Newton's laws of motion

Newton's laws of motion are three important rules that explain how things move when forces act on them. These rules help us understand why objects stay still, speed up, slow down, or change direction. They form the basis of what we call Newtonian mechanics.

The first law says that an object will keep doing what it's doing—either staying still or moving at a steady speed in a straight line—unless something pushes or pulls on it. The second law tells us that the force acting on an object equals its mass times its acceleration. This helps us figure out how much a push or pull will change an object's speed. The third law states that for every action, there is an equal and opposite reaction, meaning that when two objects push each other, they feel the same amount of force but in opposite directions.

Isaac Newton first described these laws in his book Philosophiæ Naturalis Principia Mathematica in 1687. They have been used ever since to explain the motion of many things, from falling apples to moving cars. Although we now have newer theories like quantum mechanics and relativity for extreme situations, Newton's laws still work very well for most everyday situations.

Prerequisites

Newton's laws describe how objects move when forces act on them. We often think of objects as simple points when the size doesn’t matter. For example, we can treat the Earth and the Sun as points when looking at their orbits, even though they are large.

To describe motion, we use positions and time. We can track an object’s position along a straight line using numbers. If we know where the object is at different times, we can find its average speed. By using calculus, we can also find the exact speed and acceleration at any moment. These ideas can be extended to motion in two or three dimensions using vectors, which show both speed and direction.

Laws

First law

Every object stays at rest or moves at a steady speed in a straight line unless a force acts on it. This is called the principle of inertia. It means an object's natural behavior is to keep doing what it's already doing unless something pushes or pulls it.

The modern view of this law is that no observer moving steadily in a straight line is any better or worse than any other. For example, a person standing still watching a train go by feels no motion, and a passenger on that train also feels no motion. There is no way to say which one is "really" moving and which one is standing still. There is no absolute standard for rest.

Second law

The change in an object's motion depends on the force acting on it and the object's mass. This means that a bigger force makes the object speed up more, and a bigger mass makes it harder to speed up.

In simple terms, force equals mass times acceleration. This law helps us understand how objects move when forces act on them. For example, when you push a toy car, the harder you push (more force), the faster it will go (more acceleration).

Third law

For every action, there is an equal and opposite reaction. This means if you push on a wall, the wall pushes back on you with the same force. The action and reaction always act on different objects.

This law shows that forces always come in pairs. If object A exerts a force on object B, then object B also exerts an equal force back on object A, but in the opposite direction.

Examples

Studying how objects move using Newton's laws is called Newtonian mechanics. Some important examples include:

Uniformly accelerated motion

If an object falls from rest near Earth's surface without air resistance, it will speed up at a constant rate. This is called free fall. The object will go faster the longer it falls, and the distance it travels grows quickly over time. All objects fall at the same rate no matter their mass.

Uniform circular motion

When an object moves in a perfect circle at a steady speed, it needs a force to keep it turning. This force, called centripetal force, points toward the center of the circle. Many orbits, like the Moon around Earth, can be thought of as uniform circular motion.

Harmonic motion

Some objects swing back and forth around a resting spot. This is called harmonic motion. A pendulum swinging back and forth is a good example. When the swing is small, it moves in a very regular way, repeating its path over and over.

Work and energy

Energy is a big idea that came after Newton's time, but it is very important in what we call "Newtonian" physics. There are two main types of energy: kinetic energy, which a body has because it is moving, and potential energy, which a body has because of where it is placed compared to other things. Thermal energy is the energy from heat. It is a type of kinetic energy, but it comes from the tiny movements of atoms and molecules, not the big movements of objects.

When a force pushes or pulls an object and the object moves in the same direction as the force, the force is doing work on the object. The amount of work done is the same as the change in the object's kinetic energy. If an object moves in a loop and returns to its starting point, sometimes the total work done is zero. This can happen with forces like gravity, but not with friction. When there is no friction, the object's energy changes between potential and kinetic, but the total amount of energy stays the same. When an object speeds up, it gains kinetic energy and loses potential energy.

Rigid-body motion and rotation

Main article: Rigid-body motion

A rigid body is an object that keeps its shape over time. To understand how it moves, we can think of it as having a center of mass — a point that represents where most of its mass is located. This helps us study how the object moves as a whole.

Center of mass

Main article: Center of mass

We can learn a lot about how an object moves by pretending all its mass is at one point — its center of mass. This point’s location depends on how the object’s material is spread out. If no outside force acts on the object, its center of mass will move at a steady speed in a straight line. This idea works even when objects collide.

Rotational analogues of Newton's laws

When we look at how rotating objects move, we find new ideas similar to Newton’s laws. Mass has a partner called moment of inertia, momentum has a partner called angular momentum, and force has a partner called torque. Angular momentum tells us how an object spins around a point, and torque tells us how a force makes it spin.

