How this instrument works
Newton's third law says every force is actually a pair: when object 1 pushes on object 2 with force F, object 2 pushes back on object 1 with a force of the same size, aimed the opposite way. Neither force exists without the other — they appear and vanish together, the instant contact is made or broken. This calculator does not compute two independent numbers; it enforces the identity F_reaction = −F_applied directly, because that identity is not an approximation of the physics, it is the physics.
The pairing comes straight from momentum bookkeeping on an isolated two-body system: if no outside force acts, total momentum cannot change, so whatever momentum object 1 loses, object 2 must gain, moment for moment. Differentiate that balance with respect to time and the two forces fall out equal and opposite by construction — the third law is really a statement about conserved momentum wearing a force costume. It is also why the two accelerations, each F divided by its own mass, differ whenever the masses differ.
The law holds for contact pushes, rope tension, magnetic repulsion, and gravity between two planets — anything where the interaction can be treated as instantaneous. It gets subtle for fields that carry their own momentum, such as two moving charges exchanging electromagnetic force with a measurable time lag, where momentum can briefly sit in the field rather than in either object. For ordinary mechanics problems — carts, blocks, a swimmer pushing off a wall — the equal-and-opposite rule applies exactly, with no fine print.
- Enter the push in the Force exerted by object 1 on object 2 field, in newtons.
- Enter the Mass of object 1 (exerting the force) — the body doing the pushing, in kilograms.
- Enter the Mass of object 2 (receiving the force) — the body being pushed, also in kilograms.
- Read the Reaction force (object 2 on object 1): it always matches the applied force in size, by Newton's third law.
- Compare Resulting acceleration of object 1 with Resulting acceleration of object 2 to see how one force pair produces two different accelerations.
Worked example — a 5 kg pusher against a 50 kg block
Suppose a 5 kilogram object pushes on a 50 kilogram object with 100 newtons of applied force. Newton's third law fixes the reaction force immediately, without any calculation beyond a sign flip: the 50 kilogram object pushes back on the 5 kilogram object with 100 newtons as well, so the reaction force reads 100.0 N.
The accelerations tell a different story, because each object obeys its own a = F divided by mass. The 5 kilogram object feels 100 N and accelerates at 100 ⁄ 5 = 20.0 m/s². The 50 kilogram object feels the same 100 N in return and accelerates at only 100 ⁄ 50 = 2.0 m/s². Equal forces, a tenfold difference in acceleration, purely because the masses are not equal — the everyday mix-up this instrument exists to clear up.
Questions
Why is the reaction force always exactly equal, never approximately?
Because the third law is an identity, not a measurement — it follows from momentum conservation between two interacting bodies. There is no friction or efficiency loss in the force pairing itself; any apparent inequality in a real experiment comes from a third force sneaking in, such as friction from the ground, not from the law failing.
Do the two objects always end up with the same acceleration?
No. Acceleration is force divided by mass, so unless mass1 and mass2 are equal, the two accelerations differ, sometimes hugely. In this calculator's own example, a 100 N force pair yields 20.0 m/s² for a 5 kg object and only 2.0 m/s² for a 50 kg object — same force, ten times the mass, one tenth the acceleration.
Does the third law apply to gravity as well as contact forces?
Yes. Earth pulls you downward with your weight, and by the third law you pull Earth upward with the identical force. You accelerate noticeably because your mass is small; Earth's acceleration from that same force is far too tiny to notice because its mass is enormously larger — the law does not care what kind of force is involved.
What if the two objects are already moving, not just pushing at rest?
The law still holds — it describes forces, not motion. A rocket exhaust and rocket body exchange equal and opposite thrust whether the rocket sits on a pad or is already climbing at speed, and two colliding billiard balls exert equal and opposite forces on each other for the instant they touch, regardless of their approach speeds.
Is the third law the same as saying two forces 'cancel out'?
No, and this is the classic mix-up. The two forces in a third-law pair act on different objects, so they never cancel each other — cancellation only happens when forces act on the same object, like the ground's push and gravity's pull that keep a resting book from accelerating. Here the applied force and the reaction force act on different bodies, which is exactly why both bodies accelerate.