# 2.12 Linear momentum and force

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• 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 $\text{kg}·\text{m/s}$ .

## Linear momentum

Linear momentum is defined as the product of a system’s mass multiplied by its velocity:

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

## Calculating momentum: a football player and a football

(a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.

Strategy

No information is given regarding direction, and so we can calculate only the magnitude of the momentum, $p$ . (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes

$p=\text{mv}$

when only magnitudes are considered.

Solution for (a)

To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation.

${p}_{\text{player}}=\left(\text{110 kg}\right)\left(8\text{.}\text{00 m/s}\right)=\text{880 kg}·\text{m/s}$

Solution for (b)

To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation.

${p}_{\text{ball}}=\left(\text{0.410 kg}\right)\left(\text{25.0 m/s}\right)=\text{10.3 kg}·\text{m/s}$

The ratio of the player’s momentum to that of the ball is

$\frac{{p}_{\text{player}}}{{p}_{\text{ball}}}=\frac{\text{880}}{\text{10}\text{.}3}=\text{85}\text{.}9.$

Discussion

Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.

## 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

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

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

## Newton’s 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.

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

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