Preface This Instructor’s Manual accompanies the 3rd edition of the textbook Modern Physics (John Wiley & Sons, 2012). It includes (1) explanatory material for each chapter; (2) suggested outside readings for instructor or student; (3) references to web sites or other generally available simulations of phenomena; (4) exercises that can be used in various active-engagement classroom strategies; (5) sample exam questions; and (6) complete solutions to the end-of-chapter problems in the text. Perhaps the greatest influence on my teaching in the time since the publication of the 2nd edition of this textbook (1996) has been the growth into maturity of the field of physics education research (PER). Rather than indicating specific areas of misunderstanding, PER has demonstrated that student comprehension is enhanced by any of a number of interactive techniques that are designed to engage the students and make them active participants in the learning process. The demonstrated learning improvements are robust and replicable, and they transcend differences among instructors and institutional types. In my own trajectory in this process, I have been especially influenced by the work of Lillian McDermott and her group at the University of Washington1 and Eric Mazur at Harvard University.2 I am grateful to them not only for their contributions to PER but also for their friendship over the years. With the support of a Course, Curriculum, and Laboratory Improvement grant from the National Science Foundation3, I have developed and tested a set of exercises that can be used either in class as group activities or outside of class (for example, in a Peer Instruction mode following Mazur’s format or in a Just-In-Time Teaching4 mode). These exercises are included in this Instructor’s Manual. I am grateful for the support of the National Science Foundation in enabling this project to be carried out. Two Oregon State University graduate students assisted in the implementations of these reformed teaching methods: K. C. Walsh helped with producing several simulations and illustrative materials, with implementing an interactive web site, and with corresponding developments in the laboratory that accompanies our course, and Pornrat Wattanakasiwich undertook a PER project5 for her Ph.D. that involved the observation of student reasoning about probability, which lies at the heart of most topics in modern physics. One of the major themes that has emerged from PER in the past two decades is that students can often learn successful algorithms for solving problems while lacking a fundamental understanding of the underlying concepts. The importance of the in-class or pre-class exercises is to force students to consider these concepts and to apply them to diverse situations that often cannot be analyzed with an equation. It is absolutely essential to devote class time to these exercises and to follow through with exam questions that require similar analysis and a similar articulation of the conceptual reasoning. I strongly believe that conceptual understanding is a necessary prerequisite to successful problem solving. In my own classes at Oregon State University I have repeatedly observed that improved conceptual understanding leads directly to improved problem-solving skills. In training students to reason conceptually, it is necessary to force them to verbalize their reasons for selecting a particular answer to a conceptual or qualitative question, and you will learn much from listening to or reading their arguments. A simple
multiple-choice conceptual question, either as a class exercise or a test problem, gives you insufficient insight into the students’ reasoning patterns unless you also ask them to justify their choice. Even when I have teaching assistants grade the exams in my class, I always grade the conceptual questions myself, if only to gather insight into how students reason. To save time I generally grade such questions with either full credit (correct choice of answer and more-or-less correct reasoning) or no credit (wrong choice or correct choice with incorrect reasoning). Here’s an example of why it is necessary to require students to provide conceptual arguments. After a unit on the Schrödinger equation, I gave the following conceptual test question: Consider a particle in the first excited state of a one-dimensional infinite potential energy well that extends from x = 0 to x = L. At what locations is the particle most likely to be found? The students were required to state an answer and to give their reasoning. One student drew a nice sketch of the probability density in the first excited state, correctly showing maxima at x = L/4 and x = 3L/4, and stated that those locations were the most likely ones at which to find the particle. Had I not required the reasoning, the student would have received full credit, and I would have been satisfied with the student’s understanding of the material. However, in stating the reasoning, the student demonstrated what turned out to be a surprisingly common incorrect mode of reasoning. The student apparently confused the graph of probability density with a similar sort of roller-coaster potential energy diagram from introductory physics and reasoned as follows: The particle is moving more slowly at the peaks of the distribution, so it spends more time at those locations and is thus more likely to be found there. PER follow-up work indicated that the confusion was caused in part by combining probability distributions with energy level diagrams – students were unsure of what the ordinate represented. As a result, I adopted a policy in class (and in this edition of the textbook) of never showing the wave functions or probability distributions on the same plot as the energy levels. The overwhelming