What Are Newton's Three Laws of Motion?
Inertia, F=ma, and action-reaction: the three laws of motion that became the foundation of classical mechanics and modern engineering.
What Are Newton’s Three Laws of Motion?
Isaac Newton’s three laws of motion, published in 1687 in the Philosophiæ Naturalis Principia Mathematica, are the axioms of classical mechanics. From them, and from the law of universal gravitation, the whole of pre-twentieth-century physics was derived, and on them most of modern engineering — from bridges to rockets — has been built. They are the working rules by which the trajectories of spacecraft are calculated, the stresses in buildings are analysed, the orbits of planets are predicted, and the flight of baseballs is described. This page is an article in the Laws of Motion and Universal Gravitation, under the Major Discoveries. It assumes a general familiarity with the broad context, which is sketched in the section overview. For the biographical and historical context, see the Isaac Newton and the discussion of the Principia in The Principia Mathematica. For the gravitational law that Newton combined with these three, see What Is Universal Gravitation?.
The First Law: The Law of Inertia
Newton’s first law states: Every body persists in its state of being at rest, or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. In simpler language: if no force acts on a body, the body will continue doing what it is already doing. If it is at rest, it will stay at rest. If it is moving, it will keep moving in a straight line at the same speed forever. The only thing that can change a body’s state of motion is a force. This is not how the everyday world looks. A rolling ball comes to rest, a sliding block slows down, a flying arrow drops to the ground. But the everyday world is full of forces that we do not always notice: friction from the surface, air resistance, gravity pulling things toward the ground. The first law describes what would happen in the absence of these forces. It is, in a deep sense, a law about idealised conditions — but idealisations are useful precisely because they let us separate the real causes from the incidental ones. The intellectual history of the first law runs through Galileo Galilei, who articulated a version of it in his Two New Sciences (1638) using the famous thought experiment of a ball rolling on a smooth horizontal plane, and through René Descartes, who stated the principle of inertia (with some philosophical hedges) in his Principles of Philosophy (1644). Newton’s contribution was to state the law precisely, in a form that could be used mathematically, and to build a mechanics on top of it. The first law is the most philosophically radical of the three. It contradicts the everyday intuition, going back to Aristotle, that rest is the natural state of matter and motion requires a cause. In Newton’s physics, motion at constant velocity is just as natural as rest; both are states that a body keeps unless a force acts. The first law is also, in modern physics, the closest cousin to the principle of relativity, which Einstein elevate to a fundamental postulate of all physics.
The Second Law: F = ma
Newton’s second law states: The change of motion of a body is proportional to the impressed force and occurs in the direction of the straight line along which that force is impressed. “Change of motion” is what we now call momentum: the product of a body’s mass and its velocity, written p = m v. The second law says, in modern notation, that the rate of change of momentum equals the applied force: F = dp/dt. In the common case where the mass is constant, this reduces to the famous F = ma, where a is the acceleration. The second law has two crucial features. First, it is a vector equation: the direction of the acceleration is the direction of the force, and forces add as vectors. A ball thrown horizontally off a cliff accelerates downward at 9.8 m/s² because gravity pulls it down, and continues to move forward at the same horizontal speed because no horizontal force acts on it. The resulting trajectory is a parabola, as Galileo had shown. Second, the second law is the bridge between kinematics (the description of motion) and dynamics (the explanation of motion in terms of forces). Given the forces on a body, the second law tells you how it will move. Given the motion of a body, the second law tells you the net force on it. The whole of classical mechanics is the application of this single idea to particular cases. It is worth pausing on the idea of mass, which appears in the second law. Newton defined mass — what he called “quantity of matter” — in terms of a body’s volume and density. The modern conception treats mass as a measure of a body’s resistance to acceleration (its inertial mass) and, separately, as the charge that couples to the gravitational field (its gravitational mass). The numerical equality of these two kinds of mass was, for Newton, an empirical fact to be checked by experiment. In Einstein’s general relativity, it is a fundamental principle: gravity is geometry, and all bodies fall the same way because the geometry is the same for all of them.
The Third Law: Action and Reaction
Newton’s third law states: To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. In other words, forces come in pairs. If body A pushes body B with a force F, then body B pushes body A with a force −F, equal in magnitude and opposite in direction. The two forces act on different bodies, so they do not cancel; they are the two ends of a single interaction. The third law is the easiest to state and the most counterintuitive in its consequences. When a cannon fires a cannonball, the cannonball is pushed forward by the explosion; the cannon is pushed backward by an equal and opposite force, which is what causes recoil. When a person walks, the foot pushes backward on the ground, and the ground pushes forward on the foot. When a rocket flies, the rocket pushes exhaust gases backward, and the gases push the rocket forward. In all of these cases, the third law is doing the work. The third law has important limits. It holds exactly when the forces between two bodies are contact forces or, more generally, when the interaction is mediated by a field that does not store momentum (such as the static electric field between two charges). In systems involving the electromagnetic field — particularly when charges are accelerating and radiating — the third law in its simple form can fail: the field itself carries momentum, and the force on body A is balanced by an equal and opposite force on body B only when the field momentum is taken into account.
What the Three Laws Together Accomplish
The three laws are the foundation of an extraordinarily powerful mathematical theory. Given the forces acting on a set of bodies at any instant, the second law gives a set of differential equations for their accelerations. Integrate the accelerations, and you get the velocities. Integrate the velocities, and you get the positions. The first law provides the initial case: zero force means constant velocity. The third law, by ensuring that the forces between bodies come in pairs, makes the system of equations self-consistent. In principle, every mechanical problem — from the orbit of the Moon to the swing of a pendulum, from the flight of a ball to the stability of a building — can be solved by writing down Newton’s second law, applying the third law wherever two bodies interact, and integrating. In practice, the resulting equations are often too complicated to solve exactly, and a large part of classical mechanics is the growth of methods (conservation laws, Lagrangian and Hamiltonian formulations, perturbation theory) for extracting answers from them.
From Philosophy to Engineering
The most striking thing about Newton’s three laws is that they turned physics into a predictive, quantitative, useful science. Before Newton, the physics of motion was largely a matter of philosophical interpretation: what was the natural state of a body, what was the cause of its motion, what role did the medium play in transmitting force. After Newton, the physics of motion was a tool. It could be used to design clocks, predict eclipses, aim artillery, and — eventually, after the work of Lagrange, Laplace, and their successors — design bridges, calculate the orbits of space probes, and predict the weather. The laws are also, despite their age, the laws that engineers use every day. The structural engineer calculating the load on a beam uses the second law (in its static form: forces balance). The aerospace engineer calculating the thrust needed to lift a rocket uses the third law. The mechanical engineer designing a flywheel uses the first and second laws to relate angular velocity, moment of inertia, and torque. The whole built environment of the modern world, from the smallest mechanism to the largest dam, is a working application of the three laws.