2012 Semifinal Exam 1 AAPT UNITEDSTATES PHYSICS TEAM AIP 2012 Semifinal Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor This examination consists of two parts. Part A has four questions and is allowed 90 minutes. Part B has two questions and is allowed 90 minutes. The first page that follows is a cover sheet.Examinees may keep the cover sheet for both parts of the exam. The parts are then identified by the center header on each page.Examinees are only allowed to do one part at a time,and may not work on other parts,even if they have time remaining Allow 90 minutes to complete Part A.Do not let students look at Part B.Collect the answers to Part A before allowing the examinee to begin Part B.Examinees are allowed a 10 to 15 minute break between parts A and B. Allow 90 minutes to complete Part B.Do not let students go back to Part A. Ideally the test supervisor will divide the question paper into 3 parts:the cover sheet (page 2),Part A (pages 3-4),and Part B(pages 6-7).Examinees should be provided parts A and B individually,although they may keep the cover sheet. The supervisor must collect all examination questions,including the cover sheet,at the end of the exam,as well as any scratch paper used by the examinees.Examinees may not take the exam questions.The examination questions may be returned to the students after April 1.2012. Examinees are allowed calculators,but they may not use symbolic math,programming,or graphic features of these calculators.Calculators may not be shared and their memory must be cleared of data and programs.Cell phones,PDA's or cameras may not be used during the exam or while the exam papers are present.Examinees may not use any tables,books, or collections of formulas. Please provide the examinees with graph paper for Part A. Copyright C2012 American Association of Physics Teachers
2012 Semifinal Exam 1 AAPT UNITED STATES PHYSICS TEAM AIP 2012 Semifinal Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor • This examination consists of two parts. • Part A has four questions and is allowed 90 minutes. • Part B has two questions and is allowed 90 minutes. • The first page that follows is a cover sheet. Examinees may keep the cover sheet for both parts of the exam. • The parts are then identified by the center header on each page. Examinees are only allowed to do one part at a time, and may not work on other parts, even if they have time remaining. • Allow 90 minutes to complete Part A. Do not let students look at Part B. Collect the answers to Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15 minute break between parts A and B. • Allow 90 minutes to complete Part B. Do not let students go back to Part A. • Ideally the test supervisor will divide the question paper into 3 parts: the cover sheet (page 2), Part A (pages 3-4), and Part B (pages 6-7). Examinees should be provided parts A and B individually, although they may keep the cover sheet. • The supervisor must collect all examination questions, including the cover sheet, at the end of the exam, as well as any scratch paper used by the examinees. Examinees may not take the exam questions. The examination questions may be returned to the students after April 1, 2012. • Examinees are allowed calculators, but they may not use symbolic math, programming, or graphic features of these calculators. Calculators may not be shared and their memory must be cleared of data and programs. Cell phones, PDA’s or cameras may not be used during the exam or while the exam papers are present. Examinees may not use any tables, books, or collections of formulas. • Please provide the examinees with graph paper for Part A. Copyright c 2012 American Association of Physics Teachers
