IOAA 2018, Теория — Задача 4. Height of a Hill

Theoretical Competition · 10 т.

Указания
  1. The theoretical competition will be 5 hours in duration and is marked out of a total of 300 points.
  2. There are Answer Sheets for carrying out detailed work/rough work. On each Answer Sheet, please fill in
    • Student Code
    • Question No.
    • Page no. and total number of pages.
  3. Start each problem on a separate Answer Sheet. Please write only on the printed side of the sheet. Do not use the reverse side. If you have written something on any sheet which you do not want to be marked, cross it out.
  4. Please remember that the graders may not understand your language. As far as possible, write your solutions only using mathematical expressions and numbers. If it is necessary to explain something in words, please use short phrases (if possible in English).
  5. You are not allowed to leave your working desk without permission. If you need any assistance (malfunctioning calculator, need to visit a restroom, need more Answer Sheets, etc.), please put up your hand to signal the proctor.
  6. The beginning and end of the competition will be indicated by a long sound signal. Additionally, there will be a short sound signal fifteen minutes before the end of the competition (before the final long sound signal).
  7. At the end of the competition you must stop writing immediately. Sort and put your sheets in separate stacks,
    • (a) Stack 1: answer sheets of part 1
    • (b) Stack 2: answer sheets of part 2
    • (c) Stack 3: answer sheets of part 3
    • (d) Stack 4: question papers and paper sheets that you do not want to be graded.
  8. Wait at your table until your envelope is collected. Once all envelopes are collected, your student guide will escort you out of the competition room.
  9. A list of constants for this competition is given on the next page.

Point distribution of this exam

Problem numberPoints
T110
T210
T310
T410
T510
T625
T725
T825
T925
T1075
T1175
Total300
Table of constants
Mass (M⊕M_{\oplus})5.98×10245.98 \times 10^{24} kgEarth
Radius (R⊕R_{\oplus})6.38×1066.38 \times 10^{6} m
Acceleration of gravity (gg)9.81 ms−2^{-2}
Obliquity of Ecliptic23°27′
Length of Tropical Year365.2422 mean solar days
Length of Sidereal Year365.2564 mean solar days
Albedo0.39
Mass (MM☾)7.35×10227.35 \times 10^{22} kgMoon
Radius (RR☾)1.74×1061.74 \times 10^{6} m
Mean Earth-Moon distance3.84×1083.84 \times 10^{8} m
Orbital inclination with the Ecliptic5.14°
Albedo0.14
Apparent magnitude (mean full moon)−12.74
Mass (M⊙M_{\odot})1.99×10301.99 \times 10^{30} kgSun
Radius (R⊙R_{\odot})6.96×1086.96 \times 10^{8} m
Luminosity (L⊙L_{\odot})3.83×10263.83 \times 10^{26} W
Absolute Magnitude4.80 mag
Surface Temperature5772 K
Angular diameter at Earth30′
Orbital velocity in Galaxy220 kms−1^{-1}
Distance from Galactic center8.5 kpc
1 au1.50×10111.50 \times 10^{11} mPhysical constants
1 pc206265 au
Gravitational constant (GG)6.67×10−116.67 \times 10^{-11} Nm2^{2}kg−2^{-2}
Planck constant (hh)6.62×10−346.62 \times 10^{-34} Js
Boltzmann constant (kBk_{\mathrm{B}})1.38×10−231.38 \times 10^{-23} JK−1^{-1}
Stefan-Boltzmann constant (σ\sigma)5.67×10−85.67 \times 10^{-8} Wm−2^{-2}K−4^{-4}
Hubble constant (H0H_0)67.8 kms−1^{-1}Mpc−1^{-1}
Speed of light in vacuum (cc)299792458 ms−1^{-1}
Magnetic Permeability of free space (μ0\mu_0)4π×10−74\pi \times 10^{-7} Hm−1^{-1}
1 Jansky (Jy)10−2610^{-26} W m−2^{-2}Hz−1^{-1}

Rayleigh-Jeans law is given by Bν=2kBTc2ν2B_\nu = \frac{2k_BT}{c^2}\nu^2, which is the power emitted per unit emitting area, per steradian, per unit frequency.

Условие

Two friends wanted to measure the height of the hill next to their village (latitude φ=40°). One of the friends climbed to the top of the hill and she agreed to send a light signal to her friend in the village as soon as she sees the sunset. On March 21, when they did this experiment, the friend in village received the light signal 4.1 minutes after the sunset from the village. Estimate the height of the hill and horizon distance for the person at the hill top. Ignore atmospheric refraction.

Отговори

Решение

Оригинал в Архива: Theory Problems.pdf · официални решения: Theory Solutions.pdf