chapter 9 skill building

MasteringPhysics: Assignment Print View

Conservation of Momentum in Inelastic Collisions

Learning Goal: To understand the vector nature of momentum in the case in which two objects collide and stick together.

In this problem we will consider a collision of two moving objects such that after the collision, the objects stick together and travel off as a single unit. The collision is therefore completely inelastic.

You have probably learned that "momentum is conserved" in an inelastic collision. But how does this fact help you to solve collision problems? The following questions should help you to clarify the meaning and implications of the statement "momentum is conserved."

Part A

What physical quantities are conserved in this collision?

ANSWER: the magnitude of the momentum only

the net momentum (considered as a vector) only

the momentum of each object considered individually

Part B

Two cars of equal mass collide inelastically and stick together after the collision. Before the collision, their speeds are and . What is the speed of the two-car system after the collision?

Hint B.1 How to approach the problem

Hint not displayed

ANSWER:

The answer depends on the directions in which the cars were moving before the collision.

Part C

http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1043409 (1 of 21)4/11/2006 2:56:04 PM


Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, what is the magnitude of their combined momentum?

Hint C.1

Hint not displayed

ANSWER:

The answer depends on the directions in which the cars were moving before the collision.

Part D

Two cars collide inelastically and stick together after the collision. Before the collision, their momenta are and . After the collision, their combined momentum is . Of what can one be

certain?

Hint D.1 Momentum is a vector

Hint not displayed

ANSWER:

You can decompose the vector equation that states the conservation of momentum into individual equations for each of the orthogonal components of the vectors.

Part E



Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, the magnitude of their combined momentum is . Of what can one be certain?

Hint E.1 How to approach the problem mathematically

Hint not displayed

Hint E.2 How to approach the problem empirically

Hint not displayed

ANSWER:

When the two cars collide, the magnitude of the final momentum will always be at most (a value attained if the cars were moving in the same direction before the collision) and at least

(a value attained if the cars were moving in opposite directions before the collision).

Momentum and Internal Forces

Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum.

We begin by introducing the following terms:

System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem.

Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system.

External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system.

Closed system: a system that is not subject to any external forces.



Total momentum: The vector sum of the individual momenta of all objects constituting the system.

In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions:

1. The blocks can move in only one dimension, namely, along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed.

At time , the x components of the velocity and the acceleration of block 1 are denoted by and

. Similarly, the x components of the velocity and acceleration of block 2 are denoted by

and . In this problem, you will show that the total momentum of the system is not changed by

the presence of internal forces.

Part A

Find , the x component of the total momentum of the system at time .

Express your answer in terms of , , , and .

ANSWER: =

Part B

Find the time derivative of the x component of the system's total momentum.

Hint B.1

Hint not displayed

Hint B.2

Hint not displayed

Express your answer in terms of , , , and .

ANSWER: =

Why did we bother with all this math? The expression for the derivative of momentum that we just obtained will be useful in reaching our desired conclusion, if only for this very special case.

Part C



The quantity (mass times acceleration) is dimensionally equivalent to which of the following?

ANSWER: momentum

energy

force

acceleration

inertia

Part D

Acceleration is due to which of the following physical quantities?

ANSWER: velocity

speed

energy

momentum

force

Part E

Since we have assumed that the system composed of blocks 1 and 2 is closed, what could be the reason for the acceleration of block 1?

Hint E.1

Hint not displayed

ANSWER: the large mass of block 1

air resistance

Earth's gravitational attraction

a force exerted by block 2 on block 1


Part F

What could be the reason for the acceleration of block 2?

ANSWER: a force exerted by block 2 on block 1


Part G



Let us denote the x component of the force exerted by block 1 on block 2 by , and the x

component of the force exerted by block 2 on block 1 by . Which of the following pairs

equalities is a direct consequence of Newton's second law?

ANSWER: and

and

and

and

Note that both and are internal forces.

