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**Preview Section 1 Work Section 2 Energy**

Section 3 Conservation of Energy Section 4 Power Section 5 Extra questions

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What do you think? List five examples of things you have done in the last year that you would consider work. Based on these examples, how do you define work? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Likely answers are: homework, babysitting, jobs, studying physics, and so on. After listening and discussing, let the students know that in physics, the definition of work is much more precise and many things that they consider work will not fit that definition. In physics, work produces a change in energy. Work is defined in terms of force and displacement on the next slide.

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Work In physics, work is the magnitude of the force (F) times the magnitude of the displacement (d) in the same direction as the force. W = Fd What are the SI units for work? Force units (N) distance units (m) N•m are also called joules (J). How much work is 1 joule? Lift an apple weighing about 1 N from the floor to the desk, a distance of about 1 m. Units are sometimes confusing. It would be a good idea to show students that 1 J = 1 N•m = 1 kg•m2/s2 at this time. Give them a chance to figure it out for themselves from the definition of a newton (F=ma). This is important because they will later learn that kinetic energy and potential energy are measured in joules as well, and the equations lead to kg•m2/s2. Students need to understand the fundamental SI units behind all of the derived units such as newtons, joules, watts and so on.

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Work Pushing this car is work because F and d are in the same direction. Why aren’t the following tasks considered work? A student holds a heavy chair at arm’s length for several minutes. A student carries a bucket of water along a horizontal path while walking at a constant velocity. In the first case, no work is done because the object does not move (d = 0). In the second case, no work is done because the distance moved is not in the direction of the force (the force is vertically upward while the distance is horizontal). There is no component of the force in the horizontal direction.

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**Work How would you calculate the work in this case?**

What is the component of F in the direction of d? F cos If the angle is 90°, what is the component of F in the direction of d? F cos 90° = 0 If the angle is 0°, what is the component of F in the direction of d? F cos 0° = F Discussion of the component of F along the direction of d should lead to the equation on the next slide.

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Work Students should already have deduced this equation from the last slide.

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Work is a Scalar Work can be positive or negative but does not have a direction. What is the angle between F and d in each case? Show students that the two diagrams on the left show force and distance in opposite directions, while those on the right show force and distance in the same direction. Ask the angle between the force and distance in the top left diagram. It looks like it is roughly 135°. Point out to them that the cos(135°) is a negative number. The angle on the top right is about 45° (cos is +). The angle on the bottom left is about 225° (cos is -). The angle on the bottom right is about 315° (cos is +). For the bottom pictures, it will be harder for students to determine the angle unless they draw the force and distance starting at a common point.

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**Classroom Practice Problem**

A 20.0 kg suitcase is raised 3.0 m above a platform. How much work is done on the suitcase? Answer: 5.9 x 102 J or 590 J Students may use the mass instead of the weight (20.0 kg x 9.81 m/s2). This is a good time to remind them that mass and weight are different although related quantities.

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Now what do you think? Based on the physics definition, list five examples of things you have done in the last year that you would consider work. Students should now select answers that show a force moving an object in the direction of the force.

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**What do you think? You have no doubt heard the term kinetic energy.**

What is it? What factors affect the kinetic energy of an object and in what way? You have no doubt heard the term potential energy. What factors affect the potential energy of an object and in what way? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Kinetic energy comments will likely center around the velocity and probably not mention the mass of the object. Students may mention the different types of potential energy (gravitational, electrostatic, elastic, etc). Listen to ideas about these topics and then begin the lecture/discussion.

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**Kinetic Energy Since then or**

Show students the steps and substitutions needed to derive the final equation for work. Make sure they see the use of F = ma in the first equation and the substitution for ax from the 2nd equation into the third equation. Help students see the transformation of the 3rd equation into the 4th equation. Have them note that calculating the work no longer requires knowledge of the force but, instead, can be determined by the effect of the force or the change in velocity. Mention that a name has been given to the quantity 1/2 mv2 . It is called kinetic energy. So, work is the change in KE. Then show them the next slide, which introduces the kinetic energy equation.

