Newton's Laws of Motion

Newton's Laws of Motion are three fundamental principles that form the foundation of classical mechanics. Formulated by Sir Isaac Newton in his landmark 1687 work Philosophiæ Naturalis Principia Mathematica, these laws describe how objects move under the influence of forces. They remain essential tools in physics and engineering, governing everything from a football's trajectory to the orbit of satellites. A force is a push or pull that acts on an object, measured in newtons (N). When multiple forces act on an object simultaneously, we consider the net force — the vector sum of all individual forces. When the net force is zero, the object is said to be in equilibrium: either stationary or moving at constant velocity. Understanding equilibrium is critical for analysing structures, vehicles, and any system where forces interact. Newton's three laws connect force, mass, and acceleration in precise mathematical relationships. The First Law addresses inertia and equilibrium, the Second Law quantifies how a net force produces acceleration, and the Third Law establishes that forces always occur in equal and opposite pairs. Together, these laws allow us to predict and explain the motion of virtually any object in everyday conditions.

Key takeaways

Newton's First Law — Law of Inertia
Newton's Second Law — Law of Acceleration
Newton's Third Law — Law of Action and Reaction
Equilibrium and Free-Body Diagrams

Newton's First Law — Law of Inertia

An object remains at rest, or continues moving in a straight line at constant speed, unless acted upon by a resultant (net) external force. This property of resisting changes to motion is called inertia. Mathematically, if the net force $\sum \mathbf{F} = \mathbf{0}$ , then acceleration $\mathbf{a} = \mathbf{0}$ , so velocity is constant. Example: a hockey puck sliding on frictionless ice maintains its velocity indefinitely because no net force acts on it.

Newton's Second Law — Law of Acceleration

The net force acting on an object equals the product of its mass and acceleration: $\mathbf{F}_{\text{net}} = m\mathbf{a}$ . This is a vector equation — force and acceleration share the same direction. A larger mass requires a greater force to achieve the same acceleration. For example, pushing a 2 kg trolley with 10 N produces an acceleration of 5 m/s², while the same force on a 5 kg trolley produces only 2 m/s². In component form on a straight line: $F_{\text{net}} = ma$ .

Newton's Third Law — Law of Action and Reaction

For every action force, there is an equal and opposite reaction force. If object A exerts force $\mathbf{F}_{AB}$ on object B, then object B exerts force $\mathbf{F}_{BA} = -\mathbf{F}_{AB}$ on object A. Crucially, these forces act on different objects, so they do not cancel each other. Example: when you push down on the ground while jumping, the ground pushes up on you with equal magnitude — this upward reaction force accelerates you upward.

Equilibrium and Free-Body Diagrams

A free-body diagram (FBD) isolates a single object and shows all forces acting on it as arrows. For static equilibrium in two dimensions, both conditions must hold: $\sum F_x = 0$ and $\sum F_y = 0$ . For example, a book resting on a table experiences its weight $W = mg$ downward and a normal reaction force $N$ upward. Since it is stationary, $N = mg$ . FBDs are the essential first step in applying Newton's Second Law to any problem.

The Key Equation

F_{\text{net}} = ma

F_{\text{net}} — Net Force (Newton (N = kg·m/s²)): The vector sum of all forces acting on the object.
m — Mass (kilogram (kg)): The quantity of matter in the object; a measure of its inertia.
a — Acceleration (metres per second squared (m/s²)): The rate of change of velocity, in the direction of the net force.

Worked Example

Problem. A 4 kg box sits on a smooth (frictionless) horizontal surface. Two horizontal forces act on it: a 20 N push to the right and a 8 N push to the left. (a) Find the net force. (b) Calculate the acceleration of the box. (c) If the box starts from rest, find its velocity after 3 seconds.

F_{\text{net}} = 12\text{ N},\quad a = 3\text{ m/s}^2,\quad v = 9\text{ m/s}

Answer. Net force = 12 N to the right; acceleration = 3 m/s² to the right; velocity after 3 s = 9 m/s to the right.

Figure

Diagram

Origins · 1 minSir Isaac Newton and how this was discovered

Sir Isaac Newton

1643–1727 · English

Newton formulated the three laws of motion and the law of universal gravitation, published in Principia Mathematica (1687). He also co-invented calculus, which provided the mathematical language needed to express his laws precisely. His work unified terrestrial and celestial mechanics, explaining both falling apples and planetary orbits with the same principles. Newton reportedly said he felt like 'a boy playing on the sea-shore' finding smooth pebbles while 'the great ocean of truth lay all undiscovered' before him. He also spent more years studying alchemy and biblical chronology than mathematics and physics.

Kinematics: Equations of Motion — Newton's Second Law gives acceleration; kinematic equations (v = u + at, s = ut + ½at²) use that acceleration to find displacement and velocity, linking dynamics and kinematics.
Linear Momentum and Impulse — Momentum p = mv is directly related to Newton's Second Law in its general form F = dp/dt, and Newton's Third Law underpins the law of conservation of momentum.
Friction and Inclined Planes — Friction is a force encountered when applying Newton's Laws to real surfaces; resolving forces on inclined planes is a classic application of equilibrium conditions.
Work, Energy, and Power — Work is done when a force causes displacement (W = Fd cosθ). The work-energy theorem links Newton's Second Law to kinetic energy, extending dynamics into energy methods.