majority of PER work has concerned the introductory course, but the effective pedagogic techniques revealed by that research carry over directly into the modern physics course. The collection of research directly linked to topics in modern physics is much smaller but no less revealing. The University of Washington group has produced several papers impacting modern physics, including the understanding of interference and diffraction of particles,, time and simultaneity in special relativity, and the photoelectric effect (see the papers listed on their web site, ref. 1). The PER group of Edward F. Redish at the University of Maryland has also been involved in studying the learning of quantum concepts, including the student’s prejudices from classical physics, probability, and conductivity.6 (Further work on the learning of quantum concepts has been carried out by the research groups of two of Redish’s Ph.D. students, Lei Bao at Ohio State University7 and Michael Wittmann at the University of Maine.8) Dean Zollman’s group at Kansas State University has developed tutorials and visualizations to enhance the teaching of quantum concepts at many levels (from pre-college through advanced undergraduate).9 The physics education group at the University of Colorado, led by Noah Finkelstein and Carl Wieman, is actively pursuing several research areas involving modern physics and has produced numerous research papers as well as simulations on topics in modern physics.10 Others who have conducted research on the teaching of quantum mechanics and developed interactive or evaluative materials include
Chandralekha Singh at the University of Pittsburgh11 and Richard Robinett at Pennsylvania State University.12 Classroom Materials for Active Engagement 1. Reading Quizzes I started developing the interactive classroom materials for modern physics after successfully introducing Eric Mazur’s Peer Instruction techniques into my calculus-based introductory course. Daily reading quizzes were a part of Mazur’s original classroom strategy, but recently he has adopted a system that is more like Just-in-Time Teaching. Nevertheless, I have found the reading quizzes to work effectively in both my introductory and modern physics classes, and I have continued using them. We use electronic classroom communication devices (“clickers”) to collect the responses, but in a small class paper quizzes work just as well. Originally the quizzes were intended to get students to read the textbook before coming to class, and I have over the years collected evidence that the quizzes in fact accomplish that goal. The quizzes are given just at the start of class, and I have found that they have two other salutary effects: (1) In the few minutes before the bell rings at the start of class, the students are not reading the campus newspaper or discussing last week’s football game – they are reading their physics books. (2) It takes no time at the start of class for me to focus the students’ attention or put them “in the mood” for physics; the quiz gets them settled into class and thinking about physics. The multiple-choice quizzes must be very straightforward – no complex thinking or reasoning should be required, and if a student has done the assigned reading the quiz should be automatic and should take no more than a minute or so to read and answer. Nearly all students get at least 80% of the quizzes correct, so ultimately they have little impact on the grade distribution. The quizzes count only a few percent toward the student’s total grade, so even if they miss a few their grade is not affected. 2. Conceptual Questions I spend relatively little class time “lecturing” in the traditional sense. I prefer an approach in which I prod and coach the students into learning and understanding the material. The students’ reading of the textbook is an important component of this process – I do not see the need to repeat orally everything that is already written in the textbook. (Of course, there are some topics in any course that can be elucidated only by a well constructed and delivered lecture. Separating those topics from those that the students can mostly grasp from reading the text and associated in-class follow-ups comes only from experience. Feedback obtained from the results of the conceptual exercises and from student surveys is invaluable in this process.) I usually take about 10 minutes at the beginning of class to summarize the important elements from that day’s reading. In the process I list on the board new or unfamiliar words and important formulas. These remain visible during the entire class so I can refer back to them as often as necessary. I explain any special or restrictive circumstances that accompany the use of any equation. I do not do formal mathematical derivations in class – they cause a rapid drop-off in student attention. However, I do discuss or explain mathematical processes or techniques
