USAPhO 2026, Първи кръг — Задача 1

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

In astronomy, some galactic objects appear to sweep across the sky faster than light speed, cc. This effect, called superluminal motion, comes purely from geometry and the finite travel time of light, and it has nothing to do with special relativity.

A jet moves from point AA to BB at speed v=βcv = \beta c. The jet emits a pulse of light at AA, and a second pulse a time δt\delta t later at BB. An observer sees these pulses at point OO. The angle between the jet and the line of sight is θ\theta. Assume the angle ϕ\phi is small, so the distances from OO to points BB and CC can be treated as equal.

Points O, C and A on a horizontal dashed line; point B lies directly above C. A solid arrow labelled v runs from A to B, making angle θ with line AC at A. A dashed line joins O to B, making a small angle φ with OC at O; a vertical dashed segment joins C and B.

Find the apparent transverse velocity, vTv_T, along CBCB as measured by the observer, in terms of β\beta and θ\theta. Express your answer as βT≡vTc\beta_T \equiv \frac{v_T}{c}

(A) βT=βsin⁡θ1−βcos⁡θ\beta_T = \frac{\beta \sin\theta}{1-\beta\cos\theta}

(B) βT=βsin⁡θ(1−βcos⁡θ)\beta_T = \beta\sin\theta(1-\beta\cos\theta)

(C) βT=βsin⁡θ1+βcos⁡θ\beta_T = \frac{\beta \sin\theta}{1+\beta\cos\theta}

(D) βT=βsin⁡θ1−β2\beta_T = \frac{\beta \sin\theta}{\sqrt{1-\beta^2}}

(E) βT=βtan⁡θ\beta_T = \beta\tan\theta

Отговори

Решение

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