Chapter 2 Physics in Action SECTION 8 Potential and Kinetic Energy: Energy in the Pole Vault back to the basic concept of work, and it is shown that the work done by the runner accelerating is the source of the kinetic energy that undergoes all of these transformations. It is then noted that whenever work is done, the energy of the system changes. For a closed system such as the pole-vault example, energy may be transferred from one form to another without any net loss. If work is done, then the energy of the system may increase (as in the example of the runner doing work to accelerate) or it may decrease (for example, as a driver applies the brakes to a car). Section Overview Students design experiments to investigate how energy is converted from one form to another. They blast a penny in the air from the end of a ruler to measure the maximum height to which the penny flies upward. They roll balls across a ramp, which collide with a ruler attached to a pencil, at three different speeds. In each case, students record the amount the ruler deflects as it moves back to its original position. The experiments simulate factors that determine the amount of energy stored in a pole-vaulter’s bar that helps to launch the vaulter up in the air. Students identify different forms of energy and solve problems to see how the total energy of a system remains constant. This section also details the concept of work and how energy relates to work. The Physics Talk in this section confines itself to discussing the forms of energy and how to calculate the energy stored in kinetic, gravitational potential, and spring potential energy. A more formal treatment of conservation of energy is given in Section 9. Background Information The treatment of the joule as the unit of energy in this activity is “soft” because groundwork for defining the joule in a mechanical context has not yet been established. At this stage without benefit of a definition of the joule, dimensional analysis can be used to satisfy yourself (and, only if it seems necessary, your students) that all of the equations given in the Investigate for forms of energy at least have the same unit: Concepts involving energy, introduced in this activity, include: • energy in three forms: kinetic energy, gravitational potential energy, and spring potential energy (the energy stored in a spring), • transformations among and conservation of the above-listed forms of energy, and 2 • Kinetic energy = 12 mv , Unit: ( kg )( m/s ) • the equivalence of work and energy. Students are introduced first to the concept of kinetic energy, and then to how this form of energy may be transferred to other forms of energy. The kinetic energy of the runner in a pole vault is converted into the spring potential energy of the bent pole, which is then lost and converted into the gravitational potential energy of the rising polevaulter. These forms of energy are then related Active Physics APTE_Ch2.indd 432 2 ( ) (s ) = ( kg ) m 2 2 • Gravitational potential energy = mgh, Unit: 2 ( kg ) ⎡⎢⎣m / (s ) ⎤⎥⎦ m = ( kg ) m2 s2 • Potential energy stored in a spring = 12 kx2 (the spring constant, k, has a unit N/m which is equivalent to kg/s2), Unit: ( )( ) ( kg/s ) m = ( kg ) ( m ) (s ) 2 2 2 2 432 7/27/09 9:16:25 AM Section 8 Overview to accelerate at 1 m/s2, one joule also can be expressed as 1 kg m/s2 ·1m = ( kg ) m2 s2 . Therefore, the equations are legitimate because the unit (kg)(m2)/(s2) is equivalent to the standard energy unit, the joule. The information above will prepare you to address questions about units, if they arise. As shown on the previous page, all of the forms of energy defined in the activity share the same unit, (kg)(m2)/(s2). Because 1 joule is the amount of work done when 1 newton of force is active through one meter of distance (1 joule = 1 newton·meter ), and because one newton is the force that will cause 1 kg ( ) ( )( ) Crucial Physics • Kinetic energy is energy due to motion and is given by half the mass times the square of the velocity. • Gravitational potential energy is energy due to an object’s position in Earth’s gravitational field. It equals the mass of the object times g times the vertical height from a reference point where the gravitational potential energy has been set to zero. • Elastic potential energy is energy due to a spring being compressed or stretched. It is equal to half the spring constant times the square of the compression or stretching distance. CHAPTER 2 • Energy can be transformed from one kind to another, and if there is no external force