Part H

Let us recall that we have denoted the force exerted by block 1 on block 2 by , and the force

exerted by block 2 on block 1 by . If we suppose that is greater than , which of the

following statements about forces is true?

Hint H.1 Which of Newton's laws is useful here?

Hint not displayed

ANSWER:

Both forces have equal magnitudes.

Newton's third law states that forces and are equal in magnitude and opposite in direction.

Therefore, their x components are related by

Part I



Now recall the expression for the time derivative of the x component of the system's total

momentum: . Considering the information that you now have, choose the best

alternative for an equivalent expression to .

Hint I.1 What is ?

Hint not displayed

ANSWER: 0

nonzero constant

The derivative of the total momentum is zero; hence the total momentum is a constant function of time. We have just shown that for the special case of a closed two-block system, the internal forces do not change the total momentum of the system. It can be shown that in any system, the internal forces do not change the total momentum: It is conserved. In other words, total momentum is always conserved in a closed system of objects.

PSS 9.1: Tools of the Trade

Learning Goal: To practice Problem-Solving Strategy 9.1 for problems involving conservation of momentum.

An astronaut performs maintenance work outside her spaceship when the tether connecting her to the spaceship breaks. The astronaut finds herself at rest relative to the spaceship, at a distance from it. To get back to the ship, she decides to sacrifice her favorite wrench and hurls it directly away from the spaceship at a speed relative to the spaceship. What is the distance between the spaceship and the wrench by the time the astronaut reaches the spaceship?

The mass of the astronaut is ; the mass of the wrench is .

MODEL: Clearly define the system.

■ If possible, choose a system that is isolated ( ) or within which the interactions are

sufficiently short and intense that you can ignore external forces for the duration of the interaction (the impulse approximation). Momentum is conserved.

■ If it is not possible to choose an isolated system, try to divide the problem into parts such that momentum is conserved during one segment of the motion. Other segments of the motion can be analyzed using Newton's laws or, as you'll learn in Chapters 10 and 11,



conservation of energy.

VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you are trying to find.

SOLVE: The mathematical representation is based on the law of conservation of momentum:

. In component form, this is

,

ASSESS: Check if your result has the correct units, is reasonable, and answers the question.

We start by choosing the objects that would make up the system. In this case, it is possible to identify the system that is isolated.

Part A

In addition to the astronaut, which of the following are components of the system that should be defined to solve the problem?

A. the spaceship B. the wrench C. the earth

Enter the letter(s) of the correct answer(s) in alphabetical order. Do not use commas. For example, if you think the system consists of all the objects listed, enter ABC.

ANSWER: B

Part B



Which of the following reasons best explains why the astronaut + wrench can be considered an isolated system?

ANSWER: The mass of the wrench is much smaller than that of the astronaut.

The force that the astronaut exerts on the wrench is very small.

The force that the astronaut exerts on the wrench is very large.

The force that the spaceship exerts on the wrench is very small.

The force that the spaceship exerts on the wrench is very large.

Now draw a before-and-after pictorial representation including all the elements listed in the problem-solving strategy. Be sure that your sketch is clear and includes all necessary symbols, both known and unknown. By the time the astronaut reaches the spaceship, the wrench will have covered a certain distance; on your pictorial representation, label this distance .

Part C

After the wrench is thrown, the astronaut and the wrench move

ANSWER: in opposite directions.

in the same direction.

in perpendicular directions.

Part D

Which statement about , , and is correct?

ANSWER:

Here is an example of what a good before-and-after pictorial representation might look like for this problem.



Now use the information and the insights that you have accumulated to construct the necessary mathematical expressions and to derive the solution.

Part E

Find the final distance between the spaceship and the wrench.

Part E.1 Find

Part not displayed

Hint E.2

Hint not displayed

Express the distance in terms of the given variables. You may or may not use all of them.

