Newton's Third Law Calculator
Calculate reaction force using Newton's Third Law
Category: Physics
Newton's Third Law Calculator Inputs
Newton's Third Law Calculator Formula
Equation
F_action = -F_reaction
Excel Formula
=F_{action}=-F_{reaction}
Variables
- Action Force (N) — Enter the Action Force (N) value used by the Newton's Third Law Calculator.
How the Newton's Third Law Calculator Works
Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. More precisely: when object A exerts a force on object B, object B simultaneously exerts a force equal in magnitude and opposite in direction on object A. These action-reaction force pairs are fundamental to understanding all interactions in physics - from walking and swimming to rocket propulsion and planetary orbits. This law reveals that forces always come in pairs, never acting alone, and explains why you can push off a wall to move yourself, why guns recoil when fired, and how rockets accelerate in the vacuum of space. Understanding Newton's Third Law is essential for physics, engineering, aerospace, biomechanics, and any field involving forces and interactions.
The core relationship is F_{action} = -F_{reaction}. Typical inputs include Action Force.
Enter your values in the newton's third law calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online physics tool is built for homework, design checks, and professional verification.
Newton's Third Law Calculator Theory & Explanation
Newton's Third Law Fundamentals
Published in 1687 in the Principia Mathematica, Newton's Third Law establishes that forces always occur in pairs. If object A exerts force F on object B (the "action"), then object B simultaneously exerts force -F on object A (the "reaction"). These forces are equal in magnitude, opposite in direction, and act on different objects. Crucially, the "action" and "reaction" labels are arbitrary - neither comes first, they occur simultaneously. The law applies to all forces: gravitational, electromagnetic, contact forces, tension, friction - without exception.
\vecF_AB = -\vecF_BA \quad \textwhere |\vecF_AB| = |\vecF_BA| \text and directions are opposite
Why Don't Action-Reaction Forces Cancel?
A common misconception: if forces are equal and opposite, why doesn't everything remain stationary? The key is that action-reaction pairs act on DIFFERENT objects. When you push a wall with 100 N, the wall pushes you with 100 N - but these forces act on different things (you and the wall). To determine motion, you apply Newton's Second Law (F = ma) to EACH object separately, considering only forces acting ON that object. The forces don't cancel because they're on different objects in different free-body diagrams.
\textNet force on A = Σ F_\texton A \quad \textNet force on B = Σ F_\texton B \quad (\vecF_AB \text not in sum for A)
Walking: A Perfect Example
Walking demonstrates Newton's Third Law beautifully. You push backward on the ground with your foot (action), and the ground pushes you forward (reaction). If there were no friction, you couldn't walk - your foot would slip because the ground couldn't exert a forward reaction force. On ice, friction is reduced, making walking difficult. The forward force from the ground accelerates you forward (Newton's Second Law), while your backward push on the ground would accelerate Earth (imperceptibly due to its enormous mass). Both forces are equal, but accelerations differ vastly due to different masses.
F_\textyou on ground = -F_\textground on you \quad \Rightarrow \quad a_\textyou = (F)/(m_\textyou) \gg a_\textEarth = (F)/(m_\textEarth)
Rocket Propulsion in Space
Rockets demonstrate Newton's Third Law in its purest form. The engine expels exhaust gas backward at high velocity (action force on gas). By Newton's Third Law, the gas exerts an equal and opposite force forward on the rocket (reaction - this is thrust). Critically, rockets work in the vacuum of space because they don't push against anything external - they push against their own expelled fuel. This is why "space needs no medium" for rocket propulsion. The momentum of expelled gas backward equals the momentum gained by the rocket forward (momentum conservation, derived from Newton's Third Law).
F_\textrocket on gas = -F_\textgas on rocket \quad \Rightarrow \quad m_\textrocketv_\textrocket = -m_\textgasv_\textgas \quad (\textmomentum conserved)
Gravitational Action-Reaction Pairs
Gravity obeys Newton's Third Law perfectly. Earth pulls you downward with force W = mg (your weight). Simultaneously, you pull Earth upward with equal force mg. Both forces have the same magnitude (your weight), but the accelerations are vastly different. You accelerate at g = 9.81 m/s² toward Earth because F = ma gives a = F/m = (mg)/m = g. Earth accelerates toward you at a_Earth = F/M_Earth = (mg)/M_Earth ≈ 10^-23 m/s² - completely negligible due to Earth's enormous mass (6 × 10^24 kg). The forces are equal; the effects are not.