on a system, the total energy is conserved (does not change). • Work occurs any time an object moves with a force parallel to the motion. Work is equal to the force parallel to the motion times the distance moved. • Energy is “stored work.” Work done on an object raises its energy. That energy later on can be converted back to work done by the object on another object. Learning Outcomes Location in the Section Evidence of Understanding Apply equations for kinetic energy, gravitational potential energy, and elastic potential energy. Physics Talk Sample Problems 1-3 Students apply equation of kinetic and elastic and gravitational potential energy to solve problems. Recognize that restoring forces are active when objects are deformed. Investigate Steps 1-3 and 5-9 Students recognize that when the ruler deflects from its original position energy is stored. Apply the equation for the force necessary to compress or stretch a spring. Physics Talk Sample Problem 2 Students solve problems by applying equation for the force necessary to compress the spring and the energy stored in compressed spring. Measure the transformations among the different forms of energy. Investigate Steps 1-3 and 5-9 Students recognize that the deformation of the ruler is caused by the kinetic energy of the rolling ball, and that the gravitational potential energy gained by the penny comes from the deformation of the ruler. Conduct simulations of the transformation of energy involved in the pole vault. Investigate Steps 1-9 Students design experiments to simulate how energy is transferred from one form to another. Physics to Go Questions 3 and 6-12 Physics to Go Questions 9-11 and 12.d) 433 APTE_Ch2.indd 433 Active Physics 7/27/09 9:16:25 AM Chapter 2 Physics in Action Section 8 Materials, Preparation, and Safety Teacher Preparation Materials and Equipment • Assemble the required material for the Investigate, including the coins or washers to be flipped upward by the rulers. Try out the methods the students will use to clamp the rulers to the lab tables they will use. You may need additional “filler” blocks to allow the rulers to be clamped in the vertical orientation. The rulers or plastic strips the students will use should be very flexible, and not prone to break when bent down. PLAN A Materials and Equipment Group (4 students) Ball, steel, 1 in. (diameter) 1 per group Ruler, metric, 30 cm 2 per group Meter stick, wood Class 1 per class C-clamp, steel, 3 in. 1 per group Ramp, starting, 20-degree angle 1 per group Pen, marking, felt tip 1 per group Tape, masking, 3/4 in. x 60 yds 6 per class Cards, index, unlined, 3 in. x 5 in., pkg. of 100 1 per class Coins (pennies, nickels)* Safety Requirements • Students should be cautioned to make certain their safety goggles are in place all during this Investigate. Do not use stiff plastic rulers to flip the coins upward. They may shatter into sharp pieces. Make certain the students are not above the area where the coins are being flipped as they measure the height achieved by the coin. 5 per group *Additional items needed not supplied PLAN B Materials and Equipment Group (4 students) Ball, steel, 1 in. (diameter) 1 per group Ruler, metric, 30 cm 2 per group Meter stick, wood Class • Student enthusiasm for this investigation runs very high. Caution the students that this is not a contest to see who can embed coins in the ceiling! 1 per class C-clamp, steel, 3 in. 1 per group Ramp, starting, 20-degree angle 1 per group Pen, marking, felt tip 1 per group Tape, masking, 3/4 in. x 60 yds 6 per class Cards, index, unlined, 3 in. x 5 in., pkg. of 100 1 per class Coins (pennies, nickels)* 5 per group *Additional items needed not supplied Note: Time, Preparation, and Safety requirements are based on Plan A, if using Plan B, please adjust accordingly. Time Requirement This Investigate should take approximately one class period or 40 min. Active Physics APTE_Ch2.indd 434 434 7/27/09 9:16:25 AM Section 8 Overview NOTES CHAPTER 2 435 APTE_Ch2.indd 435 Active Physics 7/27/09 9:16:25 AM Chapter 2 Physics in Action Meeting the Needs of All Students Differentiated Instruction: Augmentation and Accommodations Learning Issue Understanding vocabulary Reference Learning Outcomes Physics to Go Questions 1, 2, and 13-15 Augmentation and Accommodations Augmentation • Students are expected to explain the transformations of energy. Define transformation and provide examples of different kinds of transformations. Students should also be able to provide some good examples. Designing an experiment Investigate