SECTION 3 Objectives Elastic and Inelastic Collisions

SECTION 3

Plan and Prepare

Preview Vocabulary

Scientific Meaning Elasticity of materials is defined by their flexibility in length, shape, and volume. Have students provide examples of elastic materials. Ask students to give examples of changes in length, shape, and volume.

Teach

Demonstration

Inelastic Collisions Purpose Show the conservation of momentum in an inelastic collision. Materials two balls with the same mass, two pieces of string, tape, small piece of modeling clay, meterstick, chalkboard Procedure Tie or tape a string around each ball. Hold the two strings so that the balls hang at the same height in front of each other on the chalkboard. Place the clay on one of the balls. Hold up one of the balls and have a student mark its displacement on the chalkboard.

Release the ball. It should stick to the second ball; both balls should move together. Have a student mark the displacement of the two balls after the collision. Measure the displacements with the meterstick. If momentum is conserved, the height of the two balls together will be _41_the original height. Explain to the students that according to the conservation of momentum, m1v1,i + m2v2,i = (m1 + m2)vf for a perfectly inelastic collision. Since the second ball starts at rest, the final velocity of the two balls will be half the initial velocity of the first ball. Because the kinetic energy at the bottom of the swing equals the potential energy at the rteoapc(hm_41_ghth=e i_n12_imtiavl2h),etihghettwofotbhaellfsirsshtobualldl.

SECTION 3

Objectives

Identify different types of collisions.

Determine the changes in kinetic energy during perfectly inelastic collisions.

Compare conservation of momentum and conservation of kinetic energy in perfectly inelastic and elastic collisions.

Find the final velocity of an object in perfectly inelastic and elastic collisions.

Elastic and Inelastic Collisions

Key Terms

perfectly inelastic collision

elastic collision

Collisions

As you go about your day-to-day activities, you probably witness many collisions without really thinking about them. In some collisions, two objects collide and stick together so that they travel together after the impact. An example of this action is a collision between football players during a tackle, as shown in Figure 3.1. In an isolated system, the two football players would both move together after the collision with a momentum equal to the sum of their momenta (plural of momentum) before the collision. In other collisions, such as a collision between a tennis racket and a tennis ball, two objects collide and bounce so that they move away with two different velocities.

The total momentum remains constant in any type of collision. However, the total kinetic energy is generally not conserved in a collision because some kinetic energy is converted to internal energy when the objects deform. In this section, we will examine different types of collisions and determine whether kinetic energy is conserved in each type. We will primarily explore two extreme types of collisions: perfectly inelastic collisions and elastic collisions.

perfectly inelastic collision a collision in which two objects stick together after colliding

Perfectly inelastic collisions can be analyzed in terms of momentum.

When two objects, such as the two football players, collide and move together as one mass, the collision is called a perfectly inelastic collision. Likewise, if a meteorite collides head on with Earth, it becomes buried in Earth and the collision is perfectly inelastic.

FIGURE 3.1

Perfectly Inelastic Collision

When one football player tackles another, they both continue to fall together. This is one familiar example of a perfectly inelastic collision.

?Nathan Bilow/Getty Images

Diff2e0r4enCthiaaptteer 6d Instruction

English Learners

Students Untitled-194 204 may be familiar with the word elastic as referring to something that always returns to its original shape. A rubber band is a familiar example.

In physics, the word elastic is related to work. Work is done to change the shape of the material during a collision. The work done to change the material's shape equals the work that the material does to return to its original

shape. The term elastic involves returning to an original shape in both cases, but the use of

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the term in physics is more specific.

204 Chapter 6

Perfectly inelastic collisions are easy to analyze in terms of momentum because the objects become essentially one object after the collision. The final mass is equal to the combined masses of the colliding objects. The combination moves with a predictable velocity after the collision.

Consider two cars of masses m1 and m2 moving with initial velocities of v1,i and v2,i along a straight line, as shown in Figure 3.2. The two cars stick together and move with some common velocity, vf , along the same line of motion after the collision. The total momentum of the two cars before the collision is equal to the total momentum of the two cars after the collision.

Perfectly Inelastic Collision

m1 v1,i + m2 v2,i = (m1 + m2) vf

FIGURE 3.2

Inelastic Collision

The total momentum of the two cars before the collision (a) is the same as the total momentum of the two cars after the inelastic collision (b).