Multi-body gravitational system

Main articles: Two-body problem and Three-body problem

Newton’s law of gravity says every object pulls on every other object. The strength of this pull depends on how heavy the objects are and how far apart they are. When we try to figure out how three or more objects move under gravity, it becomes very complex and usually needs computers to solve approximately.

Chaos and unpredictability

Nonlinear dynamics

Main article: Chaos theory

Newton's laws of motion show that some physical systems can behave in surprising ways. Even small changes in how things start can lead to very different results later on. This is called sensitive dependence on initial conditions. Examples include systems with three moving objects, double pendulums, and certain bouncing balls.

Newton's laws also help us understand fluids, like water or air, by thinking of them as many tiny pieces pushing on each other. This leads to important equations that describe how fluids move and change over time.

Relation to other formulations of classical physics

Classical mechanics can be described in many different ways besides Newton’s laws. While they all explain the same physical ideas, each method offers unique ways to solve problems and understand motion. For example, Lagrangian mechanics helps connect patterns in nature to rules about what stays the same, and it’s useful for objects moving along curves or surfaces.

Hamiltonian mechanics is another approach, often used in studying energy and its changes. It looks at how positions and motions evolve over time using special math tools. Both Lagrangian and Hamiltonian methods can recreate Newton’s laws, showing that these laws are part of a bigger picture in understanding how things move.

Relation to other physical theories

Thermodynamics and statistical physics

In statistical physics, Newton's laws of motion are used to study the behavior of very large numbers of tiny particles, like atoms in a gas. These laws help explain how gases push on the walls of their containers, which we feel as pressure.

The Langevin equation is a special version of Newton's second law. It describes how very small objects move when they are hit randomly by even smaller particles, like in the wiggling motion of tiny particles in a liquid or gas, known as Brownian motion.

Electromagnetism

Newton's laws also work for electricity and magnetism, though there are some extra details to consider.

Coulomb's law, which describes the force between electric charges, looks very similar to Newton's law of gravity. It says that the force between two charges depends on how much charge they have and how far apart they are, and it acts in a straight line between them. This matches Newton's third law, which says that for every action, there is an equal and opposite reaction.

Electromagnetism uses fields to describe forces on charges. The Lorentz force law tells us how a charge moves in electric and magnetic fields. In a magnetic field, a moving charge experiences a force that makes it turn, which can result in circular or spiral motion.

Sometimes, groups of charges don't perfectly follow Newton's third law because the electromagnetic field itself can carry momentum. This is explained by the Poynting vector, which describes how momentum is stored in the field.

There is also a difference between electromagnetism and Newton's first law. Maxwell's theory says that light travels at a constant speed in empty space, which seems to prefer one viewpoint over others. This is resolved in the theory of special relativity, which changes our understanding of space and time.

Special relativity

Further information: Relativistic mechanics and Acceleration (special relativity)

In special relativity, Newton's first law still holds, but the idea of mass changes. Newton's second law also works if we change how we think about momentum. However, Newton's third law needs to be updated because, in special relativity, what happens at the same time for one observer might not for another.

Newtonian mechanics works well when things move much slower than the speed of light.

General relativity

General relativity is a new theory of gravity that goes beyond Newton's ideas. In this theory, gravity is not a force pulling objects, but instead, it is the shape of space and time that makes objects move the way they do. A famous phrase by physicist John Archibald Wheeler sums it up: "Spacetime tells matter how to move; matter tells spacetime how to curve." General relativity works better than Newton's theory when gravity is very strong or when objects move very fast.

Newton's theory of gravity is still a good approximation when gravity is weak and things move slowly compared to the speed of light.

Quantum mechanics

Quantum mechanics is a theory that helps us understand very small things, like atoms and particles. It works very differently from classical physics. Instead of knowing exactly where and how fast something is moving, we can only talk about the chances of finding it in different places or moving at different speeds.

The Ehrenfest theorem links quantum physics to Newton's second law in a general way, but because quantum physics is so different, it is hard to make exact comparisons. In quantum physics, positions and speeds are described using special math tools, and the results of measurements are based on probabilities.

History

The ideas behind Newton's laws of motion go back a long way. Early thinkers like Aristotle thought about how things move, but they had some wrong ideas. For example, they thought that objects needed a constant push to keep moving. Later, John Philoponus suggested that objects keep moving because they have a kind of inner push, called "impetus."

Galileo Galilei made big steps forward. Through experiments, he found that objects keep moving unless something stops them. This idea is called inertia. Later, René Descartes and others helped shape this idea into what we now know as Newton's first law.

Newton himself brought all these ideas together. He showed that forces can act from a distance, like gravity, without touching anything. He also explained how forces change the motion of objects, which became known as Newton's second law. Finally, he described how forces between objects always come in pairs, known as his third law.

After Newton, scientists kept building on his work. They developed new math and ideas, like energy and vectors, to better understand motion. But Newton's three laws remain the foundation of how we study movement today.