Steps 1 and 2 Augmentation • Students with attention and behavior concerns often focus better on tasks that they are motivated to complete because the task is high-interest or their idea. These students may be more successful in designing their own experiment rather than following the directions of the previously designed experiment. • When allowing students to design their own experiment, make sure that they have a sound experimental design and data-collection method before they begin. Following directions Investigate Steps 3-9 Augmentation • Provide a physical model of each experiment setup. • Set time limits for each step of the Investigate and use a timer to keep students on task. • Make a list, presented orally and visually, of the data that should be recorded at the end of the experiment. Accommodation • Provide a checklist that includes each step of the experiment so that students can mark off each task as it is completed. Understanding vocabulary Physics Talk Augmentation • Provide a definition and examples for the word “conserve.” Students could generate a class list of what people conserve and why. Physics Essential Questions Understanding energy transformation Physics Talk Augmentation • Energy transformation is difficult for students to understand because they cannot visibly see the energy “moving,” instead they see the effects of energy transformation. • Provide a drawing or graphic of a ball being thrown into the air that has the types of energy along the labeled trajectory. • Ask students to think of a new situation in which energy is transferred in a similar way, sketch the situation, and label the energy transformations. • Students might find it difficult to grasp that for work to be done, the object must move over a distance. Provide examples such as pushing or pulling as hard as one can on an object without it moving. Ask students if any work has been done. Learning many formulas at one time Physics Talk Accommodation • Provide a table that includes the variables, symbols, and units for each of the formulas. This graphic organizer will help students have more confidence in problem-solving. Students can identify the given values more easily when they become comfortable with the units. Active Physics APTE_Ch2.indd 436 436 7/27/09 9:16:26 AM Section 8 Overview Learning Issue Finding key phrases in word problems Reference Physics Talk Physics to Go Steps 3, 8-11 Augmentation and Accommodations Augmentation • Solving word problems is a difficult skill for most students, especially ninth graders. • Ask students to highlight or underline the key phrases in word problems to emphasize what the problem is asking them to find. • Generate a class list of key phrases for problem solving, including “how much,” “what is,” “calculate the,” “solve for the,” etc. • Mastering the above skill makes it easier for students to choose the correct formula needed to solve a problem. Accommodation • Highlight the key phrases that students are expected to find in word problems and fade this accommodation away as they become more proficient in problem-solving. Strategies for Students with Limited English-Language Proficiency catapult plausible elastic potential energy potential energy gravitational potential energy release joule simulate kinetic energy work law of conservation of energy Consider giving students a cloze activity when you reach the end of the section. Cloze activities are useful tools for summarizing material and for giving English-language learners opportunities to practice writing complete sentences using science vocabulary. Cloze activities are most effective when used frequently, to build students’ abilities with more complex sentences. Ask students to give you sentences describing what they did in the section, and telling what important lessons they learned. Their comments should include the vocabulary words listed above. You may wish to offer a first sentence as an example. For instance, “We investigated how much energy is stored in a pole vaulter’s pole.” Write simple sentences, and work them into paragraphs. Model using a topic sentence, supporting statements, and a closing sentence. Model the process of editing, in which students make corrections that improve the sentences. There is a lot of information in this section, so you will likely end up with a few paragraphs. Once the paragraphs are complete and students agree that they accurately summarize what they did and learned, have them copy down the paragraphs. Explain that there will be a brief quiz on the paragraphs tomorrow