USAPhO 2003, Четвъртфинал — Задача 2

AAPT AIP United States Physics Team 2003 — Preliminary Exam, Open Response Questions · 25 т.

Указания

4 Questions, 60 minutes

INSTRUCTIONS

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Show all your work. Partial credit will be given.

  • Start each question on a new sheet of paper. Be sure to put your name in the upper right-hand corner of each page, along with the question number and the page number/total pages for this problem. For example,

    Doe, Jamie

    Q1 – 1/3

  • 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. You may not use any tables, books, or collections of formulas.

  • Each of the four questions is worth 25 points. The questions are not necessarily of the same difficulty.

  • Good luck!

Possibly Useful Information
Gravitational field at the Earth’s surfaceg=9.8 N/kgg = 9.8\ \mathrm{N/kg}
Newton’s gravitational constantG=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}
Binomial expansion(1+x)n≈1+nx(1 + x)^n \approx 1 + nx for ∣x∣≪1\lvert x\rvert \ll 1

Moment of Inertia about Center of Mass – Uniform Object

(will not be provided on the second screening exam)

Disk12MR2\tfrac{1}{2} MR^2
Sphere25MR2\tfrac{2}{5} MR^2
Rod112ML2\tfrac{1}{12} ML^2
Solutions to Problems

Unless explicitly told to use a specific method, any correct solution should be awarded equivalent points. Suggested partial-credit points are presented in square brackets at the right margin. You may further break down the listed points into one point increments. If it is clear they have done an intermediate step, they should get credit for it even if they have not presented it. For example, in 2.a., if a student wrote down 2vot−12gt2=H−vot−12gt22v_o t - \frac{1}{2}gt^2 = H - v_o t - \frac{1}{2}gt^2, they should get 8 points credit. Students should not be penalized in a subsequent part for using the wrong answer to a previous part. (No double jeopardy.)

Условие

Suppose that Galileo is in the Leaning Tower of Pisa. He throws a cannonball straight down from height HH above the ground. The ball leaves his hands with an initial speed vov_o. At the same instant, his friend throws a cannonball straight up from ground level with an initial speed of 2vo2v_o. When the cannonballs collide, they are traveling in the same direction and Galileo's cannonball is traveling with 7 times the speed of his friend's. Ignore air resistance.

a. At what time does the collision occur? Express your answer in terms of HH and vov_o only. [10 т.]

b. At what height above the ground does the collision occur? Express your answer in terms of HH only. [15 т.]

Отговори

Решение

Оригинал в Архива: Quarterfinal 2003.pdf · официални решения: Quarterfinal 2003 Solutions.pdf