Newton's third law

Newton's third law: whenever one object pushes or pulls a second object, the second object pushes or pulls back on the first, at the same time, with a force that is equal in size and opposite in direction.

"If the two forces are equal and opposite, why don't they cancel?" Because they act on two different objects. Your push acts on the wall; the wall's push acts on you. Forces cancel only when they act on the same object.

See it for yourself:

• Sit on a chair with wheels and push a table: you roll backwards.
• Hook two spring balances together and pull: they always show the same reading.
• Let go of a blown-up balloon: it pushes air out, and the air pushes the balloon the other way.

Walking. Stand up and take a step. Feel your back foot push the ground backwards? Friction from the ground pushes you forwards. That is why you slip on oil or water: too little friction to push you.

More examples:

• Climbing a coconut tree: the legs push the trunk down; friction pushes the climber up.
• Rowing: the paddle pushes the water back; the water pushes the canoe forward.
• A rocket: it pushes hot gas down; the gas pushes the rocket up, with a force bigger than its weight.
• Chandrayaan-3: to land softly near the Moon's south pole, the Vikram lander fired its engines forwards to slow down.

Equal forces, very different results. When a gun fires, the same 2 N force acts on the bullet and on the gun. The 0.1 kg bullet gets a huge acceleration, 20 m s⁻²; the 5 kg gun gets only 0.4 m s⁻², felt as a kick backwards (recoil). In the same way, a falling fruit pulls the Earth up as hard as the Earth pulls it down, but the Earth is so massive that it hardly moves.