Newton’s laws of motion are three principles of classical mechanics describing how bodies move and how interactions change their motion. Formulated by Isaac Newton and published in his Philosophiæ Naturalis Principia Mathematica in 1687, they connect force, mass, and changes in motion. Together with laws specifying particular forces, they provide a framework for calculating the motion of objects, from everyday machinery to celestial bodies. Their ordinary formulation applies in inertial reference frames and is an approximation outside the classical regime. (fitzmuseum.cam.ac.uk)
Historical setting
Newton presented the laws near the beginning of the Principia, before developing their mathematical consequences. The work combined principles of motion with a quantitative account of gravitation, allowing terrestrial and celestial phenomena to be treated within one framework. Its publication became a major event in the Scientific Revolution. The motion laws and the gravitational law are distinct: the former describe the response to forces, whereas the latter specifies a particular interaction. (wwwe.lib.cam.ac.uk)
First law: inertia
The first law states that a body remains at rest or moves with constant velocity in a straight line unless acted on by a nonzero net external force. Constant velocity means unchanged speed and direction; rest is the special case of zero velocity. Thus, force is not required to sustain uniform straight-line motion. Forces are required to change it. This persistence of motion is called inertia. (openstax.org)
The law also characterizes an inertial reference frame, in which a force-free body has constant velocity. A nonrotating frame moving at constant velocity relative to an inertial frame is also inertial. An accelerating or rotating frame is generally not. The first law therefore establishes the reference-frame conditions under which the ordinary Newtonian equations apply. (openstax.org)
Zero net force does not mean that no forces act. A stationary car, for example, experiences downward gravitational force and upward support from the pavement. These forces balance. Likewise, a car traveling at constant velocity may have a forward driving force balanced by resistance. Both situations have zero acceleration. (openstax.org)
Second law: force and acceleration
In modern notation, the second law relates net external force to the rate of change of momentum:
[ \mathbf F_{\mathrm{net}}=\frac{d\mathbf p}{dt}. ]
For a body of constant mass, Newtonian momentum is (\mathbf p=m\mathbf v), giving
[ \sum_i\mathbf F_i=m\mathbf a. ]
Here (\mathbf a=d\mathbf v/dt) is acceleration, and the sum includes every force acting on the body. The time derivative expresses an instantaneous rate of change. (openstax.org)
This is a vector equation: acceleration points in the direction of the net force, not necessarily in the direction of motion. For a given mass, doubling the net force doubles the acceleration; for a given force, doubling the mass halves it. A force perpendicular to velocity can change the direction of motion without changing speed. In the International System of Units, force is measured in newtons, with (1,\mathrm N=1,\mathrm{kg,m,s^{-2}}). (openstax.org)
Application requires defining the body or system being analyzed. A free-body diagram represents the external forces acting on that selected body. Forces it exerts on other objects do not belong in its own force sum. Resolving forces into coordinate components gives separate equations, such as (\sum F_x=ma_x), which can be solved together. (openstax.org)
Third law: reciprocal forces
The third law states that if body A exerts a force on body B, body B simultaneously exerts an equal-magnitude force in the opposite direction on A:
[ \mathbf F_{A\to B}=-\mathbf F_{B\to A}. ]
These are often called an action–reaction pair. The names do not imply a time delay or that one force is more fundamental than the other. Both forces belong to the same interaction but act on different bodies. (openstax.org)
Consequently, the pair does not cancel in the force equation for either body separately. A swimmer pushing against a pool wall accelerates because the wall pushes back on the swimmer. Similarly, a rocket pushes exhaust backward and receives a forward force; it does not need surrounding air to push against. Equal interaction forces need not produce equal accelerations, because the interacting bodies may have different masses. (openstax.org)
Conservation laws and related results
For a system of particles obeying the third law, internal force pairs cancel when all particle equations are added. The total momentum (\mathbf P) therefore satisfies
[ \frac{d\mathbf P}{dt}=\mathbf F_{\mathrm{external}}. ]
If the net external force is zero, total momentum remains constant. This is conservation of momentum, applicable to collisions and recoil even when individual particles accelerate strongly during their interaction. (openstax.org)
A related consequence is the work–energy theorem. For a constant-mass particle, integrating net force along its displacement gives the net work, which equals the change in kinetic energy:
[ W_{\mathrm{net}}=\Delta K,\qquad K=\tfrac12 mv^2. ]
This relation can determine changes in speed without explicitly solving the complete trajectory. Momentum and energy methods thus complement direct force-and-acceleration calculations. (openstax.org)
Domain of validity
Newtonian mechanics describes a wide range of macroscopic motion when speeds are much smaller than the speed of light. Near that speed, special relativity changes the relationships among momentum, velocity, and energy, so the simple equation (\mathbf F=m\mathbf a) is no longer generally valid. Atomic-scale phenomena often require quantum mechanics rather than definite classical trajectories. These limitations do not remove the usefulness of Newton’s laws within the conditions where their predictions accurately approximate observations. (openstax.org)