Newton's third law

Before this

This leads to

Free-body diagrams
Newton's laws problems — pulleys and slopes

Push a wall, and the wall pushes back

Stand facing a wall and push it with your hands. The wall does not move. But you can feel something: the wall pressing into your palms. Push harder, and it presses harder.

The last lecture ended with a question it had avoided: when one object pushes another, what does the second object do back? Here is the answer. It pushes back, always, with a force of exactly the same size. This is Newton's third law. It sounds simple, and it is, but it creates a puzzle that confuses many students. If every push comes with an equal push back, why doesn't everything cancel? How does anything ever move?

The difficulty is not in the law. It is in one word: on. Keep asking "which object does this force act on?", and the puzzle disappears.

Newton's third law of motion

If object A exerts a force on object B, then object B exerts a force on object A that is equal in size and opposite in direction.

The two forces are called a third-law pair (some books say an action–reaction pair). Every such pair has four features:

1. Equal in size. Always, at every instant, whatever the objects.
2. Opposite in direction. Along the same line.
3. On different objects. One force acts on A, the other on B. Never both on the same object.
4. The same type of force. If one is a contact push, so is the other. If one is gravity, so is the other.

Feature 3 solves the puzzle. Forces can cancel only when they act on the same object, because only then do they add up to its resultant force. The two forces in a third-law pair act on different objects. Each one goes into a different object's $F = ma$ . They never cancel each other.

That is it. That is the whole idea. Forces come in pairs, but each force of a pair acts on a different object.

A book on a table: two pairs, not one

In the lecture on the first law, a 1.5 kg book rested on a table. Two forces acted on the book: its weight, 14.7 N down, and the table's push, 14.7 N up. They were equal and opposite. Are they a third-law pair?

No, and it is worth seeing exactly why. Both forces act on the same object, the book, which breaks feature 3. And they are different types of force: one is gravity, the other a contact push, which breaks feature 4. They balance because the book is not accelerating (the first law), not because of the third law.

Each of them has its own partner:

The two third-law pairs for a book on a table. The book is drawn lifted off the table so that the forces on each object can be seen. Pair 1 (contact): the table pushes the book up, and the book pushes the table down. Pair 2 (gravity): the Earth pulls the book down, and the book pulls the Earth up. All four forces are 14.7 N.

Pair 1, contact. The table pushes up on the book (14.7 N). The book pushes down on the table (14.7 N). This is the force the first-law lecture said "belongs to the third law".
Pair 2, gravity. The Earth pulls down on the book (14.7 N, its weight). The book pulls up on the Earth (14.7 N).

Yes, the book really pulls the Earth upwards with 14.7 N. The Earth does not visibly move because its mass is about $5.97 \times 1 0^{24}$ kg. By the second law, its acceleration is about $14.7 \div (5.97 \times 1 0^{24}) \approx 2.5 \times 1 0^{- 24}$ m/s². That is real, but far too small to notice.

Stop and think

A horse pulls a cart. By the third law, the cart pulls back on the horse with an equal force. So how can the horse and cart ever start moving?

Show the answer

Look at each object separately. On the cart, the horse's forward pull is larger than the friction on its wheels, so the cart has a forward resultant force and accelerates. The cart's backward pull acts on the horse, not on the cart. On the horse, there is a second horizontal force: its hooves push backwards on the ground, and by the third law the ground pushes the horse forwards. That forward push from the ground is larger than the cart's backward pull, so the horse accelerates too. The equal-and-opposite pair is on two different objects, so it never stops anything from moving.

Equal forces, unequal accelerations

Before the next box, make a prediction.

Predict — a truck and a small car collide

A 10 000 kg truck runs into a 1000 kg car. During the collision, which vehicle receives the larger force? Choose one, then read the explanation.

The forces are equal. The truck pushes the car, and the car pushes the truck, with the same size of force at every instant (the third law). The effects are different: with 10 times the mass, the truck's acceleration is only one-tenth of the car's (the second law). That is why the car, and the people in it, suffer far more.

