2.3 Newton's Third Law

Picture stories

Press Next (or Play) to walk through each story one small step at a time. The numbers come last.

1. Pushing off the rink wall

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A skater stands still at the edge of a rink, facing the wall. She puts her hands on the wall and pushes. She shoots backward across the ice, but she pushed forward, on the wall. Something must have pushed her backward. The only thing touching her hands is the wall.

Now two skaters stand palm to palm in the middle of the ice, one big and one small. Only the big one pushes. Both fly apart, the small one faster.

The question this topic answers: when you push on something, does it push back? How hard? And if the pushes are equal, why do the skaters not move the same way?

2. Palm to palm on the ice

3. How walking works

Quick check. A bug splats on the windshield of a fast car. Which force is bigger?

4. An apple pulls on the whole Earth

5. Equal and opposite, but not a pair

6. Who wins a tug-of-war?

Quick check. A cup sits on a table. The table pushes the cup up. What is the third-law partner of that push?

7. Check yourself

Think of your answer first, then tap to see it.

a) A big truck hits a small car. Which pushes harder on the other?

Show answer

Neither. The two forces are a third-law pair: the same size, opposite directions. The car just has less mass, so it speeds up or slows down far more.

b) A book rests on a table. Earth pulls the book down. What is the third-law partner of that pull?

Show answer

The book pulls Earth up with the same size force. (The table's push is not the partner: it acts on the same object, the book.)

c) A 50 kg skater and a 25 kg skater push apart. The 50 kg skater feels a 100 N push. What force does the 25 kg skater feel, and what is her acceleration?

Show answer

She also feels 100 N. a = 100 N ÷ 25 kg = 4 m/s².

Already know this?

Three quick questions. Get all three right on the first try and you can skip ahead to the simulation.

1. A big truck and a small car crash head on. During the crash, how does the force on the car compare with the force on the truck?

2. A book rests on a table. Its weight and the table's upward push are equal and opposite. Are they a third-law pair?

3. A horse pulls a cart. The cart pulls back on the horse just as hard. How can the cart still speed up?

The idea in plain words

Forces always come in pairs. If object A pushes or pulls on object B, then B pushes or pulls on A with a force of the same size in the opposite direction, at the same time.

FA on B = −FB on A

The two forces in a pair:

Trap: saying the two forces of a third law pair cancel. Instead: they act on different objects, so they sit on different diagrams; only forces on the same object can cancel.

To find the partner of any force, swap the names: the partner of "Earth pulls the book" is "the book pulls Earth".

Equal forces, different motion. The same force on a smaller mass gives a bigger acceleration (a = F/m, which is topic 2.5). That is why the small skater flies off faster, and why Earth does not visibly move when an apple falls.

Do not confuse a pair with balanced forces. A book on a table feels its weight (Earth on book) and the normal force (table on book). They are equal and opposite, but they act on the same object and are different kinds of force, so they are not a third law pair. They are equal only because the book is not accelerating.

Trap: calling the weight and the normal force on a book a third law pair. Instead: both act on the book; the partner of Earth-on-book is book-on-Earth, and the partner of table-on-book is book-on-table.

Worked numbers: two skaters push apart

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Skater A (60 kg) and skater B (40 kg) stand palm to palm on frictionless ice. They push apart with 200 N for 0.50 s. Positive is to the right; A is on the left.

  1. Force on B (by A): 200 N to the right. Force on A (by B): 200 N to the left, so −200 N. Why: third law. Same size, opposite direction, no matter who is bigger.
  2. aA = −200 N / 60 kg = −3.3 m/s²; aB = 200 N / 40 kg = +5.0 m/s². Why: same force, different masses, so different accelerations.
  3. After 0.50 s: vA = −3.33 × 0.50 = −1.67 m/s; vB = 5.0 × 0.50 = +2.5 m/s. Why: start from rest, v = at while the push lasts.
  4. After the push no sideways forces act, so each glides at constant velocity (first law, topic 2.4).
  5. Check: 60 × 1.67 = 100 and 40 × 2.5 = 100. Mass times speed is the same for both: a first look at momentum (Unit 4). Why: equal forces for equal times always give this.

Lab: two skaters push apart

Change the masses, the push and how long it lasts. Watch the two force graphs: they are always mirror images. Then watch the velocity graphs: they are not, unless the masses are equal. Positive is to the right.

Each diagram shows the forces on one skater. The horizontal pushes are the third law pair: one on each diagram, same size, opposite directions. The weight and normal force on one skater balance, but they are not a pair.

Examples

Example 1: name the pairs basic

You stand still on the floor. List the two forces on you and the third law partner of each.

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  1. Earth pulls you down (weight). Partner: you pull Earth up, with the same size force.
  2. The floor pushes you up (normal force). Partner: you push the floor down, with the same size force.
  3. Your weight and the floor's push both act on you, so they are not a pair. They balance because you are not accelerating.

