Newton's Third Law Calculator
See how two bodies push on each other with equal and opposite forces yet accelerate differently, from ice skaters to collisions to rocket thrust.
🔁 What is Newton's Third Law?
Newton's third law of motion says that forces always come in pairs. Whenever one body pushes or pulls on a second body, the second body pushes or pulls back on the first with a force of the same size and the opposite direction. It is often summed up as "every action has an equal and opposite reaction", although the wording is loose: neither force is really the cause of the other, and they exist at the same instant. The pair always has three properties. The two forces are equal in magnitude, opposite in direction, and, most importantly, they act on different bodies. A swimmer pushing off a wall pushes the wall, and the wall pushes the swimmer. A tire pushes the road backward and the road pushes the car forward. This calculator turns that pair into numbers for three situations: two bodies pushing apart, a collision, and a rocket.
The most common mistake is to ask why the forces do not cancel, so nothing could ever move. They do not cancel because cancelling is only possible between forces on the same body. Body A feels only the force from B, and body B feels only the force from A. Each body then follows Newton's second law with its own mass, a = F / m, so the lighter body accelerates more. Two ice skaters shoving each other with 120 N of force each feel the same push, yet the 60 kg skater picks up 2 m/s squared and the 80 kg skater 1.5 m/s squared. You can see exactly that in the pushing mode, together with the velocity of each body against time. Use the Newton's second law calculator for the single-body relationship F = m a.
The collision mode finds the equal and opposite forces during an impact. It computes the final velocities from momentum conservation and a restitution coefficient, takes the impulse on body 1, and divides by the contact time to get the average force on each body. The same force acts on both, which is why a small car and a large truck feel identical forces in a head-on crash although the car's occupants decelerate far more. To study the velocities after a collision in more detail, use the conservation of momentum calculator, and for the impulse and momentum relationship use the impulse and momentum calculator. A gun and bullet are a related case, covered by the recoil energy calculator.
The rocket mode shows the third law at work in propulsion. The engine accelerates exhaust gas backward, which requires a force on the gas, and the gas pushes the rocket forward with the reaction force. The thrust equals the mass flow rate times the exhaust speed relative to the rocket, T = (dm/dt) ve. Because the rocket pushes against its own exhaust and not against the air, it works in a vacuum. The mode also reports the thrust-to-weight ratio, the acceleration at the start and end of the burn as the propellant is used up, and the velocity at burnout, with an option to ignore gravity for a spacecraft in deep space. The model assumes a constant exhaust flow and speed, a vertical flight in constant gravity and no air drag. Real launches also include drag, a changing gravity and staging.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Two Skaters Pushing Apart
A 60 kg skater pushes an 80 kg skater with 120 N for 0.5 s, both starting at rest
Example 2 — Two Carts Already Moving
A 2 kg cart at 1.5 m/s and a 3 kg cart at 0.5 m/s push each other with 6 N for 0.4 s
Example 3 — A Car Hits a Parked Car
A 1000 kg car at 20 m/s hits a stationary 1500 kg car and they lock together, in a 100 ms contact
Example 4 — An Elastic Bounce Between Carts
A 2 kg cart at 3 m/s hits a stationary 1 kg cart with e = 1 in a 10 ms contact
Example 5 — A Small Rocket Launching From Earth
A 1000 kg rocket expels 10 kg/s of exhaust at 2500 m/s for 20 s, launching vertically
Example 6 — A Spacecraft in Deep Space
A 500 kg spacecraft expels 5 kg/s at 3000 m/s for 30 s with gravity ignored
❓ Frequently Asked Questions
🔗 Related Calculators
What is Newton's third law?
Newton's third law says that when body A exerts a force on body B, body B exerts a force on body A that is equal in size and opposite in direction. The two forces always come as a pair and act on different bodies.
Why do action and reaction forces not cancel?
They act on different bodies, and forces only cancel when they act on the same body. Skater A pushing skater B accelerates B, and the push back accelerates A, so each body feels only one of the two forces.
If the forces are equal, why do the accelerations differ?
Acceleration is force divided by mass. A 120 N force gives a 60 kg skater 2 m/s squared and an 80 kg skater 1.5 m/s squared. The lighter body always accelerates more, by the ratio of the masses.
How do two ice skaters pushing apart move?
They move in opposite directions with speeds inversely proportional to their masses. A 120 N push for 0.5 s gives a 60 kg skater 1 m/s one way and an 80 kg skater 0.75 m/s the other way, so the total momentum stays zero.
How is the force in a collision found?
Use the impulse. The average force equals the change in momentum divided by the contact time. A 1000 kg car that slows from 20 to 8 m/s in 100 ms feels an average force of 120,000 N, and the other vehicle feels the same force the other way.
How does a rocket work by Newton's third law?
The engine throws exhaust gas backward, so the gas pushes the rocket forward with an equal and opposite force. The thrust equals the exhaust mass flow rate times the exhaust speed. Ten kilograms per second at 2500 m/s gives 25,000 N.
Does a rocket need air to push against?
No. The rocket pushes against its own exhaust, not against the air, so it works in a vacuum. That is why the rocket mode has a switch to ignore gravity for a spacecraft far from any planet.
Is momentum conserved when two bodies push on each other?
Yes, as long as no outside force acts. The impulses on the two bodies are equal and opposite, F t each, so the total momentum before and after is the same. The calculator shows both totals.
What is the thrust-to-weight ratio?
It is the thrust divided by the weight of the rocket at liftoff. A value above 1 means the rocket can lift off. A 1000 kg rocket with 25,000 N of thrust has a ratio of 2.549 and an initial acceleration of 15.19 m/s squared.
What happens in an elastic collision compared with a sticking collision?
The contact forces are still equal and opposite. With a restitution coefficient of 0 the bodies leave with the same velocity and the most kinetic energy is lost. With 1 they bounce apart and no kinetic energy is lost.