Newton's Second Law: the Momentum Principle

This page describes Newton's second law, also known as the momentum principle, which relates net force to the change in linear momentum. This principle is used to predict the effects of forces on the motion of objects.

The Main Idea

Newton's second law, also known as the momentum principle, states that [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}_{system}}{dt} }[/math] where [math]\displaystyle{ \vec{p} }[/math] is the linear momentum of the system and [math]\displaystyle{ \vec{F}_{net} }[/math] is the net external force acting on the system from its surroundings. Often, the system in question consists of a single particle whose motion we want to predict. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

The momentum principle has no derivation, as it is considered the definition of force.

A Mathematical Model

The momentum principle states that [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]. Recall that [math]\displaystyle{ \vec{p} = m\vec{v} }[/math]. Therefore, by product rule, [math]\displaystyle{ \frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt} }[/math]. Under the assumption that mass is constant, the second term becomes 0, and the momentum principle becomes [math]\displaystyle{ \vec{F}_{net} = m\frac{d\vec{v}}{dt} }[/math], or [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math], which may be a more familiar form of Newton's second law. The form [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] is preferred for several reasons:

1. When a particle accumulates mass that was initially at rest, such as a snowball rolling downhill, the term [math]\displaystyle{ \vec{v}\frac{dm}{dt} }[/math] is not 0, and [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] is no longer accurate, while [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] is;
2. As a particle approaches the speed of light, [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] is no longer accurate, while [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] is, as long as relativistic momentum is used; and
3. [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] has a more direct rotational analogue (that is, it will be easier to accurately learn rotational physics if you learn linear physics using this form.)

The momentum principle is responsible for the relationship between impulse and momentum.

A Computational Model

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