Newton's Third Law Calculator

Compute equal and opposite force pairs, relate masses and accelerations using m₁·a₁ = – m₂·a₂. Visualize action-reaction vectors on two interacting objects.

Object 1 (Action side)
Auto-sync
F₁ = m₁·a₁ (If "Auto-sync" is checked, it will update automatically). You can uncheck it and manually enter a custom F₁. 。
Object 2 (Reaction side)
Computed or user-provided depending on mode.
F₂ = –F₁ (action-reaction).
? Rocket & Exhaust (m₁=500kg, a₁=5 m/s² , m₂=200kg)
? Gun recoil (m₁=0.01kg, a₁=400 m/s², m₂=2kg)
⛸️ Ice skater push (m₁=60kg, a₁=0.8 m/s², m₂=50kg)
? Jumping off boat (m₁=70kg, a₁=2.5 m/s², m₂=300kg)
? Collision (m₁=1000kg, a₁=-4 m/s², m₂=1500kg)
100% local computation: All calculations run in your browser. No data is sent to any server.

Understanding Newton's Third Law: Action and Reaction

Newton's third law states: For every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object. These forces always occur in pairs and act on different bodies. Mathematically: F₁→₂ = – F₂→₁.

F₁ = –F₂   ⟹   m₁·a₁ = – m₂·a₂

The product mass × acceleration for two interacting objects are equal in magnitude and opposite in direction.

Why Use This Interactive Third Law Calculator?

  • Intuitive Visualization: See force vectors on two separate objects – crucial for understanding why forces don't cancel.
  • Relate Mass & Acceleration: Given m₁, a₁, and m₂, instantly find a₂ (or vice versa) via momentum conservation.
  • Real-World Engineering: Rocket propulsion, firearm recoil, jet engines, and even walking rely on action-reaction.
  • Supports Multiple Modes: Compute opposite force, missing acceleration, or missing mass.

Mathematical Derivation & Conservation of Momentum

From Newton's second law, F = dp/dt. For two isolated interacting objects, the total momentum is constant: p₁ + p₂ = constant. Thus dp₁/dt + dp₂/dt = 0 ⇒ F₁ + F₂ = 0 ⇒ F₂ = –F₁. For constant masses, m₁·a₁ = – m₂·a₂. Our calculator uses this relationship. When solving for a₂: a₂ = – (m₁·a₁)/m₂. Note that negative sign indicates opposite direction.

Step‑by‑Step Use Guide

  1. Select calculation mode: Opposite Force, Acceleration of object 2, or Mass of object 2.
  2. Enter known values for mass₁, acceleration₁, and optionally force₁ (如果取消自动同步可手动填写)。
  3. Provide mass₂ (and acceleration₂ if solving for mass₂).
  4. Click "Calculate & Update Diagram" – results show both forces and the acceleration of object 2, plus a visual vector diagram.

Real-World Examples of Action-Reaction

Scenario Action Force Reaction Force Key insight
Rocket thrust Exhaust gases pushed backward Rocket pushed forward Rocket moves without pushing against ground
Gun recoil Bullet accelerated forward Gun moves backward m_bullet·a_bullet = – m_gun·a_gun
Swimming Swimmer pushes water backward Water pushes swimmer forward Propulsion in fluids
Walking Foot pushes ground backward Ground pushes foot forward Friction enables motion
Case Study: Rocket Launch

A rocket of mass 500 kg ejects exhaust at a rate producing acceleration of 5 m/s² upward. The exhaust gases experience an equal downward force. Using m₁·a₁ = – m₂·a₂, if the exhaust mass flow rate corresponds to an effective mass of 200 kg, then the exhaust acceleration a₂ = – (500×5)/200 = –12.5 m/s² (downward). Our calculator instantly verifies thrust and recoil effects, crucial for aerospace engineering.

Common Misconceptions About the Third Law

  • "Action and reaction cancel each other." – No, because they act on different bodies. Only forces on the same body can cancel.
  • "If one object is much heavier, the reaction force is smaller." – Forces are always equal in magnitude; accelerations differ.
  • "Action and reaction occur sequentially." – They occur simultaneously; there is no delay.

From Third Law to Conservation Laws

Newton’s third law is the foundation of momentum conservation in isolated systems. It also implies that internal forces cannot change the total momentum of a system. This principle governs collisions, explosions, and even celestial mechanics.

Authoritative Physics Foundation – Based on Sir Isaac Newton's Principia Mathematica (1687) and peer-reviewed mechanics textbooks (Halliday, Resnick, Young & Freedman). Validated with NASA educational resources and ISO 80000-4. Regularly reviewed by the GetZenQuery Tech team. Last updated May 2026.

Frequently Asked Questions

Because they act on different objects. Cancellation requires forces to act on the same object. For example, a book on a table: Earth pulls book down (gravity), table pushes book up (normal). These are not action-reaction pairs. Action-reaction: book pulls Earth up (equal to gravity on book).

Select "Calculate Acceleration of object 2". Enter mass of bullet (m₁), acceleration of bullet (a₁), mass of gun (m₂). Click calculate to find recoil acceleration (a₂). The negative sign confirms opposite direction.

In classical mechanics, yes. In electromagnetism, forces between moving charges can be more complex, but momentum conservation still holds via field momentum. For everyday macroscopic systems, the third law is accurate.

Yes. Enter m₁ and a₁, the calculator automatically computes F₁ = m₁·a₁ (if auto-sync enabled). Then using third law, F₂ = –F₁. You can also input a custom F₁ after unchecking "auto-sync".

SI units: kilograms (kg), meters per second squared (m/s²), Newtons (N). For imperial, convert manually: 1 lb = 4.448 N, 1 slug = 14.59 kg.
References: NASA Glenn – Newton's Third Law; Halliday, Resnick, Krane "Physics" Vol 1; Encyclopedia Britannica.