at the beginning of the class. The quiz will be on the same paragraphs that they wrote down, but with several blanks where some terms were. The students will need to fill in the blanks with the terms that are missing. Tell students how the quiz will be graded. Prepare this quiz by keying in the paragraphs and then going back and removing every vocabulary term and replacing it with a blank. Choose a variety of words to leave out—nouns, verbs, adjectives, etc. They can be science-content words, but they do not have to be. Score the quiz by allotting two points for every blank, one point for the correct term or word (or perhaps another word with the correct meaning), and a second point for the correct spelling of that term or word. 437 APTE_Ch2.indd 437 CHAPTER 2 Point out new vocabulary words in context and practice using the words as much as possible throughout the section. As you work through the section, have students write the terms in their Active Physics log and add the definitions in their own words. Active Physics 7/27/09 9:16:26 AM Chapter 2 Physics in Action SECTION 8 Teaching Suggestions and Sample Answers What Do You See? This illustration is full of images that students can ponder and discuss with their peers. Interrelated concepts are represented by humorous sketches that stimulate curiosity. Remind students that the main purpose of the What Do You See? is to get them to think about motion and relate it to the title of this section. As students feel more comfortable in expressing their idea about the topic, keep them engaged by asking open-ended question that draw attention to key concepts of this section. Students’ Prior Conceptions 4. If a fuel is used up, then it cannot be transferred from one form of energy to another. This section culminates in students’ understanding the relationships among forces, the concept of work, and transformations of gravitational and potential energy. 1. In general, students confuse the concepts of force, work, and energy; they may use them interchangeably. 2. Students associate energy only with animate objects; energy is linked with force and movement. 3. Preconceptions 1 and 2 may lead students to think of energy as a causal agent that can be stored in some objects as an ingredient or catalyst similar to a fuel that is used up. Active Physics APTE_Ch2.indd 438 The nature of these student preconceptions makes it paramount for teachers to interview students while they are doing the investigation to encourage them to see through observations, measurements, and mathematical modeling that kinetic energy, gravitational potential energy, and spring potential energy all involve forces acting over distances to transform the different forms of energy. In the absence of friction, energy is conserved in these transformations. In the presence of friction, energy is converted to heat which is no longer available to energy transformations within the system. 438 7/27/09 9:16:26 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault What Do You Think? A Physicist’s Response The speed at which the athlete can run and the runner’s kinetic energy determine the height a pole vaulter may attain in the pole vault. Although the arm strength of the pole vaulter may add a small amount to the height, it is the transformation of the athlete’s kinetic energy to gravitational potential energy that determines the height attained, not the pole length. Because humans have a limiting speed at which they can run, they also have a limit as to the height they may reach with the pole vault. Human limitations on the kinetic energy and the amount of input work that the vaulter is able to provide limit the height much more than the length of the pole. A vaulter using an 8-m pole would not be able to reach a height any higher than if he or she used a 5.5-m pole. CHAPTER 2 Ask students to answer the question without any hesitation, even if they are not sure about their responses. Encourage them to discuss their answers and record them in their Active Physics log book. Reassure your students that while they may not have the right answers at this stage, they will gradually learn more as they progress in the section. At this stage, students’ answers don’t have to be correct. What matters most is that students think and reflect with ease. Facilitate discussions in small groups as well the whole class. What Do You Think? In addition to the kinetic energy of the vaulter, the other factors that determine the height attained are the efficiency of the pole in converting kinetic energy into gravitational potential energy, and the work the vaulter is able to do with his arms at the top of the rise to gain a little bit of additional height. NOTES 439 APTE_Ch2.indd 439 Active Physics 7/27/09 9:16:26 AM Chapter 2 Physics in Action NOTES Investigate 1.a) The students record that the more the ruler is deflected, the higher the penny travels. 