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**Kinetic Energy What are the SI units for KE? kg•m2/s2 or N•m or J**

Ask students to determine the units from the equation before showing this on the slide. Have them see that, since N are kg•m/s2, the units of N•m are equivalent to kg•m2/s2.

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**Work and Kinetic Energy**

KE is the work an object can do if the speed changes. Wnet is positive if the speed increases. Discuss the many examples of moving objects doing work on other objects. For example, a moving baseball bat does work on a ball as it exerts a force on the ball, and the ball moves a distance in the direction of the force. Conversely, the ball does work on the bat as it exerts a force opposite to the direction the bat is moving. Work has a negative value in this case. A change in speed for an object allows it to do work on its environment.

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**Classroom Practice Problems**

A 6.00 kg cat runs after a mouse at 10.0 m/s. What is the cat’s kinetic energy? Answer: 3.00 x 102 J or 300 J Suppose the above cat accelerated to a speed of 12.0 m/s while chasing the mouse. How much work was done on the cat to produce this change in speed? Answer: 1.32 x 102 J or 132 J For the second problem, students should just use the change in KE (432 J – 300 J = 132 J). Sometimes they make the mistake of thinking that they can use the change in speed (2 m/s) in the equation for KE and end up with an answer of 12 J for the work done. This does not work because ( )2 is not equal to ( ).

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Potential Energy Energy associated with an object’s potential to move due to an interaction with its environment A book held above the desk An arrow ready to be released from the bow Some types of PE are listed below. Gravitational Elastic Electromagnetic Hold a book above the desk. The book has the potential to move due to an interaction with Earth (gravity). Stretch a rubber band with a wad of paper held in it like a sling shot. The paper has the potential to move due to an interaction with its environment (the rubber band).

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**Gravitational Potential Energy**

What are the SI units? kg•m2/s2 or N•m or J The height (h) depends on the “zero level” chosen where PEg = 0. This equation comes from W = Fd = (ma)d = mgh, so PEg is simply the work done in lifting an object. To help students understand the fact that the zero level is arbitrary, hold a book over the desk and ask them what they would use for h in order to calculate the PE. Then, maintaining the book at the same height, move it over the floor and ask the students once again what value they would use for h. Point out that, in general, our primary concern in physics involves changes in PE, not the actual amount of PE. The change in PE is always the same regardless of what zero level is assigned. Generally, the zero level is assigned to the lowest point the object will reach. For example, the desk if the book is held over the desk, and the floor if the book is held over the floor.

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**Elastic Potential Energy**

The energy available for use in deformed elastic objects Rubber bands, springs in trampolines, pole-vault poles, muscles For springs, the distance compressed or stretched = x Point out that x in the diagram is the “Distance compressed.” This will be used in the equation for elastic potential energy (slide 10). Discuss the transfer of the elastic potential energy to the block when the deformed spring returns to its original configuration.

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**Spring Constant(k) Click below to watch the Visual Concept.**

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**Elastic Potential Energy**

The spring constant (k) depends on the stiffness of the spring. Stiffer springs have higher k values. Measured in N/m Force in newtons needed to stretch a spring 1.0 meters What are the SI Units for PEelastic? Help students find the SI units as N•m or Kg•m2/s2 or J. Now would be a good time to remind them that work, KE, and PE are all measured in joules (kg•m2/s2).

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**Classroom Practice Problems**

When a 2.00 kg mass is attached to a vertical spring, the spring is stretched 10.0 cm such that the mass is 50.0 cm above the table. What is the gravitational potential energy associated with the mass relative to the table? Answer: 9.81 J What is the spring’s elastic potential energy if the spring constant is N/m? Answer: 2.00 J Point out to students that the zero level is at the table for gravitational PE. Also, they must use meters, not centimeters, in order to have joules as units in the final answers.

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**Now what do you think? What is kinetic energy?**

What factors affect the kinetic energy of an object and in what way? How are work and kinetic energy related? What is potential energy? What factors affect the gravitational potential energy of an object and in what way? What factors affect the elastic potential energy of an object and in what way? Mass and KE are directly proportional. KE is directly proportional to the velocity squared. The net work done on an object equals the change in kinetic energy. Factors affecting gravitational PE are mass, acceleration due to gravity, and height above the zero level. All are directly related. Factors affecting elastic PE are the spring constant (directly related) and the displacement from equilibrium position (directly related to the displacement squared).

