Newton's Third Law

Stand on a skateboard and shove a friend who is standing on their own skateboard. Something surprising happens: you roll backwards too. You only pushed them — so why did you move? Try the same thing swimming: you push the water backwards with your hands, and the water somehow drives you forwards. Fire a rifle and it kicks into your shoulder. Blow up a balloon and let go, and it darts across the room.

Every one of these is the same secret rule at work. Whenever one object pushes or pulls another, the second object pushes or pulls straight back, just as hard, in the opposite direction. Forces are never lonely — they always come in pairs. That is Newton's third law, and once you see it you cannot un-see it: it is behind walking, swimming, rockets, guns, and the ground holding you up right now.

The law itself: action and reaction

Isaac Newton wrote it as a beautifully short sentence. If object A pushes on object B — call that the action — then at the very same instant B pushes back on A with an equal and opposite reaction. In symbols, the force A exerts on B and the force B exerts on A are related by

\vec{F}_{A \to B} = -\,\vec{F}_{B \to A}.

The little minus sign carries the whole idea: same size, opposite direction.

To every action there is an equal and opposite reaction. Whenever one object exerts a force on a second, the second exerts a force back on the first, and that pair of forces:

acts on two different objects — one force on each (this is the key to everything);
is equal in size (magnitude);
points in opposite directions;
is the same type of force (both pushes, both pulls, both gravity, both friction, and so on).

Naming the pair out loud is a great habit: "my foot pushes the ground backwards; the ground pushes my foot forwards." Say it that way and you can never lose track of which force acts on which body.

How you walk (and swim, and fly)

You might think your legs push you forwards. They don't — not directly. When you take a step, your foot pushes backwards on the ground. By the third law, the ground pushes forwards on your foot with an equal force, and that is what drives you along. On a slippery icy floor your foot can't grip to push back on the ground, so the ground can't push you forwards, and you go nowhere. Walking is a conversation between you and the planet.

The same pattern is everywhere once you look:

Swimming — your hands push water backwards; the water pushes you forwards.
A rowing boat — the oar pushes water back; the water pushes the boat on.
A car — the tyres push the road backwards (via friction); the road pushes the car forwards.
A jet or rocket — the engine throws hot gas backwards; the gas pushes the craft forwards.
A bird or a plane wing — it pushes air downwards; the air pushes it up.

In every case you move one way by shoving something else the other way. To go forwards, push the world backwards.

A common guess is that a rocket flies by pushing against the air, like a swimmer against water — so surely it would be helpless in the vacuum of space? In fact space is exactly where rockets work best. A rocket doesn't push on the outside air at all. It carries its own fuel, burns it, and hurls the hot exhaust gas out of the back at enormous speed. The rocket pushes the gas backwards; by Newton's third law the gas pushes the rocket forwards. That reaction pair is complete without any air, so a rocket happily accelerates in the emptiness between the planets — and with no air resistance to fight, it does so more efficiently than it ever could in the atmosphere.

Equal forces, unequal results

Here is the part that trips almost everyone. The two forces in a pair are always equal in size — but that does not mean the two objects move in the same way. What an object does depends on its force and its mass, through a = \dfrac{F}{m}. The same force gives a small mass a big acceleration and a big mass a small one.

Worked example — two skaters push off. A light skater (mass 40\ \text{kg}) and a heavier skater (mass 60\ \text{kg}) stand on frictionless ice and push apart. The push between them is 120\ \text{N}. By the third law each feels the same 120\ \text{N}, in opposite directions. Their accelerations, though, are different:

a_{\text{light}} = \frac{120\ \text{N}}{40\ \text{kg}} = 3\ \text{m/s}^2, \qquad a_{\text{heavy}} = \frac{120\ \text{N}}{60\ \text{kg}} = 2\ \text{m/s}^2.

Equal forces, but the lighter skater shoots off faster. This is why, when you shove your friend on skateboards, the lighter person always ends up moving quicker — even though the push on each of you was identical.

Worked example — a rifle's recoil. When a rifle is fired, the gun pushes the bullet forwards and the bullet pushes the gun backwards with an equal force. The bullet is tiny, so it rockets out at high speed; the gun is far heavier, so the same force only shoves it back slowly — a manageable "kick" against your shoulder. Same force, wildly different speeds, because of the huge difference in mass.

See the pair: two skaters on ice

Below are two skaters, A and B, at the moment they push off each other. Skater A pushes B to the right; skater B pushes A to the left. Drag the slider to change how hard they push, and watch the two arrows: however hard they shove, the arrow on A and the arrow on B stay exactly the same length and point in opposite directions. There is no way to make one bigger than the other — that is the third law made visible.

The trap that catches everybody: the book on the table

A book lies still on a table. Two forces act on it: its weight pulling it down, and the table's normal (support) force pushing it up. They are equal in size and opposite in direction, the book doesn't move — so surely these are Newton's action–reaction pair? No. This is the most famous mistake in the whole topic.

Both of those forces act on the same object — the book. A third-law pair must act on different objects, one force on each. They are also different types: weight is gravity, the support force is a contact push. They are equal here only because the book happens to be balanced (Newton's first law), not because of the third law. The genuine reaction partners are elsewhere:

The Earth pulls the book down (its weight) ⇄ the book pulls the Earth up by the same amount.
The table pushes the book up (the normal force) ⇄ the book pushes the table down by the same amount.

Each true pair acts on two different bodies and is the same type of force. Spot the partner by swapping the two objects around in the sentence: "the table pushes the book" becomes "the book pushes the table."

If the two forces are always equal and opposite, why don't they simply cancel and leave nothing moving ever? Because they act on different objects, so they can never be added together. When you push a wall, your force acts on the wall and the wall's reaction acts on you — one force on each body, so there is nothing for them to cancel.

Three specific slips to avoid:

The book's weight and the table's push are NOT a third-law pair. They act on the same object (the book) and are different types of force. They balance because of the first law, not the third.
Equal forces do NOT mean equal accelerations. An acceleration is a = F/m, so the lighter object in a pair always speeds up more. When Earth pulls an apple down, the apple pulls Earth up by exactly as much — but Earth is so massive it barely twitches.
Only add forces that act on the SAME object. To find how something moves, add up the forces on it alone. The reaction it exerts on other things belongs on their diagrams, never its own.

Spotting the reaction, every time

You never have to guess what the reaction to a force is. Take any force and write it as "A exerts a <type> on B." The reaction is the same sentence with A and B swapped: "B exerts a <type> on A," equal in size, opposite in direction. It is that mechanical.

Hammer hits nail (a push) ⇄ nail pushes back on hammer.
Your hand pulls a door (a pull) ⇄ the door pulls back on your hand.
Rocket pushes exhaust gas back ⇄ gas pushes rocket forwards.
Earth's gravity pulls the Moon ⇄ the Moon's gravity pulls the Earth, equally.

Notice the last one: even gravity, reaching across empty space with nothing touching, still obeys the rule. The Earth pulls the Moon and the Moon pulls the Earth back with an equal force — which is exactly why the tides rise and fall.

Yes — and it is the only way to do it. Imagine an astronaut floating still in space, drifting a few metres from their ship with nothing to grab and no air to swim through. How do they get back? They throw a heavy tool (a spanner, a wrench) away from the ship as hard as they can. As they push the spanner one way, by Newton's third law the spanner pushes them the opposite way — back towards the ship. The spanner sails off into the dark, and the astronaut glides gently home. It looks like magic, but it is nothing more than action and reaction, the same law that fires every rocket.