1.b) Students will record that if the ruler is clamped toward an end that allows Active Physics APTE_Ch2.indd 440 most of the ruler to flex, the penny will not go as high for an equal deflection since the spring constant will be lower for a “longer” ruler. 2. How the ruler is clamped to the table, how far it is deflected, and the mass of the coin determine how high the penny travels. The students will form conclusions about how high the coin goes as a function of one of the factors described above. The students should list the data that will be recorded in a table and list which variables are being kept constant to present their data in the form of a graph for quick analysis. 440 7/27/09 9:16:26 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault A more careful analysis shows that if the distance up the ramp is doubled, the speed of the ball increases by the square root of 2. To double the speed of the ball at the bottom of the ramp, the ball must be placed four times further up the ramp. Doubling the speed should approximately double the ruler deflection. 5.a) The students will record their data for ruler deflection depending upon starting position on the ramp. 5.b) 6. If students clamp less than the recommended amount of the ruler, the measurement of small deflection differences will become more difficult. 3. The student should keep all variables constant except one, and then record the height as that variable is changed. There are many variations on how the next activity can be done. Any way different speeds of the ball can be achieved is fine. Likewise, any method to determine how far the ruler deflects is also fine. Teaching Tip A track of some sort may be necessary to make certain the rolling ball hits the ruler. 4. Prior to doing Step 5, the students may expect, as they linearly increase the distance up the ramp, to see a linear increase in the speed of the ball coming down the ramp and striking the ruler. 441 APTE_Ch2.indd 441 CHAPTER 2 The students will start with a simple conception that the higher the speed, the greater the deflection of the pole. Do not discourage this view at this point. Active Physics 7/27/09 9:16:26 AM Chapter 2 Physics in Action 7. Any coin will do but uniformity of coins for each group allows a better comparison of data between groups. Teaching Tip If the penny falls off the ruler easily, the students can tape a cutdown paper cup onto the ruler to hold the penny in place until it is launched. 8. Placing a second reference ruler or straightedge next to the deflecting ruler may assist the deflectionmeasurement process. 8.a) Students record the height the penny rises for the 2-cm deflection of the ruler. It might be expected that as the deflection of the ruler is doubled, the energy in the ruler increases by a factor of 4 due 2 to the equation SPE = 12 kx , and thus the penny’s height should increase by a factor of 4. The situation in practice is more complex, however, so the students should be expected only to see an increase of more than double. 9.a) Encourage the students to take several trials for each ruler deflection to ensure uniform results. 9.b) The larger the amount of deflection, the higher the coin travels. The relationship should not be expected to be linear; that is, twice the deflection may cause the coin to travel more than twice as high. Active Physics APTE_Ch2.indd 442 9.c) The more bend in the pole, the higher in the air the pole vaulter can go. Physics Talk As students read about the law of conservation of energy, have them focus on the example of a ball that travels upward and then is acted upon by a gravitational force. Emphasize that the velocity decreasing to zero means that the kinetic energy is also zero but the ball now has maximum potential energy due to gravity. Point out that when the transformation of energy takes place the energy is not being lost, but transformed to a different form. Consider drawing two columns for the gravitational potential energy and kinetic energy on the board to show what happens at different stages of a ball’s trajectory. Calculations may be 442 7/27/09 9:16:27 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault used to show that as the ball’s