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What do you think? Imagine two students standing side by side at the top of a water slide. One steps off of the platform, falling directly into the water below. The other student goes down the slide. Assuming the slide is frictionless, which student strikes the water with a greater speed? Explain your reasoning. Would your answer change if the slide were not frictionless? If so, how? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. A variety of answers are possible. Encourage them to explain their ideas so you can better understand the answer they chose. Remind the students that friction is not a consideration. Water slides are not truly frictionless, so they might want to imagine an air hockey table as a slide. Students will find out later in the lesson that, in the absence of friction, the speeds would be the same. The only difference would be the direction each is traveling at impact.

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What do you think? What is meant when scientists say a quantity is conserved? Describe examples of quantities that are conserved. Are they always conserved? If not, why? Students may be familiar with conservation of mass as a conservation law. Try to get them to explain what it means for something to be “conserved.” There are many ways to describe conservation but students often struggle. They may say that they know what it means but they can’t figure out how to say it or write it. These questions should be a good exercise for them to express their ideas in writing. Conservation may be expressed as “no change in the quantity” or “before = after” or “the total always stays the same.”

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**Mechanical Energy (ME)**

ME = KE + PEg + PEelastic Does not include the many other types of energy, such as thermal energy, chemical potential energy, and others ME is not a new form of energy. Just a combination of KE and PE Discuss ME as a useful tool for studying motion. Do not tell students yet that ME is conserved. They will determine this from the coming slides and calculations. As an example. toss a ball in the air and talk about the potential energy and kinetic energy as it rises and falls. As another example. show students a pendulum and talk about the PE and the KE changing as it swings.

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**Classroom Practice Problems**

Suppose a 1.00 kg book is dropped from a height of 2.00 m. Assume no air resistance. Calculate the PE and the KE at the instant the book is released. Answer: PE = 19.6 J, KE = 0 J Calculate the KE and PE when the book has fallen 1.0 m. (Hint: you will need an equation from Chapter 2.) Answer: PE = 9.81 J, KE = 9.81 J Calculate the PE and the KE just as the book reaches the floor. Answer: PE = 0 J, KE = 19.6 J Students should use PE = mgh to get the PE at each point. To calculate the KE they need to first find the velocity. The easiest way to get v2 is using vf2 = 2gy. (Note that the initial velocity is zero, so it was eliminated from the equation). After getting the velocity, use the equation KE = 1/2 mv2. After making these calculations, show students the chart on the next slide.

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**Table of Values for the Falling Book**

h (m) PE(J) KE(J) ME(J) 19.6 0.5 14.7 4.9 1.0 9.8 1.5 2.0 Mention the following to the students: (1) KE and PE change during the fall but ME remains the same (it is conserved). (2) You might ask students what values would change if air resistance was considered. The KE values would be smaller and the ME would gradually decrease, so ME would not be conserved. (3) No consideration was given to the path the book took to the floor, only its position. Therefore, the result would be the same even if it slid down a ramp (as long as the ramp was frictionless).

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**Conservation of Mechanical Energy**

The sum of KE and PE remains constant. One type of energy changes into another type. For the falling book, the PE of the book changed into KE as it fell. As a ball rolls up a hill, KE is changed into PE. Conservation of energy provides an easier method of solving some problems. For example, go back to the table created for the falling book, and point out the fact that they never needed to calculate the velocity. If they know the PE, then the KE is simply (19.6 J - PE).

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**Conservation of Mechanical Energy**

Click below to watch the Visual Concept. Visual Concept

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**Conservation of Energy**

Acceleration does not have to be constant. ME is not conserved if friction is present. If friction is negligible, conservation of ME is reasonably accurate. A pendulum as it swings back and forth a few times Consider a child going down a slide with friction. What happens to the ME as he slides down? Answer: It is not conserved but, instead, becomes less and less. What happens to the “lost” energy? Answer: It is converted into nonmechanical energy (thermal energy). Previously, in Chapter 2, the equations developed required that acceleration be constant. That is not the case for conservation of ME. The example with friction provides an opportunity to point out that even though ME may not be conserved, energy in general is conserved. With friction, the material becomes warmer due to the contact between the surfaces, which causes the speed of the vibrating molecules to increase.