that might be unfamiliar to students. I encourage students to e-mail me with questions about the reading before class, and at this point I answer those questions and any new questions that may puzzle the students. The remainder of the class period consists of conceptual questions and worked examples. I follow the Peer Instruction model for the conceptual questions: an individual answer with no discussion, then small group discussions, and finally a second individual answer. On my computer I can see the histograms of the responses using the clickers, and if there are fewer than 30% or more than 70% correct answers on the first response, the group discussions normally don’t provide much benefit so I abandon the question and move on to another. During the group discussion time, I wander throughout the class listening to the comments and occasionally asking questions or giving a small nudge if I feel a particular group is moving in the wrong direction. After the second response I ask a member of the class to give the answer and an explanation, and I will supplement the student’s explanation as necessary. I generally do not show the histograms of the clicker responses to the class, neither upon the first response nor the second. The daily quiz, summary, two conceptual questions or small group projects, and one or two worked examples will normally fill a 50-minute class period, with a few minutes at the end for recapitulation or additional questions. I try to end each class period with a brief teaser regarding the next class. Some conceptual questions listed for class discussion may appear similar to those given on exams. I never use the same question for both class discussion and examination during any single term. However, conceptual questions used during one term for examinations may find use for in-class discussions during a subsequent term. 3. JITT Warm-up Exercises Just-in-Time Teaching uses web-based “warm-up” exercises to assess the student’s prior knowledge and misconceptions. The instructor can use the responses to the warm-up exercises to plan the content of the next class. The reading quizzes and conceptual questions intended for in-class activities can in many cases be used equally well for JiTT warm-up exercises. Lecture Demonstrations Demonstrations are an important part of teaching introductory physics, and physics education research has shown that learning from the demos is enhanced if they are made interactive. (For example, you can ask students to predict the response of the apparatus, discuss the predictions with a neighbor, and then to reconcile an incorrect prediction with the observation.) Unfortunately, there are few demos that can be done in the modern physics classroom. Instead, we must rely on simulations and animations. There are many effective and interesting instructional software packages on the web that can be downloaded for your class, and you can make them available for the students to use outside of class. I have listed in this Manual some of the modern physics software that I have used in my classes. Of particular interest is the open-source collection of Physlets (physics applets) covering relativity and quantum physics produced by Mario Belloni, Wolfgang Christian, and Anne J. Cox.13
Sample Test Questions This Instructor’s Manual includes a selection of sample test questions. A typical midterm exam in my Modern Physics class might include 4 multiple-choice questions (no reasoning arguments required) worth 20 points, 2 conceptual questions (another 20 points) requiring the student to select an answer from among 2 or 3 possibilities and to give the reasons for that choice, and 3 numerical problems worth a total of 60 points. Students have 1 hour and 15 minutes to complete the exam. The final exam is about 1.5 times the length of a midterm exam. One point worth considering is the use of formula sheets during exams. Over the years I have gone back and forth among many different exam systems: open book, closed book and notes, and closed book with a student-generated formula sheet. I have found that in the open book format students seem to spend a lot of time leafing through the book looking for an essential formula or constant. On the other hand, I have been amazed at how many equations a student can pack onto a single sheet of paper, and I often find myself wondering how much better such students would perform on exams if they spent as much study time working on practice problems as they do miniaturizing equations. (Students often have difficulty distinguishing important formulas, which represent a fundamental concept or relationship, from mere equations which might be intermediate steps in solving a problem or deriving a formula.) I have finally settled on a closed book format in which I supply the formula sheet with each exam. I feel this has a number of advantages: (1) It equalizes the playing field. (2) Students don’t need to waste time copying equations. (3) The formula sheet, a copy of which I give to students at the beginning of the term, itself serves as a kind of study guide. (4) Students use the formula sheet when working homework problems and studying for the exams, so they know what formulas are on the sheet and where they are located. (5) I can be sure that the formulas that students need to work the exams are included on the formula sheet. A sample copy of my formula sheet is included in this Instructor’s Manual. This Instructor’s Manual is always a work in progress. I would be grateful to receive corrections or suggestions from users. Kenneth S. Krane [email protected] References 1. http://www.phys.washington.edu/groups/peg/ 2. http://mazur.harvard.edu/education/educationmenu.php. Also see E. Mazur, Peer Instruction: A User’s Manual (Prentice Hall, 1997).
3. National Science Foundation grant DUE-0340818, “Materials for Active Engagement in the Modern Physics Course” 4. http://jittdl.physics.iupui.edu/jitt/. Also see G. Novak, A. Gavrin, W. Christian, and E. Patterson, Just in Time Teaching: Blending Active Learning with Web Technology (Benjamin Cummings, 1999). 5. “Model of Student Understanding of Probability in Modern Physics,” Pornrat Wattanakasiwich, Ph.D. dissertation, Oregon State University, 2005. 6. http://www.physics.umd.edu/perg/ 7. http://www.physics.ohio-state.edu/~lbao/ 8. http://www.umaine.edu/per/ 9. http://web.phys.ksu.edu/ 10. http://www.colorado.edu/physics/EducationIssues/index.htm 11. http://www.phyast.pitt.edu/~cls/ 12. http://www.phys.psu.edu/~rick/ROBINETT/robinett.html 13. M. Belloni, W. Christian, and A. J. Cox, Physlet Quantum Physics: An Interactive Introduction (Pearson Prentice Hall, 2006).
