USAAAO 2019, II кръг — Задача 9

2019 National Astronomy Olympiad (NAO) · 15 т.

Instructions (Please Read Carefully)

(Start the exam after reading the instructions carefully and affirming that the contestant understood the instructions given.)

The top 5 eligible scorers on the NAO will be invited to represent USA at the next IOAA. In order to qualify for the national team, you must be a high school student with US citizenship or permanent residency.

This exam consists of 3 parts: Short Questions (7 points each), Medium Questions (15 points each) and Long Questions (25 points each). The maximum score is 153 points.

The test must be completed within 2.5 hours (150 minutes). The proctor should mark the start and end time of the exam on the front page.

Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. The contestant’s full name in capital letters should appear at the top of each solution page. If the contestant uses scratch papers, those should be labeled with the contestant’s name as well and marked as ”scratch paper” at the top of the page. Scratch paper will not be graded. Partial credit will be available given that correct and legible work was displayed in the solution. This exam document, solution, and all the scratch paper used should be turned in to the proctor at the end of the exam.

This is a written exam. Contestants can only use a scientific calculator for this exam. A table of physical constants will be provided. Discussing the problems with other people is strictly prohibited in any way until the end of the examination period on March 8th. Receiving any external help during the exam is strictly prohibited. This means that the only allowed items are: a scientific calculator, the provided table of constants, a pencil (or pen), an eraser, blank sheets of papers, and the exam. No books, notes, laptops, mobile phones, or any other devices are allowed while taking the exam.

After reading the instructions, please make sure to sign at the bottom of this page, affirming that:

  1. All work on this exam is mine.
  2. I took this exam under a proctor’s supervision.
  3. I did not receive any external aids besides the materials provided.
  4. I am not allowed to discuss the test with others throughout the period of this examination.
  5. Failure to follow these rules will lead to disqualification from the exam.

To be completed by the proctor:

Full name (First, Last): ______

Position (e.g. Physics Teacher at High School x): ______

Email address: ______

Student began exam at (hh:mm): ______

Student submitted the exam at (hh:mm): ______

Signature: ______ Date (mm/dd/yy): ______

To be completed by the student:

Last name: ______

First name: ______ Middle name: ______

Date of birth (mm/dd/yy): ______ Email address: ______

Signature: ______ Date (mm/dd/yy): ______

Table of Physical Constants
ConstantSymbolValue
Electron rest massmem_e9.109×10−319.109\times10^{-31} kg
Proton rest massMpM_p1.6726×10−271.6726\times10^{-27} kg
Electronic chargeee1.6022×10−191.6022\times10^{-19} C
Speed of light in free spacecc2.9979×1082.9979\times10^8 m s−1^{-1}
Permeability of free spaceμ0\mu_04π×10−74\pi\times10^{-7} H m−1^{-1}
Permittivity of free spaceϵ0\epsilon_08.854×10−128.854\times10^{-12} F m−1^{-1}
Planck’s constanthh6.626×10−346.626\times10^{-34} J s
Reduced Planck’s constantℏ=h/2π\hbar=h/2\pi1.0546×10−341.0546\times10^{-34} J s
ℏc\hbar c197.33 MeV fm
Boltzmann’s constantkBk_B1.3807×10−231.3807\times10^{-23} J K−1^{-1}
Gas constantR=kB/mH\mathcal R=k_B/m_H8.250×1038.250\times10^3 J kg−1^{-1} K−1^{-1}
Molar gas constantRR8.315 J mol−1^{-1} K−1^{-1}
Avogadro’s numberNAN_A6.022×10236.022\times10^{23} mol−1^{-1}
Standard molar volume22.414×10−322.414\times10^{-3} m3^3 mol−1^{-1}
Unified atomic mass unit (12^{12}C scale)uu931.5931.5 MeV/c2=1.660538×10−27/c^2=1.660538\times10^{-27} kg
Mass of hydrogen atommHm_H1.0078u=1.6735×10−271.0078u=1.6735\times10^{-27} kg
Bohr magnetonμB\mu_B9.274×10−249.274\times10^{-24} A m2^2 or J T−1^{-1}
Nuclear magnetonμN\mu_N5.051×10−275.051\times10^{-27} A m2^2 or J T−1^{-1}
Proton magnetic momentμp\mu_p2.7928μN2.7928\mu_N
Neutron magnetic momentμn\mu_n−1.9130μN-1.9130\mu_N
Bohr radiusa0a_05.292×10−115.292\times10^{-11} m
Fine structure constantα=e2/(4πϵ0ℏc)\alpha=e^2/(4\pi\epsilon_0\hbar c)(137.04)−1(137.04)^{-1}
Compton wavelength of electronλC=h/(mec)\lambda_C=h/(m_ec)2.4263×10−122.4263\times10^{-12} m
Rydberg’s constantR∞R_\infty1.0974×1071.0974\times10^7 m−1^{-1}
R∞hcR_\infty hc13.606 eV
Stefan-Boltzmann constantσ\sigma5.671×10−85.671\times10^{-8} W m−2^{-2} K−4^{-4}
Radiation density constanta=4σ/ca=4\sigma/c7.561×10−167.561\times10^{-16} J m−3^{-3} K−4^{-4}
Gravitational constantGG6.673×10−116.673\times10^{-11} N m2^2 kg−2^{-2}