2012 Semifinal Exam Cover Sheet 2 AAPT UNITEDSTATES PHYSICS TEAM AIP 2012 Semifinal Exam INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN Work Part A first.You have 90 minutes to complete all four problems.Each question is worth 25 points.Do not look at Part B during this time. After you have completed Part A you may take a break. Then work Part B.You have 90 minutes to complete both problems.Each question is worth 50 points.Do not look at Part A during this time. Show all your work.Partial credit will be given.Do not write on the back of any page.Do not write anything that you wish graded on the question sheets. Start each question on a new sheet of paper.Put your AAPT ID number,your name,the question number and the page number/total pages for this problem,in the upper right hand corner of each page.For example, AAPT ID# Doe,Jamie A1-1/3 A hand-held calculator may be used.Its memory must be cleared of data and programs.You may use only the basic functions found on a simple scientific calculator.Calculators may not be shared.Cell phones,PDA's or cameras may not be used during the exam or while the exam papers are present.You may not use any tables,books,or collections of formulas. Questions with the same point value are not necessarily of the same difficulty. In order to maintain exam security,do not communicate any information about the questions (or their answers/solutions)on this contest until after April 1,2012. Possibly Useful Information.You may use this sheet for both parts of the exam. g=9.8 N/kg G=6.67×10-11Nm2/kg2 k=1/4re0=8.99×109N.m2/c2 km=4o/4r=10-7T.m/A c=3.00×108m/s k3=1.38×10-23J/K NA=6.02×1023(mol)-1 R=NA=8.31J/mol.K)） o=5.67×10-8J/(s·m2.K4) e=1.602×10-19C 1eV=1.602×10-19J h=6.63×10-34J.s=4.14×10-15eV.s me=9.109×10-31kg=0.511MeV/c2(1+x)m≈1+nx for<1 sin0≈0-03forl9l≤1 cos0≈1-02forl0l<1 Copyright C2012 American Association of Physics Teachers
2012 Semifinal Exam Cover Sheet 2 AAPT UNITED STATES PHYSICS TEAM AIP 2012 Semifinal Exam INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN • Work Part A first. You have 90 minutes to complete all four problems. Each question is worth 25 points. Do not look at Part B during this time. • After you have completed Part A you may take a break. • Then work Part B. You have 90 minutes to complete both problems. Each question is worth 50 points. Do not look at Part A during this time. • Show all your work. Partial credit will be given. Do not write on the back of any page. Do not write anything that you wish graded on the question sheets. • Start each question on a new sheet of paper. Put your AAPT ID number, your name, the question number and the page number/total pages for this problem, in the upper right hand corner of each page. For example, AAPT ID # Doe, Jamie A1 - 1/3 • A hand-held calculator may be used. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones, PDA’s or cameras may not be used during the exam or while the exam papers are present. You may not use any tables, books, or collections of formulas. • Questions with the same point value are not necessarily of the same difficulty. • In order to maintain exam security, do not communicate any information about the questions (or their answers/solutions) on this contest until after April 1, 2012. Possibly Useful Information. You may use this sheet for both parts of the exam. g = 9.8 N/kg G = 6.67 × 10−11 N · m2/kg2 k = 1/4π0 = 8.99 × 109 N · m2/C 2 km = µ0/4π = 10−7 T · m/A c = 3.00 × 108 m/s kB = 1.38 × 10−23 J/K NA = 6.02 × 1023 (mol)−1 R = NAkB = 8.31 J/(mol · K) σ = 5.67 × 10−8 J/(s · m2 · K4 ) e = 1.602 × 10−19 C 1eV = 1.602 × 10−19 J h = 6.63 × 10−34 J · s = 4.14 × 10−15 eV · s me = 9.109 × 10−31 kg = 0.511 MeV/c 2 (1 + x) n ≈ 1 + nx for |x| 1 sin θ ≈ θ − 1 6 θ 3 for |θ| 1 cos θ ≈ 1 − 1 2 θ 2 for |θ| 1 Copyright c 2012 American Association of Physics Teachers