Chapter 1 This chapter presents a review of some topics from classical physics. I have often heard from instructors using the book that “my students have already studied a year of introductory classical physics, so they don’t need the review.” This review chapter gives the opportunity to present a number of concepts that I have found to cause difficulty for students and to collect those concepts where they are available for easy reference. For example, all students should know that kinetic energy is 12 mv 2 , but few are readily familiar with kinetic energy as p 2 / 2m , which is used more often in the text. The expression connecting potential energy difference with potential difference for an electric charge q, ΔU = qΔV , zips by in the blink of an eye in the introductory course and is rarely used there, while it is of fundamental importance to many experimental set-ups in modern physics and is used implicitly in almost every chapter. Many introductory courses do not cover thermodynamics or statistical mechanics, so it is useful to “review” them in this introductory chapter. I have observed students in my modern course occasionally struggling with problems involving linear momentum conservation, another of those classical concepts that resides in the introductory course. Although we physicists regard momentum conservation as a fundamental law on the same plane as energy conservation, the latter is frequently invoked throughout the introductory course while former appears and virtually disappears after a brief analysis of 2-body collisions. Moreover, some introductory texts present the equations for the final velocities in a one-dimensional elastic collision, leaving the student with little to do except plus numbers into the equations. That is, students in the introductory course are rarely called upon to begin momentum conservation problems with pinitial = pfinal . This puts them at a disadvantage in the application of momentum conservation to problems in modern physics, where many different forms of momentum may need to be treated in a single situation (for example, classical particles, relativistic particles, and photons). Chapter 1 therefore contains a brief review of momentum conservation, including worked sample problems and end-ofchapter exercises. Placing classical statistical mechanics in Chapter 1 (as compared to its location in Chapter 10 in the 2nd edition) offers a number of advantages. It permits the useful expression K av = 23 kT to be used throughout the text without additional explanation. The failure of classical statistical mechanics to account for the heat capacities of diatomic gases (hydrogen in particular) lays the groundwork for quantum physics. It is especially helpful to introduce the Maxwell-Boltzmann distribution function early in the text, thus permitting applications such as the population of molecular rotational states in Chapter 9 and clarifying references to “population inversion” in the discussion of the laser in Chapter 8. Distribution functions in general are new topics for most students. They may look like ordinary mathematical functions, but they are handled and interpreted quite differently. Absent this introduction to a classical distribution function in Chapter 1, the students’ first exposure to a distribution function will be |ψ|2, which layers an additional level of confusion on top of the mathematical complications. It is better to have a chance to cover some of the mathematical details at an earlier stage with a distribution function that is easier to interpret. 1
Suggestions for Additional Reading Some descriptive, historical, philosophical, and nonmathematical texts which give good background material and are great fun to read: A. Baker, Modern Physics and Anti-Physics (Addison-Wesley, 1970). F. Capra, The Tao of Physics (Shambhala Publications, 1975). K. Ford, Quantum Physics for Everyone (Harvard University Press, 2005). G. Gamow, Thirty Years that Shook Physics (Doubleday, 1966). R. March, Physics for Poets (McGraw-Hill, 1978). E. Segre, From X-Rays to Quarks: Modern Physicists and their Discoveries (Freeman, 1980). G. L. Trigg, Landmark Experiments in Twentieth Century Physics (Crane, Russak, 1975). F. A. Wolf, Taking the Quantum Leap (Harper & Row, 1989). G. Zukav, The Dancing Wu Li Masters, An Overview of the New Physics (Morrow, 1979). Gamow, Segre, and Trigg contributed directly to the development of modern physics and their books are written from a perspective that only those who were part of that development can offer. The books by Capra, Wolf, and Zukav offer controversial interpretations of quantum mechanics as connected to eastern mysticism, spiritualism, or consciousness. Materials for Active Engagement in the Classroom A. Reading Quizzes 1. In an ideal gas at temperature T, the average speed of the molecules: (1) increases as the square of the temperature. (2) increases linearly with the temperature. (3) increases as the square root of the temperature. (4) is independent of the temperature. 