gravitational energy is increasing, its kinetic energy is decreasing. Encourage students to think about the concept of energy remaining constant, regardless of its transformations. You could reemphasize the question, “Was there some quantity that was not changed in these situations?” Have students distinguish between the different forces that propel the ball, either upward or downward. Discuss the definition of work and ask students to write down the definition of work in their Active Physics log. To help relate the concept of work to energy, explain to the students that the work done by the thrower’s hand while it was in contact with the ball is equal to the ball’s increase in kinetic energy as it starts to rise. As students read about the conservation of energy, ask them to provide examples of 443 APTE_Ch2.indd 443 CHAPTER 2 situations where work is stored as elastic potential energy and then changes into kinetic energy. Discuss examples of different forms of energy transformations that an object goes through and have students take a short quiz to gauge their understanding. Concentrate on the general idea of energy being conserved as it passes through the various transformations between work and the different forms of energy. The Physics Talk provides several examples of calculating different forms of energy. It does not specifically provide numerical examples of using the conservation of energy principle to determine the energy at various stages when transformations occur. This aspect of conservation of energy will be dealt with more explicitly in the following section. Active Physics 7/27/09 9:16:28 AM Chapter 2 Physics in Action Active Physics APTE_Ch2.indd 444 444 7/27/09 9:16:28 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault CHAPTER 2 445 APTE_Ch2.indd 445 Active Physics 7/27/09 9:16:29 AM Chapter 2 Physics in Action Active Physics APTE_Ch2.indd 446 446 7/27/09 9:16:29 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault Checking Up 1. Work must be done on an object to change its total energy. 2. The penny gets kinetic energy from the elastic potential energy of the ruler. 3. The energy to bend the pole comes from the kinetic energy of the running pole vaulter, and the energy to go over the bar comes from the stored energy in the pole and the work done by the pole vaulter’s muscles. CHAPTER 2 4. The unit for all kinds of energy and work is joules, which means they are all equivalent forms. 447 APTE_Ch2.indd 447 Active Physics 7/27/09 9:16:30 AM Chapter 2 Physics in Action Active Physics Plus Students explore the change in energy due to work done by learning the strategies of solving sample problems. As students solve each problem, ask them to focus on how one type of energy is equated to another to solve a variable comprising the energy equation. Draw their attention to the statement “energy is stored work,” and have them explain how this concept is shown by sample problems. Active Physics APTE_Ch2.indd 448 448 7/27/09 9:16:30 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault CHAPTER 2 449 APTE_Ch2.indd 449 Active Physics 7/27/09 9:16:31 AM Chapter 2 Physics in Action Active Physics APTE_Ch2.indd 450 450 7/27/09 9:16:31 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault What Do You Think Now? Physics Essential Questions What does it mean? Although the height and the speed of the vaulter change, the sum of the gravitational potential energy, GPE, and the kinetic energy, KE, remains constant. How do you know? The spring toy had kinetic energy at one point and then had an equal amount of gravitational potential energy at another point. Why do you believe? Conservation of energy holds for all events, including all sports. Why should you care? When a pop fly is hit in baseball, the ball first has kinetic energy. There is a loss in kinetic energy as the ball rises and an equal gain in gravitational potential energy. 