# Newton's Laws of Motion

Isaac Newton's 3 laws of motion form the basis for classical mechanics. They are:

1) An object in motion will remain in motion in a straight line unless acted upon by an external force. An object at rest will remain at rest unless acted upon by an external force. (Or, alternatively, an object's velocity remains constant unless the object is acted upon by an external force.)

2) The rate of change of an object's momentum is equal to the net force acting on it ($\vec F = d{\vec p}/dt$, sometimes written as $\vec F = m \times \vec a$ when mass can be assumed to be constant).

3) For every action there is an equal and opposite reaction; or, more precisely, the total momentum of any isolated system is always constant.

## Explanation

The first law defines an inertial frame of reference as one which is acted upon by no outside forces. In general, inertial frames are far easier to understand conceptually and deal with mathematically than accelerated frames.

The second law relates force and momentum. Mathematically, $\vec F = d{\vec p}/dt = d(m \times \vec v)/dt = m \times d{\vec v}/dt + \vec v \times dm/dt$. Usually dm / dt = 0, so the law is simplified to $\vec F = m \times d{\vec v}/dt = m \times \vec a$, or mass times acceleration. A notable exception is rocket motion, where dm / dt is not 0, and so $\vec F = m \times \vec a$ does not apply. Note that the quantities F, p, v, and a are all vector quantities--that is, they have an associated direction as well as a magnitude. In general, the second law gives a way to predict the motion of an object by summing all the forces acting on that object.

The third law states that momentum is always conserved. If one object imparts a momentum p0 on another, the first object's momentum will change by -p0. This can be viewed as a consequence of Noether's Theorem; the associated symmetry is that the laws of physics do not change under spatial translations (that is, the laws of physics are the same everywhere).