USAPhO 2021, Първи кръг — Задача 25

2021 F=maF = ma Exam · 1 т.

Указания

25 QUESTIONS - 75 MINUTES

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Use g=10g = 10 N/kg throughout this contest.
  • You may write in this booklet of questions. However, you will not receive any credit for anything written in this booklet. You may only use the scratch paper provided by the proctor.
  • This test contains 25 multiple choice questions. Select the answer that provides the best response to each question. Please be sure to use a No.2 pencil and completely fill the box corresponding to your choice. If you change an answer, the previous mark must be completely erased. Only the boxes preceded by numbers 1 through 25 are to be used on the answer sheet.
  • All questions are equally weighted, but are not necessarily of the same level of difficulty.
  • Correct answers will be awarded one point; incorrect answers or leaving an answer blank will be awarded zero points. There is no additional penalty for incorrect answers.
  • 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 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.
  • The question booklet, the answer sheet and the scratch paper will be collected at the end of this exam.
  • In order to maintain exam security, do not communicate any information about the questions (or their answers or solutions) on this contest until after February 1, 2021.

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

We acknowledge the following people for their contributions to this year’s exams (in alphabetical order):

JiaJia Dong, Mark Eichenlaub, Abi Krishnan, Kye Shi, Brian Skinner, Mike Winer and Kevin Zhou

Условие

Spaceman Fred’s trusty pellet sprayer is held at rest a distance hh away from the center of Planet Orb, which has radius R≪hR \ll h. The pellet sprayer ejects pellets radially outward, uniformly in the plane of the page. These pellets are all launched with the same speed vv, so that a pellet launched directly away from Orb by the pellet sprayer can just barely escape it. What fraction of the pellets eventually lands on Orb? (Hint: you may use the small angle approximation, sin⁡θ≈θ\sin\theta \approx \theta for θ≪1\theta \ll 1, where θ\theta is in radians.)

A circle (Planet Orb) with radius R drawn from its centre, and a vertical line of length h from the centre up to a dot marking the pellet sprayer.
Left: the pellet sprayer relative to Orb
A circle (the pellet sprayer) with eight arrows pointing radially outward at equal angles; one arrow is labelled v.
Right: a close-up of the pellet sprayer

(A) 12πRh\dfrac{1}{2\pi}\dfrac{R}{h}

(B) 1πRh\dfrac{1}{\pi}\dfrac{R}{h}

(C) 2πRh\dfrac{2}{\pi}\dfrac{R}{h}

(D) 12πRh\dfrac{1}{2\pi}\sqrt{\dfrac{R}{h}}

(E) 1πRh\dfrac{1}{\pi}\sqrt{\dfrac{R}{h}}

Отговори

Решение

Оригинал в Архива: 2021_Fma_exam.pdf · официални решения: 2021_Fma_Solutions.pdf