451 APTE_Ch2.indd 451 CHAPTER 2 Have students review their answers in light of what they have learned so far in this section. Ask them questions that lead students to answer the What Do You Think Now? questions. Share the Physicist’s Response with them. Invite students to discuss their answers. Remind them that at this stage they should review and revise their original responses. While students are revising their responses, emphasize how they can use their knowledge of energy conservation to determine the maximum height pole vaulters can attain. Active Physics 7/27/09 9:16:32 AM Chapter 2 Physics in Action Reflecting on the Section and the Challenge This is the time for students to reflect on what their commentary could cover if they were sportscasting a pole-vault event. They should be able to describe the law of conservation of energy in relation to other sports and include it in their voice-over narration. Have the students write a short summary of their sportscast, so that they can discuss it with their peers and incorporate physics concepts that apply to the sport they want to include in their Chapter Challenge. Point out that their work will be evaluated on the basis of how clearly they explain the transformation of energy in a system after force is applied. Use the terms force, work, and energy frequently in your discussion so that students can relate to each term in context. Physics to Go 1. The shot putter imparts a launching speed to the projectile, which is kinetic energy. By giving the shot two speeds in the same direction, the speeds add together. One of the motions is provided by the spinning motion of the shot putter before release, and the other is provided by the thrusting action of the shot putter’s arm. In both cases, the athlete does work that is transformed into the kinetic energy that the ball has upon release. The horizontal component of the shot’s speed is maintained while the projectile is Active Physics APTE_Ch2.indd 452 in the air. The vertical component of the speed can be used to calculate the part of the kinetic energy, which will be transformed into gravitational potential energy, allowing prediction of the height to which the shot will rise at the peak of its flight. Students are more likely to write that the athlete uses muscles to transfer kinetic energy to the shot. Some of this energy makes the shot rise, converting kinetic energy into potential energy. Then the shot falls as potential energy is converted back to kinetic energy. When the shot hits the ground, the kinetic energy transforms into internal energy as the temperature of the shot and ground increases slightly. 2. In golf, the golfer does work on the golf club to give the end of the club a great deal of kinetic energy. Upon impact, the end of the golf 452 7/27/09 9:16:32 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault club does work on the golf ball, giving the golf ball kinetic energy. As with the shot after release, the golf ball has both horizontal and vertical speed components. The energy transformations are therefore the same as with the shot. 3. Lewis’ maximum speed in the dash = 12 m/s. Solving for h in the equation given in the problem: h = v 2 2 g = 7.3 m. 4. that Emma’s height was rounded to two significant figures in the above substitution. 5. 7. The amount of internal energy (due to the temperature rise) generated during the bending of the pole would “rob” part of the potential energy that can be stored in the pole by causing it to bend less than it would if no internal energy were generated and by causing the pole to straighten with less “straightening” speed than if no internal energy were generated; overall, heating of the pole would reduce the vaulter’s height. v = 2 gh = 2 10 m/s2 (6.1 m ) = 6. ( v = 2 gh = 2 10 m/s 2 ) ( 4.6 m ) = 92 m 2 /s2 = 9.6 m/s. Notice ( ) 120 m /s = 11 m/s . Therefore, Sergei’s speed was greater than Emma’s. Notice that while Sergei’s speed was only about 15% greater than Emma’s, his vault height (6.1 m compared to Emma’s 4.6 m) was about 32% higher. This is due to the squared effect of speed on kinetic energy; squaring the ratio of speeds verifies that the ratio of heights should be 1.32:1.00, as shown below: 2 2 CHAPTER 2 The vaulter’s kinetic energy is determined by the running speed and the amount of work he or she does using their arms to lift their body. This energy, which includes the work done, determines the maximum height the pole vaulter can reach, regardless of the length of the vaulting pole. The pole could, however, have an effect if it is either too short or too long. 2 ⎛ 6.1 m/s ⎞ 2 ⎜ ⎟ = (1.33) = 1.76 4 . 6 m/s ⎝ ⎠ NOTES 453 APTE_Ch2.indd 453 Active Physics 7/27/09 9:16:33 AM Chapter 2 Physics in Action NOTES 8.a) 9.b) Using the law of conservation of energy, the gravitational potential energy (GPE) at the top of the cliff is equal to the kinetic energy (KE) at the bottom. The work becomes spring potential energy (SPE). This SPE becomes kinetic energy (KE) of the arrow. GPEi + KEi = GPEf + KEf mgh + 0 = 0 + mv 1 2 ( 47 J = (0.1 kg)v 1 2 2 v= v = 2 gh v = 2 9.8 m/s KE = 12 mv 2 2 ) (100 m ) = 2(47 J) = 30.7 m/s 0.1 kg W = 12 kx2 = 8.b) 1 2 9.a) W = 12 kx2 = 1 2 mgh = 12 kx2 x= x= 10.a) 44 m/s Yes, the speed is independent of the mass. All objects fall at the same rate as long as air resistance is small enough to ignore. In the energy equations, you can see that the mass drops out. 2 The compression of the spring will store this energy as spring potential energy (SPE). You can therefore compare the GPE to the SPE and not concern yourself with the KE. 2mgh k ( ) 2 (0.04 kg ) 9.8 m/s2 (1 m ) (18 N/m ) = 0.21 m (315 N/m)(0.30 m)2 = 14.2 J 12.a) F (in newtons) = ma = 10.b) F = kx = (315 N/m)(0.30 m) = 95 N (1 kg ) (1 m/s ) = 1 N = 1 ( kg ⋅ m ) /s 11. 