(a)

Vl, i

V2, i

m1

m2

(b) Vf

This simplified version of the equation for conservation of momentum is useful in analyzing perfectly inelastic collisions. When using this equation, it is important to pay attention to signs that indicate direction. In Figure 3.2, v1,i has a positive value (m1 moving to the right), while v2,i has a negative value (m2 moving to the left).

PREMIUM CONTENT

m1 + m2

Perfectly Inelastic Collisions

Interactive Demo



Sample Problem E A 1850 kg luxury sedan stopped at a traffic light is struck from the rear by a compact car with a mass of 975 kg. The two cars become entangled as a result of the collision. If the compact car was moving at a velocity of 22.0 m/s to the north before the collision, what is the velocity of the entangled mass after the collision?

ANALYZE

Given: Unknown:

m1 = 1850 kg m2 = 975 kg v1,i = 0 m/s v2,i = 22.0 m/s to the north vf = ?

SOLVE

Use the equation for a perfectly inelastic collision.

m1v1,i + m2v2,i = (m1 + m2) vf

vf

=

_ m1v1,i +_ m2v2,i m1 + m2

vf

=

_ (1850 kg)_ (0 m/s) +_ (975 kg_ )(22.0 m/_ s north) 1850 kg + 975 kg

vf = 7.59 m/s to the north

Continued

Momentum and Collisions 205

Classroom Practice

Perfectly Inelastic Collisions An empty train car moving east at 21 m/s collides with a loaded train car initially at rest that has twice the mass of the empty car. The two cars stick together.

a. Find the velocity of the two cars after the collision.

b. Find the final speed if the loaded car moving at 17 m/s had hit the empty car initially at rest.

Answers a. 7.0 m/s to the east

PHYSICS

b. 11 m/s Spec. Number PH 99 PE C06-003-006-A

Boston Graphics, Inc.

An 617.523.1333 empty train car moving at 15 m/s collides with a loaded car of three times the mass moving in the same direction at one-third the speed of the empty car. The cars stick together. Find the speed of the cars after the collision.

Answer: 7.5 m/s

Untitled-194 205

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Momentum and Collisions 205

Teach continued

PROBLEM guide E

Use this guide to assign problems.

SE = Student Edition Textbook

PW = Sample Problem Set I (online)

PB = Sample Problem Set II (online)

Solving for:

vf

SE Sample, 1?3;

Ch. Rvw. 28?32,

PW 7?9

PB 5?7

vi

SE 4, 5*; Ch. Rvw. 39, 42

PW 4?6

PB Sample, 1?4

m

SE 5*; Ch. Rvw. 38*

PW Sample, 1?3

PB 8?10

*Challenging Problem

Answers

Practice E 1. 3.8 m/s to the south 2. 1.8 m/s 3. 4.25 m/s to the north 4. 4.2 m/s to the right 5. a. 3.0 kg

b. 5.32 m/s

Perfectly Inelastic Collisions (continued)

1. A 1500 kg car traveling at 15.0 m/s to the south collides with a 4500 kg truck that is initially at rest at a stoplight. The car and truck stick together and move together after the collision. What is the final velocity of the two-vehicle mass?

2. A grocery shopper tosses a 9.0 kg bag of rice into a stationary 18.0 kg grocery cart. The bag hits the cart with a horizontal speed of 5.5 m/s toward the front of the cart. What is the final speed of the cart and bag?

3. A 1.50 ? 104 kg railroad car moving at 7.00 m/s to the north collides with and sticks to another railroad car of the same mass that is moving in the same direction at 1.50 m/s. What is the velocity of the joined cars after the collision?

4. A dry cleaner throws a 22 kg bag of laundry onto a stationary 9.0 kg cart. The cart and laundry bag begin moving at 3.0 m/s to the right. Find the velocity of the laundry bag before the collision.

5. A 47.4 kg student runs down the sidewalk and jumps with a horizontal speed of 4.20 m/s onto a stationary skateboard. The student and skateboard move down the sidewalk with a speed of 3.95 m/s. Find the following: a. the mass of the skateboard b. how fast the student would have to jump to have a final speed of 5.00 m/s

Kinetic energy is not conserved in inelastic collisions.