F_\textEarth on you = F_\textyou on Earth = mg \quad \textbut \quad a_\textyou = g \gg a_\textEarth = (mg)/(M_\textEarth) ≈ 0
Collisions and Momentum Conservation
During collisions, Newton's Third Law ensures momentum conservation. When two objects collide, the force object A exerts on B equals the force B exerts on A at every instant (but opposite direction). Using F = dp/dt (force equals rate of momentum change), we get: dp_A/dt = -dp_B/dt, which integrates to Δp_A = -Δp_B, meaning momentum lost by A equals momentum gained by B. Total momentum p_A + p_B is conserved. This is the fundamental reason momentum conservation works - it's a direct consequence of Newton's Third Law applying throughout the collision.
F_AB = -F_BA \quad \Rightarrow \quad (dp_A)/(dt) = -(dp_B)/(dt) \quad \Rightarrow \quad Δ p_A = -Δ p_B \quad \Rightarrow \quad p_\texttotal = \textconst
Swimming and Propulsion
Swimming works by Newton's Third Law: you push water backward (action), and water pushes you forward (reaction). The faster and more water you push backward, the greater your forward thrust. Flippers and fins increase the effective surface area, allowing you to push more water backward per stroke, generating greater forward thrust. Birds fly similarly - wings push air downward and backward (downwash), and air pushes wings upward and forward (lift and thrust). Propellers and jet engines work identically: accelerate fluid backward to generate forward thrust.
F_\texton fluid = -F_\textfluid on you \quad \textand \quad |F| \propto \dotmv_\textfluid \quad (\textthrust \propto \textmass flow × \textvelocity)
Tension in Ropes and Strings
When you pull a rope, tension forces obey Newton's Third Law throughout. If you pull one end with force F, that end of the rope pulls you with force F (action-reaction). For a massless rope in equilibrium, the tension is uniform throughout - every segment of rope experiences equal and opposite forces from adjacent segments. If you pull a rope attached to a wall, you exert F on the rope, the rope exerts F on the wall, and by Newton's Third Law, the wall exerts F back on the rope (which transmits it to your hand). The rope doesn't accelerate if the forces balance.
\vecF_\textyou on rope = -\vecF_\textrope on you \quad \textand \quad \vecF_\textrope on wall = -\vecF_\textwall on rope
Gun Recoil
When a gun fires, the expanding gases push the bullet forward (action) and push the gun backward (reaction) with equal force. The bullet, being much lighter, gets much greater acceleration (a = F/m). A 0.01 kg bullet fired from a 2 kg rifle: if the bullet accelerates at 100,000 m/s² forward, the rifle accelerates at (0.01/2) × 100,000 = 500 m/s² backward - 200 times less due to 200 times greater mass. Both experience the same force and same momentum change magnitude (Δp_bullet = -Δp_gun), but vastly different velocities and kinetic energies.
F_\texton bullet = F_\texton gun \quad \Rightarrow \quad m_b a_b = m_g a_g \quad \Rightarrow \quad (a_b)/(a_g) = (m_g)/(m_b) \quad (\textinverse mass ratio)
Normal Force and Support
When you stand on the floor, gravity pulls you down with force mg (Earth on you). This is NOT the action-reaction pair with the normal force! The reaction to Earth pulling you down is you pulling Earth up (also mg). Separately, you push down on the floor with force mg (if stationary), and by Newton's Third Law, the floor pushes up on you with normal force N = mg. Four forces total: (1) Earth pulls you down (mg), (2) You pull Earth up (mg) - action-reaction pair. (3) You push floor down (mg), (4) Floor pushes you up (N = mg) - different action-reaction pair.
\textPair 1: \vecF_\textEarth on you = -\vecF_\textyou on Earth = mg\hatj \quad \textPair 2: \vecF_\textyou on floor = -\vecN_\textfloor on you
Newton's Third Law Calculator Worked Examples
Worked Example
Inputs
- actionForce: 150
Result: Reaction Force: -150.00 N
Explanation
When you exert an **action force of 150 N** on an object (such as pushing a wall, pulling a rope, or firing a rocket exhaust), Newton's Third Law guarantees that the object exerts an equal and opposite **reaction force of -150 N** on you. The negative sign indicates opposite direction.
**Real-World Context:**
If you push a wall with 150 N eastward, the wall pushes you with 150 N westward. Both forces have magnitude 150 N but opposite directions. The wall doesn't move (assuming it's attached to Earth with enormous mass), so the 150 N accelerates YOU, not the wall. If your mass is 75 kg, you'll accelerate at a = F/m = 150/75 = 2 m/s² backward (away from the wall).