2012 Semifinal Exam Part A 3 Part A Question Al A newly discovered subatomic particle,the S meson,has a mass M.When at rest,it lives for exactly T =3 x 10-8 seconds before decaying into two identical particles called P mesons(peons?) that each have a mass of aM. a.In a reference frame where the S meson is at rest,determine i.the kinetic energy, ii.the momentum,and iii.the velocity of each P meson particle in terms of M,a,the speed of light c,and any numerical constants. b.In a reference frame where the S meson travels 9 meters between creation and decay,determine i.the velocity and ii.kinetic energy of the S meson. Write the answers in terms of M,the speed of light c,and any numerical constants. Solution a.Apply conservation of four momentum.For the S meson,we have psc=(Es;0) and for the two P mesons we have ppc=(Ep,±p), where p is the magnitude of the (relativistic)three momentum of the P mesons. This yields Es =2EP We must also satisfy the relation E2 p2c2+m2c4 for each particle,so Es2=M2c4 and Ep2 p2c2+a2M2c4. Therefore,the kinetic energy of each P meson is Kp-Ep-oMe-iMle-aNe- 3-awe Copyright C2012 American Association of Physics Teachers
2012 Semifinal Exam Part A 3 Part A Question A1 A newly discovered subatomic particle, the S meson, has a mass M. When at rest, it lives for exactly τ = 3 × 10−8 seconds before decaying into two identical particles called P mesons (peons?) that each have a mass of αM. a. In a reference frame where the S meson is at rest, determine i. the kinetic energy, ii. the momentum, and iii. the velocity of each P meson particle in terms of M, α, the speed of light c, and any numerical constants. b. In a reference frame where the S meson travels 9 meters between creation and decay, determine i. the velocity and ii. kinetic energy of the S meson. Write the answers in terms of M, the speed of light c, and any numerical constants. Solution a. Apply conservation of four momentum. For the S meson, we have pSc = (Es, 0) and for the two P mesons we have pP c = (Ep, ±p), where p is the magnitude of the (relativistic) three momentum of the P mesons. This yields ES = 2EP We must also satisfy the relation E 2 = p 2 c 2 + m2 c 4 for each particle, so ES 2 = M2 c 4 and EP 2 = p 2 c 2 + α 2M2 c 4 . Therefore, the kinetic energy of each P meson is KP = EP − αM c2 = 1 2 M c2 − αM c2 = 1 2 − α M c2 . Copyright c 2012 American Association of Physics Teachers
2012 Semifinal Exam Part A 4 Square the energy conservation expression,and combine with the momentum/energy/mass relations: 2m2e4=p22+a2m2e4, a2Mc2 =pc. The velocity of each P meson will be found from the relativistic three momentum, p=myu and the relativistic energy, E=me2 so mcyu E Yme2 =. Putting in the values for the P meson, ev1-4a2 b.From relativistic kinematics, d=ut =vYT, So v d Call this k for now.Then k =By, k2=822, k2= 32 1-321 k21-32)=32, k2=(1+2)32, k2 1+k2 =32 Combine, d2 v=cV22+2 Copyright C2012 American Association of Physics Teachers
2012 Semifinal Exam Part A 4 Square the energy conservation expression, and combine with the momentum/energy/mass relations: 1 4 M2 c 4 = p 2 c 2 + α 2M2 c 4 , 1 4 − α 2 M2 c 4 = p 2 c 2 r 1 4 − α2M c2 = pc. The velocity of each P meson will be found from the relativistic three momentum, p = mγv and the relativistic energy, E = γmc2 so pc E = mcγv γmc2 = β. Putting in the values for the P meson, v = c q 1 4 − α2M c2 1 2M c2 = c p 1 − 4α2 b. From relativistic kinematics, d = vt = vγτ, so v c γ = d cτ Call this k for now. Then k = βγ, k 2 = β 2 γ 2 , k 2 = β 2 1 − β 2 , k 2 (1 − β 2 ) = β 2 , k 2 = (1 + k 2 )β 2 , k 2 1 + k 2 = β 2 . Combine, v = c r d 2 c 2τ 2 + d 2 Copyright c 2012 American Association of Physics Teachers
2012 Semifinal Exam Part A 5 SO 92 v=Vg+9=反 Then Y=1/V1-1/2=2 It isn't much work to find the kinetic energy, K=y-1)Mc2=(W2-1)Mc2. c.This is a velocity addition problem,so US +US w=1+s212 or,using the numbers from the first part of the problem, V1-4a2 U= 1-2a2 Copyright C2012 American Association of Physics Teachers
2012 Semifinal Exam Part A 5 so v = c r 9 2 9 2 + 92 = c √ 2 Then γ = 1/ p 1 − 1/2 = √ 2 It isn’t much work to find the kinetic energy, K = (γ − 1)M c2 = (√ 2 − 1)M c2 . c. This is a velocity addition problem, so v = vS + vS 1 + vS 2/c2 or, using the numbers from the first part of the problem, v = c √ 1 − 4α2 1 − 2α2 Copyright c 2012 American Association of Physics Teachers