12.b) The gravitational potential energy (GPE) of the car will become kinetic energy (KE) of the car. This KE will then do work on the spring and compress it. 2 ( 2 ) GPE = mgh = (1 kg ) 1 m/s2 (1 m ) = (1 N )(1 m ) = 1 J (1500 N/m)(0.25 m)2 = 47 J Active Physics APTE_Ch2.indd 454 454 7/27/09 9:16:33 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault energy and temperature of the water. 14. 15. The student’s voice-over should include a description of the energy changes the ball undergoes similar to the descriptions in the answers to Questions 14 and 15. It should also include an exciting description of what the outfielder should do to catch the ball, and the work done on the mitt and the temperature change as the ball lands in the player’s mitt. 12.c) (1 kg )(1 m/s ) (1 kg )(1 m/s ) = (1 kg ) (1 m /s ) = (1 kg ⋅ m/s ) ( m ) = 1 J KE = 12 mv 2 = 2 2 1 2 2 2 2 13. When the diver walks onto the diving board, she first does work jumping in the air to increase her potential energy. When she falls back to the springboard, the potential energy turns to kinetic energy, which turns to spring potential energy as the board bends. The stored potential energy in the diving board then imparts kinetic energy to the diver causing her to rise off the board and gain GPE equal to the stored SPE that was in the diving board. The gravitational potential energy turns into kinetic energy as the diver falls to the water. When the diver strikes the water, her kinetic energy is lost and converted to thermal energy, increasing the 455 APTE_Ch2.indd 455 CHAPTER 2 When the volleyball player “sets” a ball for another player to spike, she is doing work on the ball to give it kinetic energy. This causes the ball to rise and gain potential energy. If timed correctly, a second player will jump up and strike the ball at the highest point of the ball’s rise (maximum GPE), doing work on the ball to give it kinetic energy. The ball will also gain kinetic energy as it falls back to the court, with the final kinetic energy being equal to the work done by the “spiker” plus the GPE that the ball had at the peak of its rise just before being struck. 16. Preparing for the Chapter Challenge The students should choose a sport and describe the energy changes. A sport with numerous changes would be more appropriate than a simpler one. The voice-over should include a discussion of how the energy changes form between work, GPE, KE, and possibly SPE and thermal energy to be complete. Active Physics 7/27/09 9:16:33 AM Chapter 2 Physics in Action SECTION 8 QUIZ 2-8a Blackline Master 1. In an emergency stop, a 1500-kg vehicle loses 300,000 joules of energy as it comes to rest. What was the speed of the vehicle the moment the brakes were applied? a) 10 m/s b) 14 m/s c) 20 m/s d) 25 m/s 2. If the kinetic energy of an object is 16 J when its speed is 4 m/s, what is the object’s mass? a) 0.5 kg b) 2.0 kg c) 8 kg d) 19.6 kg 3. If a 5-kg mass is raised two meters vertically from the surface of Earth, its gain in potential energy will be a) 5 J. b) 10 J. c) 20 J. d) 100 J. 4. A spring has a spring constant of 120 N/m. How much energy is stored in the spring as it is stretched a distance of 0.20 meter? a) 2.4 J b) 4.8 J c) 12 J d) 24 J 5. A 0.50-kg ball is thrown vertically upward with an initial velocity of 35 m/s. Approximately, how high will the ball rise? Active Physics APTE_Ch2.indd 456 a) 35 m b) 60 m c) 85 m d) 120 m 456 7/27/09 9:16:34 AM Section 8 Potential and Kinetic Energy: Energy in the Pole Vault SECTION 8 QUIZ ANSWERS mv 2 v = 2KE = m 2(300, 000 J) = 20 m/s (1500 kg) 1 c) KE = 2 b) KE = 12 mv 2 m = 3 d) GPE = mgh = ( 5 kg ) 9.8 m/s2 ( 2 m ) = 100 J 4 a) SPE = 12 kx2 = 12 (120 N/m)(0.20 m)2 = 2.4 J 5 b) KE = GPE gives 1 2 2KE 2(16 J) = = 2.0 kg (4 m/s)2 v2 ( 1 2 ) mv 2 = mgh Solving for h we have (35 m/s ) = 61 m v2 h= = 2 g 2 10 m/s2 2 ) CHAPTER 2 ( NOTES 457 APTE_Ch2.indd 457 Active Physics 7/27/09 9:16:34 AM

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