ANSWER: =

When you work on a problem on your own, without the computer-provided feedback, only you can assess whether your answer seems right. The following questions will help you practice the skills necessary for such an assessment.

Part F



Intuitively, which of the following statements are correct?

A. For realistic values of the quantities involved, it is possible that .

B. If the astronaut threw a space pen instead of a wrench, the pen would travel further than the wrench would in the time it takes the astronaut to reach the ship. (Assume the space pen weighs less than the wrench).

C. If the astronaut were more massive, the wrench would travel further in the time it takes the astronaut to reach the ship.

Type the letters corresponding to the correct answers. Do not use commas. For instance, if you think that only expressions C and D have the units of distance, type CD.

ANSWER: BC

could only be zero if . As you can see from your answer, this would only happen if

the mass of the astronaut were zero, which is obviously unrealistic.

Part G

Which of the following mathematical expressions have the units of distance, where and are

distances?

A.

B.

C.

D.

E.

F.

Type the letters corresponding to the correct answers. Do not use commas. For instance, if you think that only expressions C and D have the units of distance, type CD.

ANSWER: ABE

The Impulse-Momentum Theorem



Learning Goal: To learn about the impulse-momentum theorem and its applications in some common cases.

Using the concept of momentum, Newton's second law can be rewritten as

, (1)

where is the net force acting on the object, and is the rate at which the object's

momentum is changing.

If the object is observed during an interval of time between times and , then integration of both sides of equation (1) gives

. (2)

The right side of equation (2) is simply the change in the object's momentum . The left side

is called the impulse of the net force and is denoted by . Then equation (2) can be rewritten as

.

This equation is known as the impulse-momentum theorem. It states that the change in an object's momentum is equal to the impulse of the net force acting on the object. In the case of a constant net

force acting along the direction of motion, the impulse-momentum theorem can be written as

. (3)

Here , , and are the components of the corresponding vector quantities along the chosen

coordinate axis. If the motion in question is two-dimensional, it is often useful to apply equation (3) to the x and y components of motion separately.

The following questions will help you learn to apply the impulse-momentum theorem to the cases of constant and varying force acting along the direction of motion. First, let us consider a particle of mass moving along the x axis. The net force is acting on the particle along the x axis. is a

constant force.



Part A

The particle starts from rest at . What is the magnitude of the momentum of the particle at

time ? Assume that .

Express your answer in terms of any or all of , , and .

ANSWER: =

Part B

The particle starts from rest at . What is the magnitude of the velocity of the particle at

time ? Assume that .

Express your answer in terms of any or all of , , and .

ANSWER: =

Part C

The particle has momentum of magnitude at a certain instant. What is , the magnitude of its momentum seconds later?

Express your answer in terms of any or all of , , , and .

ANSWER: =

Part D

The particle has momentum of magnitude at a certain instant. What is , the magnitude of its velocity seconds later?

Express your answer in terms of any or all of , , , and .

ANSWER: =

Let us now consider several two-dimensional situations.

A particle of mass is moving in the positive x direction at speed . After a certain constant force



is applied to the particle, it moves in the positive y direction at speed .

Part E

Find the magnitude of the impulse delivered to the particle.

Hint E.1 How to approach the problem

Hint not displayed

Part E.2 Find the change in momentum

Part not displayed

Express your answer in terms of and . Use three significant figures in the numerical coefficient.

ANSWER: =

Part F

Which of the vectors below best represents the direction of the impulse vector ?

ANSWER: 1

2

3

4

5

6

7

8

Part G



What is the angle between the positive y axis and the vector as shown in the figure?

ANSWER: 26.6 degrees

30 degrees

60 degrees

63.4 degrees

Part H

If the magnitude of the net force acting on the particle is , how long does it take the particle to

acquire its final velocity, in the positive y direction?

Express your answer in terms of , , and . If you use a numerical coefficient, use three

significant figures.