2. The heat capacity of molecular hydrogen gas can take values of 3R/2, 5R/2, and 7R/2 at different temperatures. Which value is correct at low temperatures? (2) 5R/2 (3) 7R/2 (1) 3R/2 Answers
B. Conceptual and Discussion Questions 1. Equal numbers of molecules of hydrogen gas (molecular mass = 2 u) and helium gas (molecular mass = 4 u) are in equilibrium in a container. (a) What is the ratio of the average kinetic energy of a hydrogen molecule to the average kinetic energy of a helium molecule? K H / K He = (1) 4 (2) 2 (3) 2 (4) 1 (5) 1/ 2 (6) 1/2 (7) 1/4
(b) What is the ratio of the average speed of a hydrogen molecule to the average speed of a helium molecule? vH / vHe = (1) 4 (2) 2 (3) 2 (4) 1 (5) 1/ 2 (6) 1/2 (7) 1/4 (C) (c) What is the ratio of the pressure exerted on the walls of the container by the hydrogen gas to the pressure exerted on the walls by the helium gas? PH / PHe = (1) 4 (2) 2 (3) 2 (4) 1 (5) 1/ 2 (6) 1/2 (7) 1/4 2. Containers 1 and 3 have volumes of 1 m3 and container 2 has a volume of 2 m3. Containers 1 and 2 contain helium gas, and container 3 contains neon gas. All three containers have a temperature of 300 K and a pressure of 1 atm.
(a) Rank the average speeds of the molecules in the containers in order from largest to smallest. (1) 1 > 2 > 3 (2) 1 = 2 > 3 (3) 1 = 2 = 3 (4) 3 > 1 > 2 (5) 3 > 1 = 2 (6) 2 > 1 > 3 (b) In which container is the average kinetic energy per molecule the largest? (1) 1 (2) 2 (3) 3 (4) 1 and 2 (5) 1 and 3 (6) All the same 3. (a) Consider diatomic nitrogen gas at room temperature, in which only the translational and rotational motions are possible. Suppose that 100 J of energy is transferred to the gas at constant volume. How much of this energy goes into the translational kinetic energy of the molecules? (1) 20 J (2) 40 J (3) 50 J (4) 60 J (5) 80 J (6) 100 J (b) Now suppose that the gas is at a higher temperature, so that vibrational motion is also possible. Compared with the situation at room temperature, is the fraction of the added energy that goes into translational kinetic energy: (1) smaller? (2) the same? (3) greater? Answers
Sample Exam Questions A. Multiple Choice 1. A container holds gas molecules of mass m at a temperature T. A small probe inserted into the container measures the value of the x component of the velocity of the molecules. What is the average value of 12 mvx2 for these molecules? (a) 32 kT (b) 12 kT (c) kT (d) 3kT 2. A container holds N molecules of a diatomic gas at temperature T. At this temperature, rotational and vibrational motions of the gas molecules are allowed. A quantity of energy E is transferred to the gas. What fraction of this added energy is responsible for increasing the temperature of the gas? (a) All of the added energy (b) 3/5 (c) 2/5 (d) 2/7 (e) 3/7 3. Two identical containers with fixed volumes hold equal amounts of Ne gas and N2 gas at the same temperature of 1000 K. Equal amounts of heat energy are then transferred to the two gases. How do the final temperatures of the two gases compare? (a) T(Ne) = T(N2) (b) T(Ne) > T(N2) (c) T(Ne) c (2) v = c (3) v v (2) v′ = v (3) v′ Curly (2) Moe Larry > Curly (6) Larry = Moe 3m 15. A particle of mass M at rest decays into two identical particles each of mass m = 0.100M that travel in opposite directions. What is the speed of these particles? (a) 0.98c (b) 0.96c (c) 0.50c (d) 0.32c 16. A certain particle has a proper lifetime of 1.00 × 10-8 s. It is moving through the laboratory at a speed of 0.85c. What distance does the particle travel in the laboratory? (a) 2.55 m (b) 4.84 m (c) 1.34 m (d) 9.19 m 17. Two particles of the same mass m and moving at the same speed v collide head-on and combine to produce only a new particle of mass M. Which of the following is correct? (a) M = 2m (b) M 2m 18. Two particles each of mass m are each moving at a speed of 0.707c directly toward one another. After the head-on collision, all that remains is a new particle of mass M. What is the mass of this new particle? (a) 0.5m (b) 1.0m (c) 2.0m (d) 2.8m (e) 4.0m