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**Classroom Practice Problems**

A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 102 N/m. What is the elastic potential energy of the slingshot before release? What is the kinetic energy of the ball right after the slingshot is released? What is the ball’s speed at the instant it leaves the slingshot? How high does the ball rise if it is shot directly upward? Remind students that J require the use of kg and m (not g and cm). Answer 1: 0.36 J. Students should use PE = 1/2 kx2 to determine the energy stored in the rubber band. Answer 2: 0.36 J. Students should use conservation of mechanical energy. PE lost = KE gained Answer 3: 8.5 m/s. Students should use the KE from the last question and KE = 1/2 mv2 to determine v Answer 4: 3.7 m. Students may revert to equations from Chapter 2. but this is most easily solved using conservation of ME. They only need to find the height required for KE lost = Peg gained = 0.36 J = mgh.

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Now what do you think? Imagine two students standing side by side at the top of a water slide. One steps off of the platform, falling directly into the water below. The other student goes down the slide. Assuming the slide is frictionless, which student strikes the water with a greater speed? Explain your reasoning. Would your answer change if the slide were not frictionless? If so, how? Both strike at the same speed. One hits the water vertically, while the other slides in nearly horizontally. The PE lost is the same for both and, therefore, the KE gained is the same as well. With friction, the student falling straight down would be moving faster, because energy is not conserved for the student on the slide. Some of the lost PE is converted into thermal energy

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Now what do you think? What is meant when scientists say a quantity is “conserved”? Describe examples of quantities that are conserved. Are they always conserved? If not, why? Conserved means that the quantity does not change. The quantity neither increases nor decreases. Common examples are the conservation of mass and the conservation of energy. Mechanical energy is considered to be conserved in cases where friction is negligible. If friction is significant, mechanical energy is not conserved.

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What do you think? Two cars are identical with one exception. One of the cars has a more powerful engine. How does having more power make the car behave differently? What does power mean? What units are used to measure power? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. For this question, students may discuss the top speed of the cars or the time required for the cars to accelerate from 0 to 60 miles per hour. See if you can get them to talk about energy consumption. How quickly would each consume a gallon of gas? They likely will mention horsepower, but it is unlikely that any will consider watts as a unit of power.

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**Power The rate of energy transfer Energy used or work done per second**

Remind students that W = Fd, and ask them to substitute this for W in the power formula. Then ask what d/t represents. At this point, move on to the next slide, which shows the alternative form of the power equation (P = Fv).

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**Power SI units for power are J/s.**

Called watts (W) Equivalent to kg•m2/s3 Horsepower (hp) is a unit used in the Avoirdupois system. 1.00 hp = 746 W Be sure students understand that P = Fv is not a new definition. It is simply a different but equivalent formula that makes calculations easier in some cases. Do not just show the units. Ask students to figure them out. It should be easy for them to get J/s but the basic units of kg•m2/s3 will be more difficult. This provides a good opportunity to review the units for joules and newtons. The horsepower was based on the work a good horse could do lifting coal out of a mine. A good horse could lift 275 pounds of coal at 2.0 ft/s, so it could do 550 ft•lb/s. This is equivalent to 746 J/s or 746 W.

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**Watts These bulbs all consume different amounts of power.**

A 100 watt bulb consumes 100 joules of energy every second. Remind students that a joule is the energy required to lift an apple a distance of a meter. Light bulbs are also rated for the amount of light they produce. Fluorescent bulbs using 22 watts can produce as much light as incandescent bulbs using 100 watts.

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**Classroom Practice Problems**

Two horses pull a cart. Each exerts a force of N at a speed of 2.0 m/s for 10.0 min. Calculate the power delivered by the horses. How much work is done by the two horses? Answers: 1.0 x 103 W and 6.0 x 105 J These two problems can be done in either order. Most students will use P = Fv, and then use P = W/t to get the work done. However, it is worth showing them that they could calculate the work done by using W = Fd, where d = 2.0 m/s s = 1.2 x 103 m. After getting the work, then they could use the fundamental definition of power (P=W/t) to get the power.