**Rocket Example:**
If a rocket engine expels exhaust gas backward with 150 N of force (action), the exhaust gas pushes the rocket forward with 150 N of thrust (reaction). This thrust accelerates the rocket. A 100 kg rocket with 150 N thrust accelerates at a = 150/100 = 1.5 m/s². In space (no air resistance or gravity), this continues indefinitely, producing constant acceleration.
**Key Insight:**
Action-reaction forces are equal in magnitude (150 N each), opposite in direction, act simultaneously, and act on DIFFERENT objects. They don't cancel because they're not on the same object. The motion of each object depends on the NET force on that object alone (Newton's Second Law).
Second Scenario
Inputs
- actionForce: 180
Result: Reaction Force: -150.00 N
Explanation
This scenario uses different inputs (actionForce = 180) to show how changing one variable affects the newton's third law result. Run the calculator above with these values to get the exact updated output with step-by-step work.
Common Newton's Third Law Calculator Use Cases
- Physics problem sets and labs
- Engineering design checks
- Unit and formula verification
- Newton's Third Law homework and study
- Newton's Third Law design and analysis
Newton's Third Law Calculator FAQs
If action and reaction forces are equal and opposite, why does anything move?
This is the most common misconception about Newton's Third Law! The key is that action-reaction pairs act on DIFFERENT objects, not the same object. Consider pushing a car: you push the car with force F (acting ON the car), and the car pushes you with force -F (acting ON you). These don't cancel because they act on different things. To find the car's acceleration, you only consider forces ON the car. If no other forces act on the car, net force on car = F, so a_car = F/m_car (it accelerates!). Similarly, net force on you = -F (backward), plus friction from ground forward. If you don't slip, ground friction cancels the car's push, and you remain stationary. The car moves because F acts on it unbalanced; you don't move because forces on you balance. Action-reaction pairs never cancel because they never act on the same object!
Why doesn't Earth move when I jump, given that I push Earth down with the same force Earth pulls me down?
Earth DOES move when you jump - just imperceptibly! When you jump, you push Earth downward with force F, and by Newton's Third Law, Earth pushes you upward with force F. Both accelerate: you accelerate at a_you = F/m_you, and Earth accelerates at a_Earth = F/M_Earth. For example, if you're 70 kg and push with 700 N (enough to accelerate yourself at 10 m/s² and jump), Earth accelerates at 700 N / (6 × 10^24 kg) ≈ 1.2 × 10^-22 m/s². This is utterly negligible - less than an atomic diameter per century! The forces are equal, but accelerations are inversely proportional to masses. Your mass is ~10^23 times less than Earth's, so your acceleration is ~10^23 times greater. That's why you jump and Earth appears stationary. Actually, Earth and you jump apart (center of mass doesn't move), but you move ~all of the distance!
How do rockets work in space if there's nothing to push against?
This is a brilliant question that confused people for centuries! Rockets don't need anything external to push against - they push against their own expelled fuel. Here's how: The rocket engine expels exhaust gas backward at high velocity (action: rocket pushes gas). By Newton's Third Law, the gas pushes the rocket forward with equal force (reaction: thrust). The rocket accelerates forward, the gas accelerates backward. This works perfectly in vacuum because the action-reaction forces are internal to the rocket-fuel system. Conservation of momentum: p_total = 0 initially, so p_rocket forward + p_exhaust backward = 0 always. The faster you expel exhaust (and the more mass you expel), the more momentum change, so the more thrust. The rocket equation is Δv = v_exhaust × ln(m_initial/m_final). This is why rockets carry huge fuel tanks - to have large mass ratio and achieve high Δv.
When I push a heavy box that doesn't move, where is the reaction force?
Great question! Even when the box doesn't move, Newton's Third Law still applies perfectly - you just need to consider all forces. When you push the box with force F, the box pushes you with force -F (action-reaction). Why doesn't the box move? Because OTHER forces act on it: static friction from the floor pushes the box backward with force equal to your push (up to a maximum of μ_s × N). Net force on box = F (from you) + (-F) (from friction) = 0, so a = 0 (no motion). You don't move because: the box pushes you with -F, and the ground pushes you with +F (friction forward on your feet), giving net zero force on you. So four forces total: (1) You push box (+F on box), (2) Box pushes you (-F on you) [action-reaction pair], (3) Box experiences friction from floor (-F on box), (4) You experience friction from floor (+F on you). Note: Forces 1 and 3 are NOT an action-reaction pair (both act on box), nor are 2 and 4 (both act on you). The action-reaction pairs are 1&2 and 3&4!
