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# Linear Momentum and Force

## Linear Momentum and

Force
Bởi:
OpenStaxCollege

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

p = mv.

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 p is a vector having the same direction as the velocity v. The SI unit for
momentum is kg·m/s .

Linear Momentum

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

velocity:

p = mv.
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

1/6
Linear Momentum and Force

example, the magnitude of momentum can be calculated directly from the definition of
momentum given in the equation, which becomes

p = mv

## Solution for (a)

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

## Solution for (b)

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

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

pplayer 880
pball = 10.3 = 85.9.

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

Δp
Fnet = Δt ,

2/6
Linear Momentum and Force

where Fnet is the net external force, Δp is the change in momentum, and Δt 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.

Δp
Fnet = Δt
Making Connections: Force and Momentum

Force and momentum are intimately related. Force acting over time can change
momentum, and Newton’s second law of motion, can be stated in its most broadly
applicable form in terms of momentum. Momentum continues to be a key concept in the
study of atomic and subatomic particles in quantum mechanics.

This statement of Newton’s second law of motion includes the more familiar Fnet=ma
as a special case. We can derive this form as follows. First, note that the change in
momentum Δp is given by

Δp = Δ(mv).

Δ(mv) = mΔv.

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

Δp mΔv
Fnet = Δt = Δt .

Δv
Because Δt = a, we get the familiar equation

Fnet=ma

## 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.

## Calculating Force: Venus Williams’ Racquet

3/6
Linear Momentum and Force

During the 2007 French Open, Venus Williams hit the fastest recorded serve in a
premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average
force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that
the ball’s speed just after impact is 58 m/s, that the initial horizontal component of
the velocity before impact is negligible, and that the ball remained in contact with the
racquet for 5.0 ms (milliseconds)?

Strategy

This problem involves only one dimension because the ball starts from having no
horizontal velocity component before impact. Newton’s second law stated in terms of
momentum is then written as

Δp
Fnet = Δt .

## Δp = mΔv = m(vf − vi).

In this example, the velocity just after impact and the change in time are given; thus,
Δp
once Δp is calculated, Fnet = Δt can be used to find the force.

Solution

To determine the change in momentum, substitute the values for the initial and final
velocities into the equation above.

Δp = m(vf – vi)
= (0.057 kg)(58 m/s – 0 m/s)
= 3.306 kg · m/s ≈ 3.3 kg · m/s

Δp
Now the magnitude of the net external force can determined by using Fnet = Δt :

Δp 3.306 kg⋅m/s
Fnet = Δt =
5.0 × 10 − 3 s

= 661 N≈660 N,

where we have retained only two significant figures in the final step.

Discussion

4/6
Linear Momentum and Force

This quantity was the average force exerted by Venus Williams’ racquet on the tennis
ball during its brief impact (note that the ball also experienced the 0.56-N force of
gravity, but that force was not due to the racquet). This problem could also be solved by
first finding the acceleration and then using Fnet = ma, but one additional step would be
required compared with the strategy used in this example.

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 p is defined to be
p = mv,
where m is the mass of the system and v is its velocity.
• The SI unit for momentum is kg·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
Δp
Fnet = Δt ,
Fnet is the net external force, Δp is the change in momentum, and Δt is the
change time.

Conceptual Questions

An object that has a small mass and an object that has a large mass have the same
momentum. Which object has the largest kinetic energy?

An object that has a small mass and an object that has a large mass have the same kinetic
energy. Which mass has the largest momentum?

Professional Application

Football coaches advise players to block, hit, and tackle with their feet on the ground
rather than by leaping through the air. Using the concepts of momentum, work, and
energy, explain how a football player can be more effective with his feet on the ground.

How can a small force impart the same momentum to an object as a large force?

## (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of

7.50 m/s. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg

5/6
Linear Momentum and Force

tranquilizer dart fired at a speed of 600 m/s. (c) What is the momentum of the 90.0-kg
hunter running at 7.40 m/s after missing the elephant?

(b) 625 to 1

## (c) 6.66 × 102 kg⋅m/s

(a) What is the mass of a large ship that has a momentum of 1.60 × 109 kg·m/s , when
the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the
momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s.

(a) At what speed would a 2.00 × 104-kg airplane have to fly to have a momentum of
1.60 × 109 kg·m/s (the same as the ship’s momentum in the problem above)? (b) What
is the plane’s momentum when it is taking off at a speed of 60.0 m/s? (c) If the ship is
an aircraft carrier that launches these airplanes with a catapult, discuss the implications
of your answer to (b) as it relates to recoil effects of the catapult on the ship.

## (b) 1.20 × 106 kg·m/s

(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the
ship, the ship will not recoil very much. The recoil would be − 0.0100 m/s, which is
probably not noticeable.

(a) What is the momentum of a garbage truck that is 1.20 × 104 kg and is moving at
10.0 m/s? (b) At what speed would an 8.00-kg trash can have the same momentum as
the truck?

A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a
track. Compute the time required for a force of 1500 N to bring the car to rest.

54 s

The mass of Earth is 5.972 × 1024 kg and its orbital radius is an average of
1.496 × 1011 m. Calculate its linear momentum.

6/6
The text passage is a section from a physics textbook that covers the concepts of linear momentum and force. It includes definitions, formulas, and several example problems with solutions.
multiple_questions_with_answers

the passage contains multiple mathematical questions related to calculating momentum, force, and other physics concepts, and provides answers to these questions.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Vous êtes sur la page 1sur 6

# Linear Momentum and Force

## Linear Momentum and

Force
Bởi:
OpenStaxCollege

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

p = mv.

