USAPhO 2022, Първи кръг, вариант B — Задача 13

2022 F = ma Exam B · 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 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, answer sheet and 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 25, 2022.

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Acknowledgements

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

Tengiz Bibilashvili, Abi Krishnan, Andrew Lin, Kris Lui, Kye Shi, Brian Skinner, Mike Winer and Kevin Zhou

Условие

A ballistic pendulum is designed as shown below.

A rigid rod hangs from the ceiling and holds a ball of mass m and radius r; the distance ℓ from the ceiling to the ball's center is marked. A horizontal tunnel runs from the right side of the ball to its center. A small block m moves left toward the tunnel with velocity v.

A uniform ball of mass mm and radius rr is attached to a massless rigid rod, so that its center is a distance ℓ\ell from the ceiling. The ball has a small tunnel hollowed out. A small block of mass mm and speed vv goes into the tunnel and collides at the center of the ball, perfectly inelastically. Subsequently, to what maximum height does the block rise?

(A) v28g\dfrac{v^2}{8g}

(B) v28g(1+r2/5ℓ2)2\dfrac{v^2}{8g(1 + r^2/5\ell^2)^2}

(C) v28g(1+2r2/5ℓ2)2\dfrac{v^2}{8g(1 + 2r^2/5\ell^2)^2}

(D) v28g(1+r2/5ℓ2)\dfrac{v^2}{8g(1 + r^2/5\ell^2)}

(E) v28g(1+2r2/5ℓ2)\dfrac{v^2}{8g(1 + 2r^2/5\ell^2)}

Отговори

Решение

Оригинал в Архива: 2022_Fma_Exam_B.pdf · официални решения: 2022_Fma_Exam_B_Solutions.pdf