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Now what do you think? Two cars are identical with one exception. One of the cars has a more powerful engine. How does having more power make the car behave differently? What does power mean? What units are used to measure power? Having more power will allow the car to consume more energy per second. This will produce greater accelerations and higher energy costs. Automobiles generally measure the power in hp. However, small motors are sometimes rated in watts and sometimes rated in hp.

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Preview Multiple Choice Short Response Extended Response

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Multiple Choice 1. In which of the following situations is work not being done? A. A chair is lifted vertically with respect to the floor. B. A bookcase is slid across carpeting. C. A table is dropped onto the ground. D. A stack of books is carried at waist level across a room.

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**Multiple Choice, continued**

Use the graph below to answer questions 3–5. The graph shows the energy of a 75 g yo-yo at different times as the yo-yo moves up and down on its string.

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**Multiple Choice, continued**

3. By what amount does the mechanical energy of the yo-yo change after 6.0 s? A. 500 mJ B. 0 mJ C. –100 mJ D. –600 mJ

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**Multiple Choice, continued**

4. What is the speed of the yo-yo after 4.5 s? F. 3.1 m/s G. 2.3 m/s H. 3.6 m/s J. 1.6 m/s

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**Multiple Choice, continued**

5. What is the maximum height of the yo-yo? A m B m C m D m

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**Multiple Choice, continued**

6. A car with mass m requires 5.0 kJ of work to move from rest to a final speed v. If this same amount of work is performed during the same amount of time on a car with a mass of 2m, what is the final speed of the second car?

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**Multiple Choice, continued**

Use the passage below to answer questions 7–8. A 70.0 kg base runner moving at a speed of 4.0 m/s begins his slide into second base. The coefficient of friction between his clothes and Earth is His slide lowers his speed to zero just as he reaches the base. 7. How much mechanical energy is lost because of friction acting on the runner? A J B. 560 J C. 140 J D. 0 J

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**Multiple Choice, continued**

Use the passage below to answer questions 7–8. A 70.0 kg base runner moving at a speed of 4.0 m/s begins his slide into second base. The coefficient of friction between his clothes and Earth is His slide lowers his speed to zero just as he reaches the base. 8. How far does the runner slide? F m G m H m J. 1.2 m

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**Multiple Choice, continued**

Use the passage below to answer questions 9–10. A spring scale has a spring with a force constant of 250 N/m and a weighing pan with a mass of kg. During one weighing, the spring is stretched a distance of 12 cm from equilibrium. During a second weighing, the spring is stretched a distance of 18 cm.

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Extended Response Base your answers to questions 14–16 on the information below. A projectile with a mass of 5.0 kg is shot horizontally from a height of 25.0 m above a flat desert surface. The projectile’s initial speed is 17 m/s. Calculate the following for the instant before the projectile hits the surface: 14. The work done on the projectile by gravity. 1200J

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**Extended Response, continued**

Base your answers to questions 14–16 on the information below. A projectile with a mass of 5.0 kg is shot horizontally from a height of 25.0 m above a flat desert surface. The projectile’s initial speed is 17 m/s. Calculate the following for the instant before the projectile hits the surface: 15. The change in kinetic energy since the projectile was fired. 1200 j

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**Extended Response, continued**

Base your answers to questions 14–16 on the information below. A projectile with a mass of 5.0 kg is shot horizontally from a height of 25.0 m above a flat desert surface. The projectile’s initial speed is 17 m/s. Calculate the following for the instant before the projectile hits the surface: 16. The final kinetic energy of the projectile. 1900 J

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**Extended Response, continued**

17. A skier starts from rest at the top of a hill that is inclined at 10.5° with the horizontal. The hillside is m long, and the coefficient of friction between the snow and the skis is At the bottom of the hill, the snow is level and the coefficient of friction is unchanged. How far does the skier move along the horizontal portion of the snow before coming to rest? Show all of your work. 290 m

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Chapter 6: Conservation of Energy

Chapter 6: Conservation of Energy

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