Newton’s Third Law Calculator finds the acceleration or mass of a second object, or the reaction force on it, from an equal and opposite force pair, with steps, impulse and energy.
Object 1 and object 2 push on each other. Use a minus sign for the backward direction.
Result Unit and Contact Time
Step-by-Step Working
- Known values: m₁ = 90 kg, a₁ = 20 m/s², m₂ = 30 kg
- Action force on object 1: F₁ = m₁ × a₁ = 90 × 20 = 1,800 N
- Newton’s third law: F₂ = −F₁ = −1,800 N on object 2
- Newton’s second law for object 2: a₂ = F₂ ÷ m₂ = −1,800 ÷ 30 = −60 m/s²
- Result: a₂ = −60 m/s²
- Impulse, velocity changes, kinetic energy and the table use the contact time Δt = 0.2 s from the options below the inputs
Same Force on Other Masses
| Object 2 | Mass (kg) | a₂ (m/s²) | Velocity after Δt (m/s) |
|---|---|---|---|
| ¼ m₂ | 7.5 | −240 | −48 |
| ½ m₂ | 15 | −120 | −24 |
| m₂ (yours) | 30 | −60 | −12 |
| 2 m₂ | 60 | −30 | −6 |
| 10 m₂ | 300 | −6 | −1.2 |
| Earth | 5.972 × 10²⁴ | −3.014 × 10⁻²² | −6.028 × 10⁻²³ |
What the Newton’s Third Law Calculator Solves
The Newton’s Third Law Calculator works on two objects that push or pull on each other along one line. It takes the force acting on object 1 and returns the equal and opposite force on object 2. From that pair it finds the acceleration or the mass of the second object.
The Solve For menu has three settings. Acceleration of object 2 needs both masses and the first acceleration. Mass of object 2 needs the first mass and both accelerations. Reaction force needs the action force and both masses, and the force can be typed in newtons, kilonewtons, pounds-force or kilograms-force.
Direction is carried by the sign of each number. A positive value points one way along the line and a minus sign points the other way. With the starting entries the main result reads −60 m/s² Opposite Direction, so object 2 speeds up backward while object 1 speeds up forward.
Masses accept kilograms, grams, pounds, slugs and metric tons. Accelerations accept m/s², ft/s² and multiples of standard gravity, g. Each entry is converted to SI units first, and the Result Unit option then shows the answer in the unit a homework sheet or lab report asks for.
Newton’s Third Law Formula
A common question is whether Newton’s 3rd law has a formula at all. It does, and it is one of the shortest in physics. The reaction force on object 2 equals the action force on object 1 with its sign reversed.
$$F_{2} = -F_{1}$$
OpenStax University Physics writes the same law for two bodies A and B as FAB = −FBA. Newton’s second law then turns each force into a mass times an acceleration. That step gives the equation the Newton’s Third Law Calculator works from.
$$m_{1} a_{1} = -m_{2} a_{2}$$
In this equation m1 and a1 are the mass and acceleration of object 1, and m2 and a2 belong to object 2. With masses in kilograms and accelerations in m/s², each side comes out in newtons. Rearranging for the unknown gives the two forms behind the first two Solve For settings.
$$a_{2} = -\frac{m_{1} a_{1}}{m_{2}}$$
$$m_{2} = -\frac{m_{1} a_{1}}{a_{2}}$$
These forms hold when the pair is the only unbalanced force on each object. Two skaters pushing apart on ice or two carts on a low-friction track fit that picture well. If friction or another push also acts, the measured accelerations will not match the pair alone.
How to Calculate Newton’s Third Law by Hand
Take a 90 kg box and a 30 kg box pushing apart on a frictionless floor. The 90 kg box accelerates at 20 m/s². Its action force is 90 × 20 = 1,800 N, so the reaction force on the 30 kg box is −1,800 N.
Dividing that reaction by the second mass gives −1,800 ÷ 30 = −60 m/s². The Step-by-Step Working panel in the Newton’s Third Law Calculator prints these same lines with your own numbers. It adds the unit conversions whenever an entry is not in SI units.
Finding a mass works the same way. Two skaters push off each other, and a 60 kg skater accelerates at 2 m/s² while the partner accelerates at −1.5 m/s². The force on the first skater is 120 N, and 120 ÷ 1.5 puts the partner’s mass at 80 kg.
In the mass setting the two accelerations must be nonzero and carry opposite signs, and the tool flags the entry if they do not. In the other two settings a zero acceleration or zero action force returns a zero result with the note No Force Between the Objects.