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 p is a vector having the same direction as the velocity v. The SI unit for
momentum is kg·m/s .

Linear Momentum

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

velocity:

p = mv.
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

1/6
Linear Momentum and Force

example, the magnitude of momentum can be calculated directly from the definition of
momentum given in the equation, which becomes

p = mv

## Solution for (a)

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

## Solution for (b)

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

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

pplayer 880
pball = 10.3 = 85.9.

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

Δp
Fnet = Δt ,

2/6
Linear Momentum and Force

where Fnet is the net external force, Δp is the change in momentum, and Δt 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.

Δp
Fnet = Δt
Making Connections: Force and Momentum

Force and momentum are intimately related. Force acting over time can change
momentum, and Newton’s second law of motion, can be stated in its most broadly
applicable form in terms of momentum. Momentum continues to be a key concept in the
study of atomic and subatomic particles in quantum mechanics.

This statement of Newton’s second law of motion includes the more familiar Fnet=ma
as a special case. We can derive this form as follows. First, note that the change in
momentum Δp is given by

Δp = Δ(mv).

Δ(mv) = mΔv.

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

Δp mΔv
Fnet = Δt = Δt .

Δv
Because Δt = a, we get the familiar equation

Fnet=ma

## 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.

## Calculating Force: Venus Williams’ Racquet

3/6
Linear Momentum and Force

During the 2007 French Open, Venus Williams hit the fastest recorded serve in a
premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average
force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that
the ball’s speed just after impact is 58 m/s, that the initial horizontal component of
the velocity before impact is negligible, and that the ball remained in contact with the
racquet for 5.0 ms (milliseconds)?

Strategy

This problem involves only one dimension because the ball starts from having no
horizontal velocity component before impact. Newton’s second law stated in terms of
momentum is then written as

Δp
Fnet = Δt .

## Δp = mΔv = m(vf − vi).

In this example, the velocity just after impact and the change in time are given; thus,
Δp
once Δp is calculated, Fnet = Δt can be used to find the force.

Solution

To determine the change in momentum, substitute the values for the initial and final
velocities into the equation above.

Δp = m(vf – vi)
= (0.057 kg)(58 m/s – 0 m/s)
= 3.306 kg · m/s ≈ 3.3 kg · m/s

Δp
Now the magnitude of the net external force can determined by using Fnet = Δt :

Δp 3.306 kg⋅m/s
Fnet = Δt =
5.0 × 10 − 3 s

= 661 N≈660 N,

where we have retained only two significant figures in the final step.

Discussion

4/6
Linear Momentum and Force

This quantity was the average force exerted by Venus Williams’ racquet on the tennis
ball during its brief impact (note that the ball also experienced the 0.56-N force of
gravity, but that force was not due to the racquet). This problem could also be solved by
first finding the acceleration and then using Fnet = ma, but one additional step would be
required compared with the strategy used in this example.

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 p is defined to be
p = mv,
where m is the mass of the system and v is its velocity.
• The SI unit for momentum is kg·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
Δp
Fnet = Δt ,
Fnet is the net external force, Δp is the change in momentum, and Δt is the
change time.

Conceptual Questions

An object that has a small mass and an object that has a large mass have the same
momentum. Which object has the largest kinetic energy?

An object that has a small mass and an object that has a large mass have the same kinetic
energy. Which mass has the largest momentum?

Professional Application

Football coaches advise players to block, hit, and tackle with their feet on the ground
rather than by leaping through the air. Using the concepts of momentum, work, and
energy, explain how a football player can be more effective with his feet on the ground.

How can a small force impart the same momentum to an object as a large force?

## (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of

7.50 m/s. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg

5/6
Linear Momentum and Force

tranquilizer dart fired at a speed of 600 m/s. (c) What is the momentum of the 90.0-kg
hunter running at 7.40 m/s after missing the elephant?

(b) 625 to 1

## (c) 6.66 × 102 kg⋅m/s

(a) What is the mass of a large ship that has a momentum of 1.60 × 109 kg·m/s , when
the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the
momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s.

(a) At what speed would a 2.00 × 104-kg airplane have to fly to have a momentum of
1.60 × 109 kg·m/s (the same as the ship’s momentum in the problem above)? (b) What
is the plane’s momentum when it is taking off at a speed of 60.0 m/s? (c) If the ship is
an aircraft carrier that launches these airplanes with a catapult, discuss the implications
of your answer to (b) as it relates to recoil effects of the catapult on the ship.

## (b) 1.20 × 106 kg·m/s

(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the
ship, the ship will not recoil very much. The recoil would be − 0.0100 m/s, which is
probably not noticeable.

(a) What is the momentum of a garbage truck that is 1.20 × 104 kg and is moving at
10.0 m/s? (b) At what speed would an 8.00-kg trash can have the same momentum as
the truck?

A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a
track. Compute the time required for a force of 1500 N to bring the car to rest.

54 s

The mass of Earth is 5.972 × 1024 kg and its orbital radius is an average of
1.496 × 1011 m. Calculate its linear momentum.

6/6
user
The text passage is a section from a physics textbook that covers the concepts of linear momentum and force. It includes definitions, formulas, and several example problems with solutions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

The passage contains multiple mathematical questions related to calculating momentum, force, and other physics concepts, and provides answers to these questions.