In an inelastic collision, the total kinetic energy does not remain constant when the objects collide and stick together. Some of the kinetic energy is converted to sound energy and internal energy as the objects deform during the collision.

This phenomenon helps make sense of the special use of the words elastic and inelastic in physics. We normally think of elastic as referring to something that always returns to, or keeps, its original shape. In physics, an elastic material is one in which the work done to deform the material during a collision is equal to the work the material does to return to its original shape. During a collision, some of the work done on an inelastic material is converted to other forms of energy, such as heat and sound.

The decrease in the total kinetic energy during an inelastic collision can be calculated by using the formula for kinetic energy, as shown in Sample Problem F. It is important to remember that not all of the initial kinetic energy is necessarily lost in a perfectly inelastic collision.

Misconception Alert!

Students may think that elastic materials can undergo only elastic collisions. Consider a large brass bell with a clapper. The material, brass, is very elastic. After the collision, the bell continues to vibrate and give off sound (energy!) for a long time afterwards. The collision isn't elastic even though the materials are. Inelastic materials undergo only inelastic collisions. Elastic materials may undergo either elastic or inelastic collisions.

Diff2e0r6enCthiaaptteer 6d Instruction

Inclusion

Visual and Untitled-194 206 kinesthetic learners will benefit from demonstrations of inelastic collisions that use toy trains or other objects. When one train car collides with another, they become coupled together and move as a unit. Plan several demonstrations of different types of collisions as you teach this section.

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206 Chapter 6

Kinetic Energy in Perfectly Inelastic Collisions

Sample Problem F Two clay balls collide head-on in a perfectly inelastic collision. The first ball has a mass of 0.500 kg and an initial velocity of 4.00 m/s to the right. The second ball has a mass of 0.250 kg and an initial velocity of 3.00 m/s to the left. What is the decrease in kinetic energy during the collision?

PREMIUM CONTENT

Interactive Demo



ANALYZE

Given: Unknown:

m1 = 0.500 kg m2 = 0.250 kg v1,i = 4.00 m/s to the right, v1,i = +4.00 m/s v2,i = 3.00 m/s to the left, v2,i = -3.00 m/s KE = ?

PLAN SOLVE

Choose an equation or situation: The change in kinetic energy is simply the initial kinetic energy subtracted from the final kinetic energy.

KE = KEf - KEi

Determine both the initial and final kinetic energy.

Initial:

KEi

=

KE1,i

+

KE2,i

=

_1_ 2

m1v

2 1,i

+

_1_ 2

m2

v

2 2,i

Final:

KEf

=

KE1,f

+

KE2,f

=

_12_(m1

+

m2)v

2 f

As you did in Sample Problem E, use the equation for a perfectly inelastic

collision to calculate the final velocity.

vf

=

_ m1v1,i+_ m2v2,i m1 + m2

Substitute the values into the equation and solve: First, calculate the final velocity, which will be used in the final kinetic energy equation.

vf

=

_ (0.500 kg_ )(4.00 m/_ s) + (0.25_ 0 kg)(-3_ .00 m/s) 0.500 kg + 0.250 kg

vf = 1.67 m/s to the right

Next calculate the initial and final kinetic energy.

KEi = _12_(0.500 kg)(4.00 m/s)2 + _12_(0.250 kg)(-3.00 m/s)2 = 5.12 J KEf = _12_(0.500 kg + 0.250 kg)(1.67 m/s)2 = 1.05 J

Finally, calculate the change in kinetic energy.

KE = KEf - KEi = 1.05 J - 5.12 J

KE = -4.07 J

CHECK

The negative sign indicates that kinetic energy is lost.

YOUR WORK

Continued

Problem Solving

Momentum and Collisions 207

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Take It Further

Tell students that they can combine the formulas for initial kinetic energy and final kinetic energy before replacing the values, and then substitute the known measures in the combined formula. The combination process is as follows:

KEf = _21(m1 + m2)v2f KEi = _21m 1v12,i+ _21m 2v22,i

Subtract the equations side by side, and

expand the right side.

