8.1 Linear Momentum and Force

Learning Objectives

By the end of this section, you will be able to:

• Define linear momentum.
• Explain the relationship between momentum and force.
• State Newton’s second law of motion in terms of momentum.
• Calculate momentum given mass and velocity.

Linear Momentum

The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as

$$\mathbf{p}=m\mathbf{\text{v}}.$$

Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum $\mathbf{p}$ is a vector having the same direction as the velocity $\mathbf{\text{v}}$. The SI unit for momentum is $\text{kg}·\text{m/s}$.

Momentum and Newton’s Second Law

The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is

$${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta }\mathbf{p}}{\mathrm{\Delta }t},$$

where ${\mathbf{F}}_{\text{net}}$ is the net external force, $\mathrm{\Delta }\mathbf{p}$ is the change in momentum, and $\mathrm{\Delta }t$ is the change in time.

This statement of Newton’s second law of motion includes the more familiar ${\mathbf{F}}_{\text{net}}\text{=}m\mathbf{a}$ as a special case. We can derive this form as follows. First, note that the change in momentum $\mathrm{\Delta }\mathbf{p}$ is given by

$$\mathrm{\Delta }\mathbf{p}=\mathrm{\Delta }\left(m\mathbf{v}\right).$$

If the mass of the system is constant, then

$$\mathrm{\Delta }\left(m\mathbf{v}\right)=m\mathrm{\Delta }\mathbf{v}.$$

So that for constant mass, Newton’s second law of motion becomes

$${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta }\mathbf{p}}{\mathrm{\Delta }t}=\frac{m\mathrm{\Delta }\mathbf{v}}{\mathrm{\Delta }t}\text{.}$$

Because $\frac{\mathrm{\Delta }\mathbf{v}}{\mathrm{\Delta }t}=\mathbf{a}$, we get the familiar equation

$${\mathbf{F}}_{\text{net}}\text{=}m\mathbf{a}$$

when the mass of the system is constant.

Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.

Test Prep for AP Courses

Section Summary

• Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
• In symbols, linear momentum $\mathbf{p}$ is defined to be

  $$\mathbf{p}=m\mathbf{v},$$

  where $m$ is the mass of the system and $\mathbf{v}$ is its velocity.
• The SI unit for momentum is $\text{kg}·\text{m/s}$.
• Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
• In symbols, Newton’s second law of motion is defined to be

  $${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta }\mathbf{p}}{\mathrm{\Delta }t}\text{,}$$

  ${\mathbf{F}}_{\text{net}}$ is the net external force, $\mathrm{\Delta }\mathbf{p}$ is the change in momentum, and $\mathrm{\Delta }t$ is the change time.

Conceptual Questions

Problems & Exercises