Rest masses of some leptons and hadrons in MeV/c2c^2:

e±e^{\pm} 0.5110, μ±\mu^{\pm} 105.66, τ±\tau^{\pm} 1777, π0\pi^0 134.98, π±\pi^{\pm} 139.57, K±K^{\pm} 493.7, K0K^0 497.7, η\eta 547, D0D^0 1865, D±D^{\pm} 1869, p 938.3, n 939.6, Λ0\Lambda^0 1115.7, Σ+\Sigma^+ 1189, Σ0\Sigma^0 1193, Σ−\Sigma^- 1197, Ξ0\Xi^0 1315, Ξ−\Xi^- 1321, Ω−\Omega^- 1672, Z0Z^0 91.187×10391.187\times10^3, W±W^{\pm} 80.41×10380.41\times10^3.

QuarkChargeI3I_3SSCCBBTTMass (GeV/c2c^2)
u+2/3+2/3+1/2+1/20000∼0.003\sim0.003
d−1/3-1/3−1/2-1/20000∼0.006\sim0.006
c+2/3+2/300+100∼1.25\sim1.25
s−1/3-1/30−1000∼0.11\sim0.11
t+2/3+2/30000+1174.3
b−1/3-1/3000−104.2
Astrophysical Data — Other data and conversion factors — Trigonometrical identities — Prefixes

Astrophysical Data

QuantitySymbolValue
Astronomical unitAU1.496×10111.496\times10^{11} m
Parsecpc3.086×10163.086\times10^{16} m
Luminosity of SunL⊙L_\odot3.85×10263.85\times10^{26} W
Mass of SunM⊙M_\odot1.989×10301.989\times10^{30} kg
Radius of SunR⊙R_\odot6.96×1086.96\times10^8 m
Mass of EarthMEM_E5.9742×10245.9742\times10^{24} kg
Radius of EarthRER_E6.3781×1066.3781\times10^6 m

Other data and conversion factors

QuantitySymbolValue
1 ångstromÅ10−1010^{-10} m
1 fermifm10−1510^{-15} m
1 barnb10−2810^{-28} m2^2
1 pascalPa1 N m−2^{-2}
1 standard atmosphere1.0132×1051.0132\times10^5 Pa
Standard acceleration due to gravitygg9.807 m s−2^{-2}
1 electron volteV1.6022×10−191.6022\times10^{-19} J
eV/hchc8.065×1058.065\times10^5 m−1^{-1}
eV/kBk_B1.1604×1041.1604\times10^4 K
Wavelength of 1 eV photon1.2399×10−61.2399\times10^{-6} m

Trigonometrical identities

sin⁡(θ+ϕ)=sin⁡θcos⁡ϕ+cos⁡θsin⁡ϕ\sin(\theta+\phi)=\sin\theta\cos\phi+\cos\theta\sin\phi
cos⁡(θ+ϕ)=cos⁡θcos⁡ϕ−sin⁡θsin⁡ϕ\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi
sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
cos⁡α−cos⁡β=2sin⁡α+β2sin⁡β−α2\cos\alpha-\cos\beta=2\sin\frac{\alpha+\beta}{2}\sin\frac{\beta-\alpha}{2}

In a triangle ABC, a/sin⁡A=b/sin⁡B=c/sin⁡Ca/\sin A=b/\sin B=c/\sin C

and a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A.

Prefixes

SymbolPrefixFactor
Ttera101210^{12}
Ggiga10910^9
Mmega10610^6
kkilo10310^3
ccenti10−210^{-2}
mmilli10−310^{-3}
μ\mumicro10−610^{-6}
nnano10−910^{-9}
ppico10−1210^{-12}
ffemto10−1510^{-15}
Archive constants sheet

The archive retains this general constants sheet alongside the USAAAO collection. The visible sheet does not identify a year or an issuing body. Its values differ from some constants explicitly used in the 2019 solutions; the archive does not establish that this was the exact sheet given to the 2019 contestants.

Условие

a) Find the shortest distance from Boston (42.3601∘ N42.3601^{\circ}\ N, 71.0589∘ W71.0589^{\circ}\ W) to Beijing (39.9042∘ N39.9042^{\circ}\ N, 116.4074∘ E116.4074^{\circ}\ E)traveling along the Earth’s surface. Assume that the Earth is a uniform sphere of radius 6371 km.

b) What fraction of the path lies within the Arctic circle (north of 66.5608∘ N66.5608^{\circ}\ N)?

Отговори

Решение

Оригинал в Архива: nao_2019.pdf · официални решения: nao_2019_solution.pdf · допълнителни данни: physical-constants.pdf