Almost everyone's first instinct is that the bigger object pushes harder. The third law says no. What the bigger mass changes is the acceleration, through the second law. The third law tells you the forces; the second law tells you what each force does to its own object. You need both.

See it with two skaters on ice, where friction is small enough to ignore. They stand face to face and push each other apart for half a second.

Simulation — two skaters push apart

Choose the two masses. The skaters push each other with 140 N for 0.5 s, then glide. Press Play, or Step. Compare the forces, the accelerations and how far each one travels.

Skater A's mass: 50 kg

Skater B's mass: 70 kg

Each skater receives 140 N for 0.5 s, in opposite directions. Distances are measured 3 s after the push starts.

Masses (A, B)	Accelerations	Distances at 3 s
50 kg, 70 kg	2.8, 2.0 m/s²	3.85 m, 2.75 m
60 kg, 60 kg	2.33, 2.33 m/s²	3.21 m, 3.21 m
30 kg, 100 kg	4.67, 1.4 m/s²	6.42 m, 1.93 m

The forces are always equal. The lighter skater always has the larger acceleration and travels further.

Whatever masses you chose, the two arrows were always the same length. Only the accelerations differed. Make A very light and B very heavy, and A flies away while B barely moves. That is the truck and the car again, and it is also why you do not move the Earth when you jump.

Walking, swimming and rockets

The third law explains how anything moves itself forwards. You cannot push yourself: a force from you on you would act on the same object and change nothing. You must push something else, and let it push you.

Walking. Your foot pushes backwards on the ground (by friction). The ground pushes forwards on you. On smooth ice, the ground cannot push back much, so your foot slips.
Swimming. Your hands push water backwards. The water pushes you forwards.
Rockets. A rocket pushes hot gas out backwards at high speed. The gas pushes the rocket forwards. Nothing outside is needed: the rocket and its gas push on each other, so rockets work in empty space.

And the wall from the start? When you push the wall, the wall pushes you backwards. You stay still because the floor's friction pushes your feet forwards: the forces on you balance. The wall pushes on the Earth through its foundations, and the Earth's mass is so huge that its acceleration, for a push of 200 N, is about $200 \div (5.97 \times 1 0^{24}) \approx 3 \times 1 0^{- 23}$ m/s².

Worked examples

Worked Example 1 — Finding the partner

For each force, state its third-law partner: (a) the Sun's pull on the Earth; (b) a hammer's push on a nail; (c) the road's forward push on a car's tyres.

(a) The Earth's pull on the Sun: equal in size, gravitational, acting on the Sun. (b) The nail's push on the hammer: equal, a contact force, acting on the hammer, which is why the hammer stops. (c) The tyres' backward push on the road: equal, a contact (friction) force, acting on the road.

The method: swap the two objects. "A on B" becomes "B on A", with the same type of force.

Worked Example 2 — Two skaters

Skater A (50 kg) pushes skater B (70 kg) with a force of 140 N. Find the acceleration of each skater.

By the third law, B pushes A with 140 N in the opposite direction.

A: $a = 140 \div 50 = 2.8$ m/s². B: $a = 140 \div 70 = 2.0$ m/s². They accelerate in opposite directions.

Check: equal forces, but the lighter skater has the larger acceleration, as the second law says.

Worked Example 3 — The horse and the cart

A 500 kg horse pulls a 300 kg cart with a force of 400 N. Friction on the cart's wheels is 100 N. Both accelerate together. Find their acceleration, and the forward force of the ground on the horse.

On the cart: resultant $= 400 - 100 = 300$ N forwards, so $a = 300 \div 300 = 1.0$ m/s².

On the horse: the cart pulls it back with 400 N (the third-law partner of the horse's pull). The horse also accelerates at 1.0 m/s², so its resultant must be $500 \times 1.0 = 500$ N forwards. So the ground's forward push on the horse is $400 + 500 = 900$ N.

Check: the 400 N pair appears once in each calculation, on different objects, and never cancels.

