Newton's laws of motion

In everyday language a force is a push or a pull. Physics needs something we can measure. The MIT notes define force operationally, through the acceleration it produces on a chosen standard body of mass $m_s$:

A force has a size and a direction, the direction of the acceleration it causes. Experiments then show something that is not obvious: when two forces act together, the accelerations add as vectors, so the forces add as vectors too. That is why we can find the net force on a body by vector addition.

Apply the same force to the standard body and to an unknown body. The unknown body's inertial mass follows from the ratio of the accelerations:

A body that accelerates half as much under the same push has twice the mass. Mass is a measure of inertia, of how hard it is to change a body's motion.

With no net force acting on it, a body stays at rest or keeps moving in a straight line at constant speed.

Before Galileo, it seemed natural that moving things need a push to keep going, because on Earth friction always slows them down. The first law says the opposite. Motion at constant velocity needs no cause; only changes in velocity do. Any change in speed or direction means a net force is acting.

The first law also tells us which reference frames to use. A frame in which it holds is called an inertial frame. Sitting in a train moving smoothly at constant velocity, no experiment inside the carriage can tell you whether you are moving or the station is. Both frames are inertial. A train that is braking or turning is not: objects in it start to move with no force pushing them.

MIT's segments 4.3 and 4.4 make this precise. If one frame moves at constant velocity relative to another, both measure the same acceleration for any object, so $\vec F = m\vec a$ holds equally in both: every frame moving at constant velocity relative to an inertial frame is itself inertial. If a frame accelerates with $\vec A$, objects in it seem to feel an extra fictitious force $-m\vec A$, the push back into your seat when a car speeds up. No real body exerts that force; it only appears because the frame is accelerating.

Newton described motion through the momentum $\vec p = m\vec v$. A force acting for a short time $\Delta t$ changes the momentum by $\Delta\vec p = \vec F\,\Delta t$, the impulse. In the limit of a very short time:

Here $\vec F$ is the total force, the vector sum of every force on the body. This equation is more than a definition, because each kind of force (gravity, springs, electric forces) has its own force law found by experiment. Put a force law into $\vec F = m\vec a$ and you can predict motion.

Here $\vec F_{1,2}$ means the force on body 2 due to body 1. Every force is one half of an interaction: if you push on a wall, the wall pushes back on you just as hard. The two forces act on different bodies, so they never cancel each other out in one body's equation of motion.

Push a crate, add friction, and watch the net force, the acceleration and the speed change.

The MIT notes stress that Newtonian mechanics is a model with limits. At speeds approaching the speed of light it is replaced by special relativity. For electrons in atoms it is replaced by quantum mechanics. And for the structure of the universe on the largest scales, it is replaced by general relativity. Those are exactly the subjects of later years on this site.

• A force is defined by the acceleration it produces; forces add as vectors.
• Mass measures inertia: under the same force, $m_\text{in}/m_s = a_s/a_\text{in}$.
• First law: no net force means constant velocity (and defines inertial frames).
• Second law: $\vec F = d\vec p/dt = m\vec a$, with $\vec F$ the total force.
• Third law: $\vec F_{1,2} = -\vec F_{2,1}$; the pair acts on two different bodies.