Why Action and Reaction Do Not Cancel
Equal and opposite forces sound like they should add up to zero. They do not, because the two forces act on different objects. The 1,800 N acts on the first box and the −1,800 N acts on the second, so each box has an unbalanced force and each one accelerates.
The OpenStax text uses a swimmer pushing off a pool wall to make the point. Her feet push on the wall and the wall pushes back on her feet. Only the wall’s push acts on the swimmer, so she moves away from the wall.
Balanced forces are a different case. A book resting on a table has its weight pulling down and the table pushing up, and both act on the book. That is equilibrium under the first law. The third-law partner of the book’s weight is the book’s own gravitational pull on Earth.
A collision follows the same rule. A small car and a truck feel the same size of force during the impact, and the lighter vehicle takes the larger acceleration. The Car Crash Calculator turns a collision into the force on a person inside the car.
Newton’s Third Law Calculator Results Explained
Below the main answer, the Newton’s Third Law Calculator fills four result cards and two fold-out panels. The first card states the action force on object 1 in newtons, then the reaction on object 2 in pounds-force and kilograms-force. The other three cards cover acceleration, impulse and energy.
Same Force, Different Acceleration
The forces match but the accelerations do not, because a = F ÷ m. In the example the 30 kg box accelerates three times as hard as the 90 kg box, and the second card shows that as a 3× acceleration ratio. It also gives both accelerations in g, here 2.039 g and −6.118 g.
The Same Force on Other Masses panel takes this further. It applies your reaction force to a quarter, half, twice and ten times the second mass, and then to the whole Earth. For the −1,800 N example with a 0.2 s contact, the rows read as follows.
| Object 2 | Mass (kg) | a2 (m/s²) | Velocity after 0.2 s (m/s) |
|---|---|---|---|
| ¼ m2 | 7.5 | −240 | −48 |
| ½ m2 | 15 | −120 | −24 |
| m2 | 30 | −60 | −12 |
| 2 m2 | 60 | −30 | −6 |
| 10 m2 | 300 | −6 | −1.2 |
| Earth | 5.972 × 1024 | −3.014 × 10−22 | −6.028 × 10−23 |
The Earth row uses a mass of 5.9722 × 1024 kg from the NASA Earth Fact Sheet. It shows why the ground never seems to move when you jump. Earth feels the same force you do, but its acceleration is far too small to measure.
Gravity forms a third-law pair as well. Earth pulls on an apple, and the apple pulls on Earth with the same force. The Gravitational Force Calculator finds that force from G, the two masses and the distance between them, which is the GMm formula.
Impulse and Speed Change
The third card needs a contact time. It is set under Result Unit and Contact Time, in seconds or milliseconds. Impulse is the force multiplied by that time, and it equals each object’s change in momentum.
$$J = F \Delta t = m \Delta v$$
At the default 0.2 s, the 1,800 N pair delivers 360 N·s to each box. The 90 kg box gains 4 m/s and the 30 kg box gains −12 m/s. The two momentum changes are +360 and −360 kg·m/s, so the total does not change. That is conservation of momentum.
The force is treated as constant over the contact, so the number you enter in the Newton’s Third Law Calculator stands for the average force of a real push. For problems that start from a velocity change, the Impulse and Momentum Calculator works in that direction. The Conservation of Momentum Calculator handles full collisions.
Kinetic Energy Split
Both objects get the same impulse but not the same energy. Starting from rest, the two boxes gain 2,880 J together, and the lighter box takes 75% of it. The energies split in the inverse ratio of the masses.
$$\frac{KE_{2}}{KE_{1}} = \frac{m_{1}}{m_{2}}$$
The fourth card also gives the separation speed, 16 m/s here, which is how fast the gap between the boxes opens. The same split explains recoil. A bullet is far lighter than the rifle, so it leaves with most of the kinetic energy while the rifle moves back slowly.
When a compressed spring between two carts supplies the push, the spring force is the action force to enter. The Hooke’s Law Calculator gives that force from the spring constant and the compression.