3. b 4. c 5. c 12. d 13. d 14. c
7. c 8. a 16. b 17. c
1. A particle of mass m is moving at a speed of v = 0.80c. It collides with and merges with another particle of the same mass m that is initially at rest. Is the mass of the resulting combined particle greater than, less than, or equal to 2m? EXPLAIN YOUR ANSWER. 2. A particle of mass M moving with velocity v decays into two photons of energies E1 and E2. Is the rest energy of the original particle equal to E1 + E2, less than E1 + E2, or greater than E1 + E2? EXPLAIN YOUR ANSWER. 3. Particle X1 of mass m1 is moving with speed v1 > 0.5c and kinetic energy K1. It collides with particle X2 of mass m2 that is initially at rest. The collision produces ONLY a new particle X3 of mass m3 and kinetic energy K3 (that is, X1 + X2 → X3). Is m3 greater than, less than, or equal to the sum of m1 + m2? EXPLAIN YOUR ANSWER. 4. Two spaceships A and B are approaching a space station from opposite directions. An observer on the station reports that both ships are approaching the station at the same speed v. According to classical physics, each ship would see the other moving at a speed of 2v. According to special relativity, does each ship sees the other moving at a speed that is greater than 2v, less than 2v, or equal to 2v? EXPLAIN YOUR ANSWER. Answers
1. A photon of energy 1.52 MeV collides with and scatters from an electron that is initially at rest. After the collision, the electron is observed to be moving with a speed of 0.937c at an angle of 64.1° relative to its original direction. (a) Find the energy of the scattered photon. (b) Find the direction of the scattered electron. 2. A particle of rest energy 547 MeV is moving in the x direction with a speed of 0.624c. It decays into 2 new particles, each of rest energy 106 MeV. One of the decay particles has a kinetic energy of 301 MeV and is moving at an angle of 38o relative to the x axis. (a) What is the kinetic energy of the second decay particle? (b) What is the direction of the second decay particle relative to the x axis?
3. Particle A has a rest energy of 1192 MeV and is moving through the laboratory in the positive x direction with a speed of 0.45c. It decays into particle B (rest energy = 1116 MeV) and a photon; particle A disappears in the decay process. Particle B moves at a speed of 0.40c at an angle of 3.03o with the positive x axis. The photon moves in a direction at an angle θ with the positive x axis. (a) Find the energy of the photon. (b) Find the angle θ.
4. A star is at rest relative to the Earth and at a distance of 1500 light-years. An astronaut wishes to travel from Earth to the star and age no more than 30 years during the entire round-trip journey. (a) Assuming that the journey is made at constant speed and that the acceleration and deceleration intervals are very short compared with the rest of the journey, what speed is necessary for the trip? (b) According to the astronaut, what is the distance from Earth to the star? (c) According to someone on Earth, how long does it take the astronaut to make the round trip? (d) It takes light 1500 years to travel from Earth to the star, but the astronaut makes the trip in 15 years. Does this mean that the astronaut travels faster than light? Explain your answer. 5. A particle of mass M is moving in the positive x direction with speed v. It spontaneously decays into 2 photons, with the original particle disappearing in the process. One photon has energy 233 MeV and moves in the positive x direction, and the other photon has energy 21 MeV and moves in the negative x direction. (a) What is the total relativistic energy of the particle before its decay? (b) What is the momentum of the particle before its decay? (c) Find the mass M of the particle, in units of MeV/c2. (d) Find the original speed of the particle, expressed as a fraction of the speed of light. 