Does Newton's Third Law apply to gravity between Earth and Moon?
Absolutely! Gravitational forces are perfect examples of Newton's Third Law. Earth attracts the Moon with gravitational force F = GMm/r², pulling it toward Earth (preventing Moon from flying off tangent to its orbit). By Newton's Third Law, the Moon attracts Earth with exactly the same force F = GMm/r², pulling Earth toward the Moon. Both forces have identical magnitude but opposite direction. Because both are accelerated by these forces, they both orbit their common center of mass (barycenter), which lies inside Earth (about 4,700 km from Earth's center, or ~75% of Earth's radius). Earth "wobbles" as Moon orbits, but with much smaller radius because Earth is 81 times more massive (a_Earth = F/M_Earth = (1/81) × a_Moon). This wobble is detectable and used to discover exoplanets - stars wobble due to planets' gravitational pull (Third Law in action)!
Why does punching a wall hurt your hand if the forces are equal?
This reveals a subtle but important distinction: equal forces don't mean equal effects! When you punch a wall with force F, the wall punches your hand with force -F (Newton's Third Law - equal magnitude). Why does your hand hurt but not the wall? Several reasons: (1) **Material strength:** Bone and tissue are much weaker than concrete/brick, so the same force causes more damage to your hand. (2) **Deformation:** Your hand deforms more under the force, concentrating stress and causing injury. The rigid wall barely deforms. (3) **Energy absorption:** Your hand absorbs kinetic energy as it stops, dissipated as deformation, heat, and damage. (4) **Acceleration:** Your hand experiences large deceleration (a = F/m_hand). The wall, attached to Earth, has negligible acceleration (a = F/M_Earth+wall ≈ 0). Equal forces, vastly different consequences due to different masses and material properties!
How does swimming work according to Newton's Third Law?
Swimming is Newton's Third Law in action! When you swim, your hands and feet push water backward and downward (action force on water). By Newton's Third Law, the water pushes your hands and feet forward and upward (reaction force on you). This reaction force propels you forward and keeps you afloat. The magnitude of thrust depends on how much water you push and how fast: F ≈ ṁ × v (force ≈ mass flow rate × velocity). Larger hands/feet or faster strokes = more thrust. Flippers help because they increase surface area, allowing you to push more water per stroke. The water accelerates backward: momentum given to water backward equals momentum you gain forward (momentum conservation, derived from Third Law). Interestingly, you're not pushing against stationary water - you're pushing against water you're accelerating backward, which is why swimming technique matters so much!
What is the reaction force when a car accelerates forward?
This is beautifully subtle! When a car accelerates forward, many forces are involved. The key action-reaction pair is between the tires and the road: (1) The tires push the road BACKWARD with force F (action), (2) The road pushes the tires FORWARD with force F (reaction - this is friction!). This forward friction accelerates the car: a = F/m. Note: It's friction that accelerates the car, not the engine directly! The engine spins the tires, which push backward on the road; the road's reaction force (friction) on the tires accelerates the car forward. If there were no friction (ice), the tires would spin but the car wouldn't accelerate - the road couldn't push back. Meanwhile, by Newton's Third Law, the road is pushed backward by force F. This would accelerate Earth backward at a_Earth = F/M_Earth ≈ 10^-23 m/s² (negligible). It's the same reason you can walk - you push ground backward, ground pushes you forward via friction!
Is the normal force always a Newton's Third Law reaction to weight?
No! This is a very common misconception. The normal force and weight are NOT a Third Law action-reaction pair - they act on the SAME object (you), and action-reaction pairs must act on DIFFERENT objects. Here's the correct analysis: **Pair 1 (gravitational):** (1a) Earth pulls you down with force W = mg (your weight), (1b) You pull Earth up with force mg (reaction). **Pair 2 (contact):** (2a) You push floor down with force N, (2b) Floor pushes you up with normal force N (reaction). If you're stationary on flat ground, it happens that N = mg (magnitudes equal) because net force = 0 (equilibrium). But they're not action-reaction pairs! Proof: In an elevator accelerating upward at a m/s², your weight is still mg (Earth pulls you down), but normal force is N = m(g + a) > mg. If weight and normal were action-reaction, they'd always be equal (they're not). Weight is gravitational; normal is electromagnetic (contact force). Different forces entirely!