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KEf - KEi = _21(m1 + m2)vf2-

( ) _21m 1v12,i+ _21m 2v22,i

-KE_21f -m 1Kv12E,ii-=_21_21mm 2v1v 222f,i+ _21m 2vf2

Next, factor out as below:

( ) ( ) KEf - KEi = _21m 1 vf2- v12,i + _21m 2 vf2- v22,i

Classroom Practice

Kinetic Energy in Perfectly Inelastic Collisions A clay ball with a mass of 0.35 kg hits another 0.35 kg ball at rest, and the two stick together. The first ball has an initial speed of 4.2 m/s.

a. What is the final speed of the balls?

b. Calculate the decrease in kinetic energy that occurs during the collision.

c. What percentage of the initial kinetic energy is converted to other forms of energy?

Answers a. 2.1 m/s b. 1.6 J c. 52 percent (This is actually 50 percent. The difference is due to rounding.)

A 0.75 kg ball moving at 3.8 m/s to the right strikes an identical ball moving at 3.8 m/s to the left. The balls stick together after the collision and stop. What percentage of the initial kinetic energy is converted to other forms?

Answer: 100 percent

Momentum and Collisions 207

Teach continued

PROBLEM guide F

Use this guide to assign problems.

SE = Student Edition Textbook

PW = Sample Problem Set I (online)

PB = Sample Problem Set II (online)

Solving for:

KE

SE Sample, 1?3; Ch. Rvw. 30?31,

PW Sample, 1?7

PB Sample, 1?10

*Challenging Problem

Kinetic Energy in Perfectly Inelastic Collisions (continued)

1. A 0.25 kg arrow with a velocity of 12 m/s to the west strikes and pierces the center of a 6.8 kg target. a. What is the final velocity of the combined mass? b. What is the decrease in kinetic energy during the collision?

2. During practice, a student kicks a 0.40 kg soccer ball with a velocity of 8.5 m/s to the south into a 0.15 kg bucket lying on its side. The bucket travels with the ball after the collision. a. What is the final velocity of the combined mass of the bucket and the ball? b. What is the decrease in kinetic energy during the collision?

3. A 56 kg ice skater traveling at 4.0 m/s to the north meets and joins hands with a 65 kg skater traveling at 12.0 m/s in the opposite direction. Without rotating, the two skaters continue skating together with joined hands. a. What is the final velocity of the two skaters? b. What is the decrease in kinetic energy during the collision?

Answers

Practice F 1. a. 0.43 m/s to the west

b. 17 J 2. a. 6.2 m/s to the south

b. 3.9 J 3. a. 4.6 m/s to the south

b. 3.9 ? 103 J

Key Models and Analogies

Just as friction is often disregarded to simplify situations, the decrease in kinetic energy in a nearly elastic collision can be disregarded to create an ideal case. This ideal case can then be used to obtain a very close approximation to the observed result.

elastic collision a collision in which the total momentum and the total kinetic energy are conserved

Elastic Collisions

When a player kicks a soccer ball, the collision between the ball and the player's foot is much closer to elastic than the collisions we have studied so far. In this case, elastic means that the ball and the player's foot remain separate after the collision.

In an elastic collision, two objects collide and return to their original shapes with no loss of total kinetic energy. After the collision, the two objects move separately. In an elastic collision, both the total momentum and the total kinetic energy are conserved.

Most collisions are neither elastic nor perfectly inelastic.

In the everyday world, most collisions are not perfectly inelastic. Colliding objects do not usually stick together and continue to move as one object. Most collisions are not elastic, either. Even nearly elastic collisions, such as those between billiard balls, result in some decrease in kinetic energy. For example, a football deforms when it is kicked. During this deformation, some of the kinetic energy is converted to internal elastic potential energy. In most collisions, some kinetic energy is also converted into sound, such as the click of billiard balls colliding. In fact, any collision that produces sound is not elastic; the sound signifies a decrease in kinetic energy.

Elastic and perfectly inelastic collisions are limiting cases; most collisions actually fall into a category between these two extremes. In this third category of collisions, called inelastic collisions, the colliding objects bounce and move separately after the collision, but the total kinetic energy decreases in the collision. For the problems in this book, we will

Diff2e0r8enCthiaaptteer 6d Instruction

Below Level

Discuss Untitled-194 208 a variety of examples of collisions with students. For each example, ask whether the collision is closer to an elastic collision or to a perfectly inelastic collision. Also ask students where kinetic energy is converted to other forms of energy in each of the different examples.

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208 Chapter 6

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