6. In your laboratory, you observe particle A of mass 498 MeV/c2 to be moving in the positive x direction with a speed of 0.462c. It decays into 2 particles B and C, each of mass 140 MeV/c2. Particle B moves in the negative x direction with a speed of 0.591c. (a) Find the relativistic total energy of each of the three particles. (b) Find the velocity (magnitude and direction) of particle C. (c) Your laboratory supervisor is watching this experiment from a spaceship that is moving in the positive x direction with a speed of 0.635c. What values would your supervisor measure for the velocities of particles B and C? 7. The pi meson is a particle that has a rest energy of 135 MeV. It decays into two gamma-ray photons and no other particles. (The pi meson disappears after the decay.) Suppose a pi meson is moving through the laboratory in the positive x direction at a speed of v = 0.90c. One of the decay photons moves in the positive x direction and
the other in the negative x direction. (a) What are the applicable conservation laws in this problem? Set up the equations for each applicable conservation law using the numerical values for this problem. THE ONLY UNKNOWNS IN YOUR EQUATIONS SHOULD BE THE ENERGIES OF THE TWO PHOTONS. Other than the unknown energies, all numerical factors in each equation should be evaluated. You don’t need to solve the equations, just set them up. (b) One of the photons has energy 15.5 MeV. Find the energy of the other photon, and show how all applicable conservation laws are satisfied. (c) What are the speeds of the two photons in the rest frame of the pi meson and in the laboratory frame of reference? Explain your answer. 8. A pi meson (rest energy = 140 MeV) is moving through the laboratory with a kinetic energy of 405 MeV. (a) Expressed as a fraction of the speed of light, what is the speed of the pi meson? (b) At this speed, how long a track will the pi meson leave in the laboratory during its lifetime? The lifetime of a pi meson at rest in the laboratory is 1.0 × 10-16 s. 9. A particle of rest energy 266.0 MeV is moving, according to a laboratory observer, in the x direction with a speed of 0.720c. It decays into 2 photons. Photon 1 has an energy of 260.6 MeV and travels at an angle of 26.2° with the direction of the motion of the original particle. (a) Find the energy and direction of photon 2 in this frame of reference. (b) A second observer is moving at a speed of 0.720c, so that the original particle appears to be at rest. According to this observer, what are the energy and direction of travel of photon 1? 10. A pi meson (m = 135 MeV/c2) moving through the laboratory at a speed of v = 0.998c decays into two gamma-ray photons. The two photons have equal energies Eγ and move at equal angles θ on opposite sides of the direction of motion of the original pi meson. Find Eγ and θ. 11. Space pilot Jim measures the length of his space ship to be 2450 m. The ship drifts at a constant velocity of 0.740c past a space platform from which Mary (at rest on the platform) observes its passage. (a) What is the length of the space ship according to Mary? (b) According to Mary, what is the time interval between the bow (front) of the ship passing her and the stern (rear) of the ship passing her? (c) According to Jim, what is the time interval between the bow (front) of the ship passing Mary and the stern (rear) of the ship passing her? Answers 1. (a) 0.568 MeV (b) -21.9° 2. (a) 187 MeV (b) 62.3° 3. (a)117 MeV (b) 12.7° 4. (a) 0.99995c (b) 15 l.y. (c) 3000.15 y (d) no 5. (a) 254 MeV (b) 212 MeV/c (c) 140 MeV/c2 (d) 0.834c
6. (a) 561.5 MeV, 173.6 MeV, 388.0 MeV (b) +0.933c (c) -0.891c, +0.731c 7. (a) E1 + E2 = 310 MeV, E1 − E2 = 279 MeV (b) 294.5 MeV (c) c 8. (a) 0.966c (b) 0.113 μm 9: (a) 122.7 MeV, -69.7° (b) 133.0 MeV, 60.0° 10. 1069 MeV, 3.6° 11. (a) 1650 m (b) 7.4 μs (c) 11 μs
1. Your air speed in still air is (750 km)/(3.14 h) = 238.8 km/h. With the nose of the plane pointed 22° west of north, you would be traveling at this speed in that direction if there were no wind. With the wind blowing, you are actually traveling due north at an effective speed of (750 km)/(4.32 h) = 173.6 km/h. The wind must therefore have a north-south component of (238.8 km/h)(cos 22°) − 173.6 km/h = 47.8 km/h (toward the south) and an east-west component of (238.8 km/h)(sin 22°) = 89.5 km/h (toward the east). The wind speed is thus v = (47.8 km/h)2 + (89.5 km/h)2 = 101 km/h in a direction that makes an angle of
89.5 km/h = 62° east of south 47.8 km/h
95 m = 179 s 0.53 m/s
95 m = 54 s 1.24 m/s + 0.53 m/s
95 m = 49 s 2.48 m/s − 0.53 m/s Δt = tup + tdown − 2tacross =