USAPhO 2026, Първи кръг — Задача 13

2026 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 13, 2026.

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

We acknowledge The US Physics Team coaches and other contributors to this year’s exams:

Tengiz Bibilashvili, Eddie Chen, Orhun Ciftcioglu, Kellan Colburn, Evan Erickson, Harry Kroft, Natalie LeBaron, Mai Qi, Elena Yudovina

Условие

A block of mass mm slides along a surface with a coefficient of kinetic friction μk\mu_k. The block is confined to move between two walls separated by a distance LL. Attached to each wall is an ideal, perfectly elastic spring with a very large force constant kk.

The block is launched from the midpoint between the walls with an initial speed v0v_0 directed toward one of the springs. It bounces back and forth between the springs, losing speed only due to friction with the surface.

Assume the time spent in contact with the springs is negligible compared to the travel time between them. Which of the following expressions correctly gives the total time TT required for the block to come to a permanent rest?

(A) T=v02μkgLT = \frac{v_0}{2\mu_k gL}

(B) T=v0μkg(1−e−μkgv0)T = \frac{v_0}{\mu_k g}\left(1 - e^{-\frac{\mu_k g}{v_0}}\right)

(C) T=v0μkgT = \frac{v_0}{\mu_k g}

(D) T=voμkgL∑n=0∞(12)nT = \frac{v_o}{\mu_k gL}\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^n

(E) T=2LμkgT = \sqrt{\frac{2L}{\mu_k g}}

Отговори

Решение

Оригинал в Архива: 2026_FMA_exam.pdf · официални решения: 2026_FMA_solutions.pdf