and create
Representation 1
Paired interaction arrows connecting two labeled objects.
Representation 2
Two separate free-body diagrams, one for each interacting object.
Representation 3
Combined-system FBD where the internal pair disappears from the external force list.
Core relationships and calculus
Variables: same interaction; different objects
Use: Newton’s third law
Variables: only when summing both internal forces in one larger system
Use: cancellation of internal pair in system force sum
Variables: matched force components for the same interaction pair
Use: apply Newton’s third law component by component
The third law is an algebraic vector relation, but it becomes powerful in calculus-based system analysis because internal force pairs cancel when summing m_i d²r⃗_i/dt² over all particles in the system, leaving the center-of-mass equation driven only by external forces.
Write Newton’s second law for every constituent. Sum the equations. Every internal pair F⃗_ij and F⃗_ji contributes zero by Newton’s third law, so the remaining force sum is external.
Newton’s third-law forces act on different interacting objects, so they do not cancel on a single-object free-body diagram. They cancel only when both objects are included in an appropriate larger system and the pair is internal.
Physical examples
Example 1
A truck and car exert equal and opposite forces on each other during a collision even though their accelerations are very different because their masses differ.
Example 2
A person pushes backward on the ground; the ground pushes forward on the person. The forward ground-on-person force can accelerate the person.
Example 3
Earth pulls a falling apple downward while the apple pulls Earth upward with equal force. Earth’s acceleration is tiny because its mass is enormous.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
A third-law pair must be the same interaction type and act on two different objects.
Reasoning pattern 2
Equal interaction forces can coexist with unequal accelerations because a=F/m is evaluated separately for each object.
Reasoning pattern 3
If both objects enter one system, show how the pair becomes internal and cancels before applying whole-system dynamics.
Deeper calculus and limiting-case check
The third law is especially powerful in many-particle calculus because Σ_iΣ_j F⃗_ij removes internal pairs. This is the mathematical reason center-of-mass motion depends only on external force for the standard constant-mass system model.
Model-selection workflow: decide what the problem is before choosing an equation
For Newton's Third Law, the fastest reliable route is not equation hunting. Start by identifying the system and interaction, encode it with the free-body diagram, choose the Newton-law or momentum-rate equation, and then test the inertial-frame and force-model assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat third-law pair and action-reaction language as physical quantities first and symbols second.
Choose a representation
Build Paired interaction arrows connecting two labeled objects.. If the prompt supplies Two separate free-body diagrams, one for each interacting object., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is F⃗_A on B = −F⃗_B on A. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Newton’s third-law forces act on different interacting objects, so they do not cancel on a single-object free-body diagram. They cancel only when both objects are included in an appropriate larger system and the pair is internal. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Paired interaction arrows connecting two labeled objects. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | F⃗_A on B = −F⃗_B on A | Newton’s third law. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | F⃗_AB + F⃗_BA = 0 | cancellation of internal pair in system force sum. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | The third law is an algebraic vector relation, but it becomes powerful in calculus-based system analysis because internal force pairs cancel when summing m_i d²r⃗_i/dt² over all particles in the system, leaving the center-of-mass equation driven only by external forces. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A 60 kg skater pushes a 40 kg skater. If the interaction force magnitude is 120 N, each skater experiences 120 N from the other in opposite directions. Their acceleration magnitudes are 2.0 m/s² and 3.0 m/s² respectively. Equal forces do not imply equal accelerations. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 2.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship F⃗_A on B = −F⃗_B on A should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Paired interaction arrows connecting two labeled objects. and Two separate free-body diagrams, one for each interacting object.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Place two force sensors between carts and record both readings during a push or collision. Plot F_AonB(t) and −F_BonA(t) on the same axes and compare their differences with calibration uncertainty.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Calling two opposite forces on the same object a third-law pair. | Return to the free-body diagram, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Thinking the larger or faster object exerts the larger interaction force. | Return to the Newton-law or momentum-rate equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Saying one force occurs first and the reaction comes later. | Return to the free-body diagram, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.1 uses internal-pair cancellation to justify center-of-mass dynamics. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 2.5 combines third-law reasoning with different masses and accelerations. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Unit 4 uses internal third-law pairs to explain conservation of momentum for isolated systems. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write F⃗_A on B = −F⃗_B on A and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 2.3 mastery ladder
Experimental connection
Place two force sensors between carts and record both readings during a push or collision. Plot F_AonB(t) and −F_BonA(t) on the same axes and compare their differences with calibration uncertainty.
Connections across Physics C
Connection 1
Topic 2.1 uses internal-pair cancellation to justify center-of-mass dynamics.
Connection 2
Topic 2.5 combines third-law reasoning with different masses and accelerations.
Connection 3
Unit 4 uses internal third-law pairs to explain conservation of momentum for isolated systems.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Calling two opposite forces on the same object a third-law pair.
Avoid this
Thinking the larger or faster object exerts the larger interaction force.
Avoid this
Saying one force occurs first and the reaction comes later.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why do equal collision forces cause unequal damage or motion?
Force equality does not make the objects identical. Different masses, structures, contact times, and energy transformations can produce very different outcomes.
Do third-law forces cancel?
They cancel only when both interacting objects are included inside the same system force sum.
Is weight paired with the normal force?
No. Weight is Earth-on-object. Its third-law partner is object-on-Earth. The normal force is a separate surface interaction.