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. 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 Hint B.1 How to appro approach ach the prob problem lem

Hint not displayed ANSWER:

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


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


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 Hint D.1 D.1 Mome Moment ntum um is a vec vecto torr


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


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 ANSWER:

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


the large mass of block 1 air resistance Earth's gravitational attraction a force exerted by block 2 on block 1 a force exerted by block 1 on block 2

Part F

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

a force exerted by block 2 on block 1 a force exerted by block 1 on block 2

Part G


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

, and the x

. 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 exerted by block 2 on block 1 by

. If we suppose that

is greater than

, and the force , which of the

following statements about forces is true? Hint H.1 Which of Newton's laws is useful here?


Both forces have equal magnitudes. Newton's third law states that forces

and

Therefore, their x components are related by Part I

are equal in magnitude and opposite in direction.


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

.

?


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 MODEL:

; the mass of the wrench is

.

Clearly define the system.

s

If possible, choose a system that is isolated (

) or within which the interactions are

s

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.

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. VISUALIZE:

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: Part B

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)

is the net force

where

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 sides of equation (1) gives

and

, then integration of both

. (2)

The right side of equation (2) is simply the change in the object's momentum is called the impulse of the net force and is denoted by

. The left side

. 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 time ? Assume that

. What is the magnitude

.

Express your answer in terms of any or all of ANSWER:

of the momentum of the particle at

,

, and .

=

Part B

The particle starts from rest at time ? Assume that

. What is the magnitude

of the velocity of the particle at

.

Express your answer in terms of any or all of

,

, and .

ANSWER:

= Part C

The particle has momentum of magnitude momentum seconds later? Express your answer in terms of any or all of

at a certain instant. What is

,

,

, and

, the magnitude of its

.

ANSWER:

= Part D

The particle has momentum of magnitude velocity seconds later? Express your answer in terms of any or all of

at a certain instant. What is

,

,

, and

, the magnitude of its

.

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 coefficient. ANSWER:

and . Use three significant figures in the numerical

=

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

ANSWER:

between the positive y axis and the vector

as shown in the figure?

26.6 degrees 30 degrees 60 degrees 63.4 degrees

Part H

If the magnitude of the net force acting on the particle is acquire its final velocity,

, how long does it take the particle to

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. Find the speed

of the particle at

seconds and

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


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 ANSWER:

and

.

=

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 ANSWER:

=

,

, and

.


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

collides with block 2, of mass after the collision.

, moves along a frictionless air track with speed = 17.0

= 27.0

. It

, which was initially at rest. The blocks stick together


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

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

. Olaf's mass is 73.5

.


Part B

If the ball hits Olaf and bounces off his chest horizontally at 7.40 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

in the opposite direction,

Chapter 9 Skill Building

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