Newton’s 1st, 2nd and 3rd Laws of Motion
Newton set out three laws of motion in the Principia. NASA’s Glenn Research Center files them under inertia, force, and action and reaction. Each law has its own formula.
| Law | Also called | Formula | What it says |
|---|---|---|---|
| 1st | Law of inertia | Fnet = 0, so v is constant | An object stays at rest or keeps a steady speed in a straight line unless an unbalanced force acts on it. |
| 2nd | Law of acceleration | F = ma | Acceleration grows with the net force and shrinks as mass grows. |
| 3rd | Law of action and reaction | F2 = −F1 | Two objects push on each other with equal and opposite forces. |
Newton’s famous formula is the second law, F = ma. NASA gives its fuller form as force equal to the change in momentum over time, which reduces to F = ma when mass stays constant. NCERT’s Class 9 chapter on force and laws of motion teaches the second law through momentum in the same way.
To calculate Newton’s second law, multiply the mass by the acceleration. The Newton’s Second Law Calculator does this for net force, mass, acceleration, final velocity or time. The Newton’s Third Law Calculator depends on it too, because a force pair only becomes an acceleration through F = ma.
The phrase “3rd formula of motion” usually points to something else. It is the third equation of motion from kinematics, which has nothing to do with action and reaction.
$$v^{2} = u^{2} + 2 a s$$
It links final speed v, starting speed u, acceleration a and distance s under constant acceleration. The SUVAT Calculator solves this and the other kinematic equations. The Acceleration Calculator finds a from a speed change, a distance or a force.
How Much Force Is a Newton?
One newton is the force that gives a 1 kg mass an acceleration of 1 m/s². It is a small force. A mass of about 102 g weighs one newton at standard gravity, which is why 1 N is often compared to the weight of a small apple.
A newton is not equal to a kilogram, because one measures force and the other measures mass. A 1 kg mass weighs 9.80665 N on Earth. Going the other way, 10 N is the weight of 1.02 kg, and the kg to Newtons Calculator converts any weight to force.
| Force (N) | Pounds-force (lbf) | Weight of this mass (kg) |
|---|---|---|
| 1 | 0.2248 | 0.102 |
| 10 | 2.248 | 1.02 |
| 1,000 | 224.8 | 102 |
| 20,000 | 4,496 | 2,039 |
So 1,000 newtons is roughly the weight of a 102 kg person, and 20,000 N is the weight of a little over two metric tons. The factors are 4.448222 N per pound-force and 9.80665 N per kilogram-force, from the NIST Guide to the SI.
The reaction force mode of the Newton’s Third Law Calculator shows the pair in these same units. For other routes to a force, the Force Calculator covers pressure, springs and velocity changes. Every other dynamics calculator on the site is listed under Dynamics and Statics Calculators.
Third Law and Energy Questions
What is Newton’s 3rd law in simple words?
When you push on something, it pushes back on you just as hard in the opposite direction. That is why the law is nicknamed the law of action and reaction. For kids, a good picture is a balloon let go without a knot. The balloon pushes air out the back, and the air pushes the balloon forward.
The same thing happens when you walk. Your foot pushes backward on the ground, and the ground pushes you forward. A rocket works this way too. It pushes exhaust gas backward, and the gas pushes the rocket forward, even in empty space.
Is there a fourth law of motion?
No. Newton’s Principia states three laws of motion, and physics courses still teach three. His law of universal gravitation is sometimes loosely called a fourth law, but it describes one particular force. Lists of a fifth or sixth law of motion do not come from Newton’s work.
Are Newton’s laws still true?
They remain accurate for everyday objects at everyday speeds, which covers cars, aircraft, sports and most engineering. They lose accuracy for objects moving near the speed of light, where special relativity takes over. They also fail at the scale of atoms, where quantum mechanics applies.
Is inertia a force?
No. Inertia is the tendency of an object to resist a change in its motion, and mass is the measure of it. A force is a push or pull that comes from another object. In the box example both boxes feel 1,800 N, and the 90 kg box accelerates less only because it has more inertia.
What is 1 joule equal to?
One joule is the energy transferred when a force of 1 N acts through 1 m, so 1 J = 1 N·m = 1 kg·m²/s². The joule is the SI unit of energy, and energy is the capacity to do work. A joule is not equal to a kilogram, since one measures energy and the other measures mass.
Ten joules lifts a 1 kg mass about 1.02 m against gravity. One thousand joules is called a kilojoule. The 2,880 J that the Newton’s Third Law Calculator shows for the box example is 2.88 kJ, or about 0.69 food calories.
How powerful is 1 billion joules?
One billion joules is a gigajoule. It equals about 278 kilowatt-hours of electricity, or the energy released by roughly 0.24 tons of TNT. The joule measures energy and not power, because power is energy per second. A gigajoule delivered in one second would be one gigawatt.