Newton's Laws of Motion/Third Law

Physical Law
Newton's third law of motion is one of three physical laws that forms the basis for classical mechanics.

Statement of Law

 * To every force there is always an equal and opposite force. That is, the forces of two bodies on each other are always equal and are directed in opposite directions.

This law is also referred to as the law of action and reaction.

Proof of third law in the special case of one-dimensional motion
It is actually possible to almost prove the third law, provided we restrict the discussion to one dimension, and provided we accept two axioms:

Axiom 1: Newtonian mechanics not only applies to particles, but to objects made up of particles in such a way that can be easily understood for a nearly infinite number of connected particles.

This is where the almost comes in: If Newtonian physics is to be useful for entire planets, the forces on these planets need to be understood without understanding all the internal forces between the many, many particles involved. For that reason, the next axiom is almost a corollary to the first axiom:

Axiom 2: When Newton's second law is applied to collections of particles, only external forces need be considered.

Without this second axiom, Newtonian physics would be so complicated that it is virtually useless; planets would go where their internal forces send them.

Consider two masses, m1 and m2 that are held at a fixed distance from each other. It is assumed that they exert "internal" forces on each other, but we do not assume that these forces are equal and opposite.

Let F12 be the "internal" force on m1 by m2 and let F21 be the "internal" force on m2 by m1. Also, let a force, Fext act only on m2. Since the objects are held at a fixed distance from each other, they both have the same velocity and the same acceleration. Let this acceleration be a, and note that the following three equations hold:

1)  Fext = (m1 + m2)a

2)  Fext + F21 = m2a

3)  F12 = m1a

Adding equations (2) and (3) and comparing with equation (1) yields Newton's third law:


 * F12 + F21 = 0

Also see

 * Newton's First Law
 * Newton's Second Law

As he put it:
 * Whenever one body exerts a force on a second body, the second body exerts a force on the first body. These forces are equal in magnitude and opposite in direction.