Example 2: truck and car medium

A 4000 kg truck and a 1000 kg car collide. At one instant the car pushes on the truck with 40 000 N. Find the force on the car and the size of each vehicle's acceleration at that instant.

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Read the steps as text
  1. Third law: the truck pushes the car with 40 000 N, opposite direction.
  2. Truck: a = 40 000 / 4000 = 10 m/s². Car: a = 40 000 / 1000 = 40 m/s². Why: equal forces, but the car has one quarter of the mass, so four times the acceleration. That is why the people in the car get hurt more.
Trap: saying the bigger truck pushes the car harder than the car pushes the truck. Instead: the forces are equal (40 000 N each); the car's acceleration is bigger because its mass is smaller.

Example 3: how can a horse pull a cart? medium

"The cart pulls back on the horse as hard as the horse pulls the cart, so they cancel and nothing can move." What is wrong?

Show solution
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  1. The two pulls act on different objects: one on the cart, one on the horse. Forces cancel only when they act on the same object.
  2. Cart: the horse's pull forward is bigger than the ground's friction backward on the wheels, so the cart speeds up.
  3. Horse: the ground pushes its hooves forward (the partner of the hooves pushing the ground backward). That push is bigger than the cart's pull back, so the horse speeds up too.

Example 4: two boxes pushed together AP

On a frictionless floor you push a 3.0 kg box with 12 N; it pushes a 1.0 kg box in front of it. Find the acceleration and both contact forces between the boxes.

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Read the steps as text
  1. Both boxes as one system: a = 12 N / 4.0 kg = 3.0 m/s².
  2. 1.0 kg box alone: the only sideways force is the push from the 3.0 kg box: F = 1.0 × 3.0 = 3.0 N forward.
  3. Third law: the 1.0 kg box pushes back on the 3.0 kg box with 3.0 N backward.
  4. Check on the 3.0 kg box: net = 12 − 3.0 = 9.0 N, and 3.0 kg × 3.0 m/s² = 9.0 N. It works. Why the check matters: it proves the backward 3.0 N belongs on the 3.0 kg box's diagram.

Practice (AP style)

1. A heavy truck and a small car collide head-on. During the collision, how do the forces they exert on each other compare?

Show answer

Third law: the forces are a pair, so they are always equal in size and opposite in direction, whatever the masses or speeds.

2. In the same collision, which vehicle has the larger acceleration?

Show answer

Equal forces, so a = F/m is larger for the smaller mass: the car.

3. A book rests on a table. What is the third law partner of the book's weight (Earth pulling the book down)?

Show answer

Swap the names: "Earth on book" becomes "book on Earth", and both are gravity. The table's upward push is equal in size here but it is a different kind of force (contact) between different objects, so it is not the partner.

4. When you start walking forward, which force pushes you forward?

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Your foot pushes backward on the floor (a force on the floor). Its partner is the floor pushing forward on you (a friction force on you). Muscles pull on parts of your own body, which are internal forces and cannot move your center of mass.

5. A 50 kg skater pushes a 75 kg skater with 150 N. What force does the 50 kg skater feel?

Show answer

The 75 kg skater pushes back with the same 150 N, in the opposite direction. "Who pushes" does not matter.

6. A rocket in deep space fires its engine and speeds up. What exerts the forward force on the rocket?

Show answer

The rocket pushes the hot gases backward out of the nozzle. By the third law the gases push the rocket forward. No air is needed.

7. Earth pulls a 1.0 kg apple down with 9.8 N. Earth's mass is 6.0 × 1024 kg. Which is true about the apple's pull on Earth?

Show answer

Pairs are equal: 9.8 N. Earth's acceleration = 9.8 N / 6.0 × 1024 kg ≈ 1.6 × 10−24 m/s², far too small to notice.

8. (Short free response: argument) A student says: "When I push a heavy couch and it does not move, it is because the couch pushes back on me just as hard, so the forces cancel." Identify the error and give a correct explanation for why the couch stays still.

Show answer

Error: the student's push (on the couch) and the couch's push back (on the student) act on different objects, so they cannot cancel.

Correct: look only at forces on the couch. Sideways, those are your push forward and the floor's static friction backward. The couch stays still because the friction from the floor grows to match your push, so the net force on the couch is zero.

9. (Short free response: experimental design) You have two force sensors that record force against time, and two carts on a track. Describe an experiment to test whether the forces two objects exert on each other are equal and opposite, even when one cart is much heavier or moving. Say what graphs you expect.

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Mount a force sensor on the front of each cart, facing each other. Load one cart with extra mass. Run trials: (1) push the light cart into the heavy one at rest, (2) push the heavy cart into the light one, (3) both moving toward each other. Record both forces against time during each collision, with the same positive direction for both sensors.

Expected: in every trial the two force-time graphs are mirror images: same shape, same peak size at the same instant, opposite signs. Masses and speeds change how big and long the force is, but never make one force bigger than the other.