Three classic mistakes

Trap 1 — Weight and the normal force as a pair

A book's weight and the table's push are equal and opposite, but they act on the same object and are different types of force. They are balanced forces, not a third-law pair. The partner of the weight is the book's pull on the Earth.

Trap 2 — "The pair cancels"

The two forces of a pair act on different objects, so they cannot cancel. Only forces on the same object add up to a resultant.

Trap 3 — "The bigger object pushes harder"

A truck and a car, a person and the Earth, a hammer and a nail: in every case the two forces are equal. The bigger mass changes the acceleration, not the force.

Try it yourself

Take $g = 9.8$ N/kg and the Earth's mass as $5.97 \times 1 0^{24}$ kg. Work each one on paper first.

A footballer's boot pushes a ball forwards with 300 N. Describe the third-law partner of this force.

Hint 1
Swap the two objects.

Hint 2
"Boot on ball" becomes "ball on boot".

Full answer
The ball pushes the boot backwards with 300 N. It is a contact force acting on the boot (and so on the foot), which is why kicking a heavy ball hurts.
Two astronauts float in space, 80 kg and 60 kg. They push each other apart with a force of 120 N. Find each astronaut's acceleration.

Hint 1
Each astronaut receives 120 N.

Hint 2
$a = F \div m$ for each one.

Full answer
80 kg: $120 \div 80 = 1.5$ m/s². 60 kg: $120 \div 60 = 2.0$ m/s². In opposite directions. Check: the lighter astronaut has the larger acceleration.
A 0.2 kg apple falls from a tree. Find the force of the Earth on the apple and the force of the apple on the Earth. Then find the Earth's acceleration.

Hint 1
The Earth's pull on the apple is its weight.

Hint 2
Divide the same force by the Earth's mass.

Full answer
Weight $= 0.2 \times 9.8 = 1.96$ N downwards on the apple. The apple pulls the Earth upwards with 1.96 N. The Earth's acceleration is $1.96 \div (5.97 \times 1 0^{24}) \approx 3.3 \times 1 0^{- 25}$ m/s²: equal forces, wildly unequal accelerations.
A 1200 kg car tows a 400 kg trailer with a rope. Both accelerate at 1.5 m/s². Friction on the trailer is 200 N, and resistive forces on the car are 300 N. Find the tension in the rope, the force of the trailer on the car, and the driving force the car needs.

Hint 1
Start with the trailer: the rope's pull minus friction gives its $ma$ .

Hint 2
The trailer pulls the car back with the same force as the rope pulls the trailer (third law).

Full answer
Trailer: $T - 200 = 400 \times 1.5 = 600$ , so $T = 800$ N. The trailer pulls the car back with 800 N. Car: driving force $- 800 - 300 = 1200 \times 1.5 = 1800$ , so the driving force is 2900 N. Check, treating car and trailer as one object: $2900 - 200 - 300 = 2400$ N, and $1600 \times 1.5 = 2400$ N. The 800 N pair disappears when both objects are taken together, because then both forces act inside the same object.

The bigger idea

The three laws now work as a team. The first says what happens with no resultant force. The second says how much a resultant force accelerates an object. The third says where forces come from: always from another object, and always in pairs.

The last answer in Try it yourself shows something deep. When two objects are taken together, their third-law pairs act inside the combined object and cancel out. Only outside forces can change the motion of a whole system. That is why a rocket must throw gas away, and it leads, in a later topic, to one of the great conservation laws of physics. The next lecture turns all of this into a reliable method: the free-body diagram, which shows only the forces on one object at a time.

Key concepts in this lesson

Newton's third law says that if A exerts a force on B, then B exerts an equal and opposite force on A.
The two forces of a third-law pair are equal in size, opposite in direction, of the same type, and act on different objects.
Third-law pairs never cancel each other, because only forces on the same object combine into a resultant force.
Equal forces on objects of different mass give different accelerations, so the lighter object accelerates more.
Walking, swimming and rockets all work by pushing something else, which pushes back on the mover.