ANSWER: =

So far, we have considered only the situation in which the magnitude of the net force acting on the particle was either irrelevant to the solution or was considered constant. Let us now consider an example of a varying force acting on a particle.

Part I



A particle of mass kilograms is at rest at seconds. A varying force

is acting on the particle between seconds and

seconds. Find the speed of the particle at seconds.

Hint I.1

Hint not displayed

Part I.2

Part not displayed

Express your answer in meters per second to three significant figures.

ANSWER: = 43.0

Colliding Balls

Balls A and B roll across a table, then collide elastically. The paths of the two balls are pictured (viewed from above) in the diagram.

Part A

Which set of arrows best represents the change in momentum for balls A and B?

Hint A.1

Hint not displayed



ANSWER: A B C D E

Part B

Which of the following arrows indicates the direction of the impulse applied to ball A by ball B?

Hint B.1 Definition of impulse

Hint not displayed

ANSWER: A B C D

E

A Game of Frictionless Catch

Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, , is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest.

Chuck then picks up a ball of mass and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is . The speed of the thrown ball relative to the ground is

.

Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie's speed relative to the ground after she catches the ball is .

When answering the questions in this problem, keep the following in mind:

1. The original mass of Chuck and his cart does not include the mass of the ball. 2. The speed of an object is the magnitude of its velocity. An object's speed will always be a

nonnegative quantity.



Part A

Find the relative speed between Chuck and the ball after Chuck has thrown the ball.

Hint A.1

Hint not displayed

Express the speed in terms of and .

ANSWER: =

Make sure you understand this result; the concept of "relative speed" is important. In general, if two objects are moving in opposite directions (either toward each other or away from each other), the relative speed between them is equal to the sum of their speeds with respect to the ground. If two objects are moving in the same direction, then the relative speed between them is the absolute value of the difference of the their two speeds with respect to the ground.

Part B

What is the speed of the ball (relative to the ground) while it is in the air?

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Initial momentum of Chuck, his cart, and the ball

Hint not displayed

Part B.3 Find the final momentum of Chuck, his cart, and the thrown ball

Part not displayed

Express your answer in terms of , , and .

ANSWER: =

Part C

What is Chuck's speed (relative to the ground) after he throws the ball?

Hint C.1

Hint not displayed

Express your answer in terms of , , and .

ANSWER: =



Part D

Find Jackie's speed (relative to the ground) after she catches the ball, in terms of .

Hint D.1

Hint not displayed

Hint D.2

Hint not displayed

Part D.3

Part not displayed

Express in terms of , , and .

ANSWER: =

Part E

Find Jackie's speed (relative to the ground) after she catches the ball, in terms of .

Hint E.1

Hint not displayed

Express in terms of , , and .

ANSWER: =

A One-Dimensional Inelastic Collision

Block 1, of mass = 2.10 , moves along a frictionless air track with speed = 27.0 . It

collides with block 2, of mass = 17.0 , which was initially at rest. The blocks stick together

after the collision.



Part A

Find the magnitude of the total initial momentum of the two-block system.

Hint A.1

Hint not displayed

Express your answer numerically.

ANSWER: = 56.7

Part B

Find , the magnitude of the final velocity of the two-block system.

Hint B.1

Hint not displayed

Express your answer numerically.

ANSWER: = 2.97

Catching a Ball on Ice

Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass

0.400 that is traveling horizontally at 10.1 . Olaf's mass is 73.5 .

Part A

If Olaf catches the ball, with what speed do Olaf and the ball move afterward?

Hint A.1

Hint not displayed

Part A.2

Part not displayed

Express your answer numerically in centimeters per second.

ANSWER: = 5.47



Part B

If the ball hits Olaf and bounces off his chest horizontally at 7.40 in the opposite direction,

what is his speed after the collision?

Hint B.1

Hint not displayed

Part B.2

Part not displayed

Express your answer numerically in centimeters per second.

ANSWER: = 9.52