10. (Short free response: mathematical routines) Box A (2.0 kg) and box B (6.0 kg) touch on a frictionless floor. You push box B with 24 N, and B pushes A ahead of it. (a) Calculate the acceleration of the boxes. (b) Calculate the force B exerts on A. (c) State the size and direction of the force A exerts on B, and use it to check box B's motion. (d) Now you push A with 24 N from the other side so A pushes B. Calculate the new contact force.

Show answer
  1. (a) System: both boxes. a = 24 N / 8.0 kg = 3.0 m/s².
  2. (b) Box A alone: the only horizontal force is B's push. F = 2.0 × 3.0 = 6.0 N forward.
  3. (c) Third law: A pushes B with 6.0 N backward. Check B: 24 − 6.0 = 18 N, and 6.0 kg × 3.0 m/s² = 18 N. It works.
  4. (d) Same acceleration, 3.0 m/s², but now the contact force must speed up B: F = 6.0 × 3.0 = 18 N.

Point guide (4 points):

  • 1 point: a = 3.0 m/s² from the two-box system.
  • 1 point: 6.0 N from a one-box system.
  • 1 point: equal and opposite partner with a correct check on B.
  • 1 point: 18 N when the push is reversed.

11. (Short free response: translation between representations) A 1.0 kg book rests on a 5.0 kg table that stands on the floor. (a) List every force on the book and every force on the table, naming the object that exerts each one (a free body diagram in words). (b) Give the size of each force. (c) For each force on the book, name its third law partner and say on which diagram (book, table, or neither) the partner appears.

Show answer
  1. (a) Book: weight (Earth on book) down; normal force (table on book) up. Table: weight (Earth on table) down; book on table, down; floor on table (on its legs), up.
  2. (b) Book: weight 1.0 × 9.8 = 9.8 N; table on book 9.8 N. Table: weight 5.0 × 9.8 = 49 N; book on table 9.8 N; floor on table 49 + 9.8 = 58.8 N.
  3. (c) Partner of Earth-on-book is book-on-Earth: on neither diagram (it acts on Earth). Partner of table-on-book is book-on-table: on the table's diagram.

Point guide (4 points):

  • 1 point: correct forces on the book and the table, with sources, and no extra forces.
  • 1 point: correct sizes, including 58.8 N from the floor.
  • 1 point: both partners named with the right objects.
  • 1 point: says which diagram each partner is on (neither / table).

12. (Short free response: qualitative/quantitative translation) A 0.20 kg apple falls from a tree. Earth's mass is 5.97 × 1024 kg. (a) Compare the force Earth exerts on the apple with the force the apple exerts on Earth. (b) Calculate the acceleration of the apple and of Earth caused by this pair. (c) Explain, using your numbers, why we say "the apple falls to Earth" and not "Earth falls to the apple".

Show answer
  1. (a) Equal in size (third law pair), opposite in direction: each is 0.20 × 9.8 = 1.96 N.
  2. (b) Apple: a = 1.96 / 0.20 = 9.8 m/s². Earth: a = 1.96 / (5.97 × 1024) = 3.3 × 10−25 m/s².
  3. (c) Same force, but Earth's mass is about 3.0 × 1025 times bigger, so its acceleration is that many times smaller. Earth's motion is far too small to notice; nearly all the visible motion is the apple's.

Point guide (4 points):

  • 1 point: equal sizes, opposite directions, with 1.96 N.
  • 1 point: apple's acceleration 9.8 m/s².
  • 1 point: Earth's acceleration about 3.3 × 10−25 m/s².
  • 1 point: explains with a = F/m that the equal force gives Earth a tiny acceleration.

Common mistakes

The mistake: "Third law forces cancel, so nothing can accelerate."

Why it is wrong: the two forces of a pair act on different objects. Only forces on the same object can cancel.

How to spot it: check the "on" word. "A on B" and "B on A" are on different objects, so they go on different diagrams.

The mistake: "The bigger (or faster, or stronger) object pushes harder."

Why it is wrong: pair forces are always equal in size. What differs is the effect: a = F/m.

How to spot it: if your answer says one force in a pair is larger, it is wrong. Look for a difference in acceleration instead.

The mistake: calling weight and normal force a third law pair.

Why it is wrong: both act on the same object and are different kinds of force. They are equal only when the vertical acceleration is zero (they differ in an accelerating elevator, topic 2.5).

How to spot it: a real pair involves exactly two objects, with their names swapped.

The mistake: "First comes the action, then the reaction."

Why it is wrong: the two forces exist at the same instants: they start together and stop together.

How to spot it: in the lab, both force graphs switch on and off at exactly the same time.

The mistake: thinking a wall or floor cannot push because it does not move.

Why it is wrong: surfaces squeeze a tiny bit and push back. The wall pushes the skater; that is the only horizontal force on her.

How to spot it: if something speeds up, some object must push or pull on it. Name it.

More explanations: free resources.