USAPhO 2026, Първи кръг — Задача 6

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

Newton’s Impact Depth Approximation provides an estimate on how far a projectile will penetrate a material at high speeds. If a projectile of length LL and density ρA\rho_A impacts a wall of density ρB\rho_B, Newton’s approximation states that the projectile will penetrate to a depth of D=LρAρBD = L\frac{\rho_A}{\rho_B}. Assuming this is true for all velocities, how does the average force the wall exerts on the projectile FF scale with the initial projectile speed vv?

(A) F∝v1/2F \propto v^{1/2}

(B) F∝vF \propto v

(C) F∝v3/2F \propto v^{3/2}

(D) F∝v2F \propto v^2

(E) F∝v5/2F \propto v^{5/2}

Left: a grey ball of size L and density ρ_A moving down with velocity v toward a horizontal surface of density ρ_B. Right: the ball at the bottom of a U-shaped channel it has dug into the wall, with the penetration depth marked D.

Отговори

Решение

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