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Newton's Third Law Calculator

Free Newton's third law calculator — equal and opposite forces, resulting accelerations and recoil speed, with an animated action-reaction pair.

Every force comes in a pair: equal in size, opposite in direction, and acting on two different bodies.

Mass of body 1 (kg)Mass of body 2 (kg)Interaction force (N)Push duration (s) — optional

The two arrows are identical. The two recoils are not — that is the whole point.

Force on each

200.00 N

equal and opposite

Acceleration of body 1

4 m/s²

Acceleration of body 2

2.5 m/s²

Acceleration ratio

0.63×

the mass ratio, inverted

Speed of body 1 after the push2 m/s

Speed of body 2 after the push1.25 m/s

Momentum each way100 kg·m/s

Net momentum of the pair0 kg·m/s — always zero

Kinetic energy split100.00 J / 62.50 J

Instant insight

Both bodies feel exactly 200.00 N — in opposite directions
That is the law itself, not a result of it: forces come in pairs, equal in size and opposite in direction, acting on two *different* bodies. The pair never cancels, precisely because the two halves act on different things — which is why you can push a wall and move, and why the pair does not add up to nothing.
Equal forces, but 0.63× the acceleration
a = F/m, so the same force produces 4 m/s² on the 50 kg body and 2.5 m/s² on the 80 kg one. The acceleration ratio is the mass ratio the other way up. This is the whole answer to "if the forces are equal, why does the small one fly off faster?" — the forces are equal, the masses are not, and acceleration is what you actually see.
Momentum divides exactly equally — energy does not
After 500.00 ms they carry 100 kg·m/s each, in opposite directions, so the pair's total momentum is still zero. But the kinetic energy splits 100 J to 62.5 J — the lighter body takes 38.5% of it, because for a given momentum, energy is p²/2m.
That asymmetry is why a rifle can be held against the shoulder while the bullet it fired is lethal: they share momentum equally and energy anything but.
Why the pair does not cancel out
The classic objection — "if every force has an equal and opposite reaction, nothing can ever accelerate" — fails because the two forces act on different bodies. To work out what one body does, you add up only the forces acting *on that body*. The reaction acts on something else entirely, and it is that something else that accelerates the other way.
What the model leaves out
Two bodies, one interaction, a constant force and nothing else acting. Real situations add friction, gravity and contact with the ground — all of which are themselves third-law pairs with the Earth. Walking works because you push the Earth backwards and it pushes you forwards; the Earth does accelerate, by an utterly unmeasurable amount, because its mass is on the bottom of a = F/m.

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Everyday Uses

How walking actually works

You push the ground backwards and it pushes you forwards. On ice friction is too weak for that push, which is why you go nowhere fast.

Recoil

Rifle and bullet receive identical impulses. The mass ratio is what turns one into a bruise and the other into a supersonic projectile.

Rocket propulsion in a vacuum

There is nothing outside to push against, so a rocket pushes against its own exhaust. It is why jets need air and rockets do not.

Stepping out of a small boat

Push forward onto the jetty and the boat is pushed backwards just as hard. The classic dockside soaking is a third-law demonstration.

Skaters pushing apart

Two skaters shove each other and drift away at speeds inversely proportional to their masses. Momentum before was zero and stays zero.

Crash forces between unequal vehicles

A car and a lorry hit each other with exactly the same force. The car occupants fare worse because the same force acts on far less mass.

Frequently Asked Questions

What is Newton's third law?

For every action there is an equal and opposite reaction. More precisely, if body A exerts a force on body B, then B exerts a force on A of the same size along the same line but in the opposite direction. The two forces always form a pair, always act on two different objects, and always appear and disappear together. There is no delay and no exception at everyday scales.

If the forces are equal, why does one object move more?

Because equal forces produce very different accelerations on different masses, which is Newton's second law doing the work. Push a supermarket trolley and you push the trolley exactly as hard as it pushes you, but you weigh far more than it does, so it accelerates away while you barely move. Fire a rifle and the bullet and the rifle receive identical impulses; the bullet is a thousand times lighter, so it leaves at high speed while the rifle merely thumps your shoulder.

Why don't the action and reaction forces cancel out?

Because they act on different objects. Forces only cancel when they act on the same body. A book resting on a table pushes down on the table and the table pushes up on the book — those are a third-law pair acting on two different things. What holds the book still is a separate balance: the table's upward push on the book cancels the Earth's downward pull on the book. Mixing these two ideas up is the single most common error in the topic.

How does a rocket work in a vacuum with nothing to push against?

It pushes against its own exhaust. The engine throws mass backwards at high speed, and the third law gives an equal forward force on the rocket. Nothing external is needed, which is why rockets work in space and jet engines do not. This is also momentum conservation viewed from a different angle: the total momentum of rocket plus exhaust stays constant, so as the exhaust gains momentum one way the rocket gains it the other.

How do you walk, according to Newton's third law?

You push backwards on the ground with your foot, and the ground pushes forwards on you with an equal force. That forward push from the ground is what accelerates you. The reason ice is difficult is not that the third law stops applying but that friction is too low to let you push backwards hard enough — the pair still exists, it is just very small. Swimming, rowing and driving all work the same way.