USAPhO 2024, Първи кръг — Задача 18

2024 F=maF = ma Exam · 1 т.

Указания

25 QUESTIONS - 75 MINUTES

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Use g=10 N/kgg = 10\ \mathrm{N/kg} throughout, unless otherwise specified.
  • You may write in this question booklet and the scratch paper provided by the proctor.
  • This test has 25 multiple choice questions. Select the best response to each question, and use a No.2 pencil to completely fill the box corresponding to your choice. If you change an answer, completely erase the previous mark. Only use the boxes numbered 1 through 25 on the answer sheet.
  • All questions are equally weighted, but are not necessarily equally difficult.
  • You will receive one point for each correct answer, and zero points for each incorrect or blank answer. There is no additional penalty for incorrect answers.
  • You may use a hand-held calculator. 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 external references, such as books or formula sheets.
  • The question booklet, answer sheet and scratch paper will be collected at the end of this exam.
  • To maintain exam security, do not communicate any information about the questions or their solutions until after February 24, 2024.

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):

Tengiz Bibilashvili, Orhun Ciftcioglu, Kellan Colburn, Natalie LeBaron, Brian Skinner, Elena Yudovina and Kevin Zhou

Условие

A satellite is initially in a circular orbit of radius RR around a planet of mass MM. It fires its rockets to instantaneously increase its speed by Δv\Delta v, keeping the direction of its velocity the same, so that it enters an elliptical orbit whose maximum distance from the planet is 2R2R.

A planet at the centre of a circle of radius R and a larger circle of radius 2R; an ellipse starts tangent to the inner circle at the point marked start and reaches the outer circle at the diametrically opposite point marked end; a dotted line joins start and end through the planet.

What is the value of Δv\Delta v? (Hint: when the satellite is in an elliptical orbit with semimajor axis aa, its total energy per unit mass is −GM/2a-GM/2a.)

(A) 0.08GMR0.08\sqrt{\dfrac{GM}{R}}

(B) 0.15GMR0.15\sqrt{\dfrac{GM}{R}}

(C) 0.22GMR0.22\sqrt{\dfrac{GM}{R}}

(D) 0.29GMR0.29\sqrt{\dfrac{GM}{R}}

(E) 0.41GMR0.41\sqrt{\dfrac{GM}{R}}

Отговори

Решение

Оригинал в Архива: 2024_F-ma_Exam.pdf · официални решения: 2024_F-ma_Solutions.pdf