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Newton's Third Law Calculator

Last updated: March 28, 2026

Calculator

Results

Reaction Force

-50

N

Acceleration of Object A

0.7143

m/s²

Acceleration of Object B

-8.3752e-24

m/s²

Force Ratio

1

Results

Reaction Force

-50

N

Acceleration of Object A

0.7143

m/s²

Acceleration of Object B

-8.3752e-24

m/s²

Force Ratio

1

Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction: $$\vec{F}_{12} = -\vec{F}_{21}$$. When object A exerts a force on object B, object B simultaneously exerts a force of equal magnitude but opposite direction back on object A. These are called an action-reaction pair, and they always act on different objects.

This principle explains why rockets work in the vacuum of space (exhaust pushes backward, rocket pushes forward), why you feel a recoil when firing a gun, and why walking is possible (your foot pushes Earth backward, Earth pushes you forward). The forces in an action-reaction pair are always equal in magnitude — what differs is the acceleration each object experiences, because acceleration depends on mass via $$a = F/m$$.

Our calculator demonstrates this law by computing the reaction force and the resulting accelerations of both objects. Even though the forces are equal and opposite, a lighter object accelerates much more than a heavier one, which is why you move when you push against the Earth but Earth's acceleration is imperceptibly tiny.

Visual Analysis

How It Works

Newton's Third Law is expressed as: $$\vec{F}_{12} = -\vec{F}_{21}$$

The reaction force is simply the negative of the action force: $$F_{reaction} = -F_{action}$$

Using Newton's Second Law, the acceleration of each object is: $$a_A = \frac{F_{action}}{m_A}$$ $$a_B = \frac{F_{reaction}}{m_B} = \frac{-F_{action}}{m_B}$$

The force ratio is always exactly -1, confirming the forces are equal in magnitude and opposite in direction. The acceleration ratio, however, is $$\frac{a_A}{a_B} = -\frac{m_B}{m_A}$$, showing that the lighter object always accelerates more.

Understanding Your Results

The reaction force magnitude always equals the action force — this is a law of nature, not a simplification. The negative sign indicates opposite direction. The accelerations differ inversely with mass: if object B is 10 times heavier than A, A accelerates 10 times more. This explains everyday experiences like why a ball bounces back when thrown against a wall (the wall barely moves due to enormous mass).

Worked Examples

Person Pushing a Wall (50 N)

Inputs

action force50
action mass70
reaction mass5.97e+24

Results

reaction force-50
accel a0.7143
accel b0
force ratio-1

The wall pushes back with exactly 50 N. You accelerate at 0.71 m/s²; Earth's acceleration is negligible.

Two Ice Skaters Pushing Apart

Inputs

action force40
action mass60
reaction mass80

Results

reaction force-40
accel a0.6667
accel b-0.5
force ratio-1

Both experience 40 N but the lighter skater (60 kg) accelerates faster (0.67 vs 0.50 m/s²).

Frequently Asked Questions

Action-reaction forces act on different objects, so they cannot cancel each other. Forces cancel only when they act on the same object. When you push a box, your force on the box and the box's force on you are an action-reaction pair acting on two separate bodies.

Yes. Earth pulls you downward with gravitational force $$F = mg$$, and you pull Earth upward with exactly the same force. However, because Earth's mass is approximately $$5.97 \times 10^{24}$$ kg, its acceleration is negligibly small (about $$10^{-23}$$ m/s² for a 70 kg person).

A rocket expels exhaust gases at high speed. The rocket pushes the exhaust backward (action), and the exhaust pushes the rocket forward (reaction). No external medium (like air) is needed — the interaction is purely between the rocket and its exhaust, which is why rockets work perfectly in vacuum.

No. Action-reaction pairs also exist for non-contact (field) forces such as gravity, electromagnetic force, and even the strong nuclear force. Earth and the Moon exert equal gravitational forces on each other despite being 384,000 km apart.

Because both objects experience the same magnitude of force, Newton's Second Law $$a = F/m$$ dictates that the object with less mass undergoes greater acceleration. This is why a rifle recoils much less than the bullet it fires, even though the forces are identical.

Yes. The sign of the force indicates direction. If the action force is negative (e.g., -50 N pointing left), the reaction force is positive (+50 N pointing right). The magnitudes are always equal; only directions differ.

Sources & Methodology

Newton, I. — Principia Mathematica (1687); Kleppner & Kolenkow — An Introduction to Mechanics, 2nd Ed. (2014); Serway & Jewett — Physics for Scientists and Engineers, 10th Ed. (2019)
R

Roboculator Team

The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.

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