USAPhO 2026, Първи кръг — Задача 12
2026 Exam · 1 т.
Указания
25 QUESTIONS - 75 MINUTES
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use 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 wake surfer of total mass (surfer plus board) is being towed at a constant horizontal velocity across a flat lake. The wakeboard has a specific geometry such that, at this speed, it is partially submerged and provides a static buoyant force (where ).
To support the remainder of the weight, the board moves with an angle of attack relative to the horizontal water surface. Assume that the water exerts a reaction force strictly normal to the bottom surface of the board. Which of the following expressions represents the horizontal tension in the tow rope required to maintain this constant velocity?
(A)
(B)
(C)
(D)
(E)
Отговори
- B
Решение
The surfer is moving at a constant velocity, meaning the net force is zero in both the vertical and horizontal directions.
Lets first consider the vertical direction. The weight of the surfer and board is supported by two upward forces: the static buoyant force () and the vertical component of the hydrodynamic reaction force from the water (), that is:
Therefore the vertical lift required from the hydro-planing action:
As stated, the water exerts a force strictly normal to the bottom surface of the board. Since the board is tilted at an angle of attack relative to the horizontal, the normal force vector is tilted backward by the angle relative to the vertical axis.
Using vector decomposition, the horizontal component (, (drag)) and the vertical component (, (lift)) are related by the tangent of the angle:
Through simple trigonometry we get:
Now, the surfer is being towed at a constant speed, so the forward tension () in the rope must balance the drag component ().
Substituting the expression for the drag we get:
Now, substitute the expression for the lift () above we get:
Оригинал в Архива: 2026_FMA_exam.pdf · официални решения: 2026_FMA_solutions.pdf