USAPhO 2015, Полуфинал — Задача 1. Question A1

USA Physics Olympiad Exam — Part A / Part B · 25 т.

Внимание: Бележка към темата

Original question and matching key disagree at A4: the question uses Tmax and X=C; repeated key statement uses400K and says X need not coincide with C, but actual worked key graph/argument still place C/X/Y together. Both native versions are explicitly retained. B1 question says three independent parts although only a/b exist; repeated key says two. Original mathematical/sign/symbol defects remain printed, with evidence notes.

Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

  • 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 minutes break between parts A and B.

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  • Ideally the test supervisor will divide the question paper into 4 parts: the cover sheet (page 2), Part A (pages 3-7), Part B (pages 9-11), and several answer sheets for one of the questions in Part A (pages 13-13). Examinees should be provided parts A and B individually, although they may keep the cover sheet. The answer sheets should be printed single sided!

  • 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 15, 2015.

  • 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.

USA Physics Olympiad 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 15, 2015.
Possibly Useful Information. You may use this sheet for both parts of the exam.
InformationValue
gg9.8 N/kg9.8\ \mathrm{N/kg}
k=1/(4πε0)k=1/(4\pi\varepsilon_0)8.99×109 N⋅m2/C28.99\times10^9\ \mathrm{N\cdot m^2/C^2}
cc3.00×108 m/s3.00\times10^8\ \mathrm{m/s}
NAN_A6.02×1023 (mol)−16.02\times10^{23}\ (\mathrm{mol})^{-1}
σ\sigma5.67×10−8 J/(s⋅m2⋅K4)5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}
1 eV1\ \mathrm{eV}1.602×10−19 J1.602\times10^{-19}\ \mathrm J
mem_e9.109×10−31 kg=0.511 MeV/c29.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2
sin⁡θ\sin\theta≈θ−16θ3\approx\theta-\frac16\theta^3 for ∣θ∣≪1\vert \theta\vert \ll1
GG6.67×10−11 N⋅m2/kg26.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}
km=μ0/(4π)k_m=\mu_0/(4\pi)10−7 T⋅m/A10^{-7}\ \mathrm{T\cdot m/A}
kBk_B1.38×10−23 J/K1.38\times10^{-23}\ \mathrm{J/K}
R=NAkBR=N_Ak_B8.31 J/(mol⋅K)8.31\ \mathrm{J/(mol\cdot K)}
ee1.602×10−19 C1.602\times10^{-19}\ \mathrm C
hh6.63×10−34 J⋅s=4.14×10−15 eV⋅s6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}
(1+x)n(1+x)^n≈1+nx\approx1+nx for ∣x∣≪1\vert x\vert \ll1
cos⁡θ\cos\theta≈1−12θ2\approx1-\frac12\theta^2 for ∣θ∣≪1\vert \theta\vert \ll1

Условие

Part A

Consider a particle of mass mm that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is EE and the acceleration of free fall is gg. Treat the particle as a point mass and assume the motion is non-relativistic.

a.

An estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce. Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle, determine the value of the energy EqE_q where quantum effects become important. Write your answer in terms of some or all of gg, mm, and Planck's constant hh.

b.

A second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height HH, we can write

2∫0Hp dx=(n+12)h2\int_0^H p\,dx=\left(n+\frac12\right)h

where pp is the momentum as a function of height xx above the ground, nn is a non-negative integer, and hh is Planck's constant.

i. Determine the allowed energies EnE_n as a function of the integer nn, and some or all of gg, mm, and Planck's constant hh.

ii. Numerically determine the minimum energy of a bouncing neutron. The mass of a neutron is mn=1.675×10−27 kg=940 MeV/c2m_n=1.675\times10^{-27}\ \mathrm{kg}=940\ \mathrm{MeV}/c^2; you may express your answer in either Joules or eV.

iii. Determine the bounce height of one of these minimum energy neutrons.

c.

Let E0E_0 be the minimum energy of the bouncing neutron and ff be the frequency of the bounce. Determine an order of magnitude estimate for the ratio E/fE/f. It only needs to be accurate to within an order of magnitude or so, but you do need to show work!

Решение

Оригинал в Архива: USAPhO-2015.pdf · официални решения: USAPhO-2015-Solutions.pdf