USAPhO 1997, Полуфинал — Задача 1. A1

Entia non multiplicanda sunt praeter necessitatem

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

1. Please show all work, and do not hesitate to use explanatory words in addition to equations. Partial credit will be awarded.

2. Begin each problem on a fresh sheet of paper. In the upper right-hand corner of each page, write your name, problem number, and the page number/total number of pages for this problem (e.g., "A2, 3/4" for "Problem A2, page 3 of four").

3. You may use a non-programmable calculator, but may not use any tables, books, notes, or collections of formulas.

4. Work Part A first. You have ninety minutes to complete all four problems.

5. After you have completed Part A, you may take a break.

6. Then work Part B. You have 90 minutes to complete both problems.

Gravitational field on Earth’s surface: $g = 9.8\ \mathrm{N/kg}$

Gravitational constant: $G = 6.67\times10^{-11}\ \mathrm{N\text{-}m^2 / kg^2}$

Coulomb constant: $k = 1 / 4\pi\epsilon_o = 8.99\times10^{9}\ \mathrm{N\text{-}m^2 / C^2}$

Speed of light in vacuum: $c = 3.00\times10^{8}\ \mathrm{m/s}$

Permeability of empty space: $\mu_o$, where $\mu_o\epsilon_o = 1/c^2$

Ideal gas constant $R = 8.31\ \mathrm{J / mol\text{-}K}$

Boltzmann’s constant: $k_B = 1.38\times10^{-23}\ \mathrm{J/K}$

Avagadro’s number: $N_A = 6.02\times10^{23}$

1 electron volt = $1\ \mathrm{eV} = 1.60\times10^{-19}\ \mathrm{J}$

Planck’s constant: $h = 6.63\times10^{-34}\ \mathrm{J\text{-}s} = 4.14\times10^{-15}\ \mathrm{eV\text{-}s}$

$(1 - x)^{-1} = 1 + x + x^2 + x^3 + \ldots$ for $|x| < 1$ $\qquad$ $\cos\theta \approx 1 - \theta^2/2$ for $\theta \ll 1$

$(1 + x)^n = 1 + nx + (1/2!)n(n-1)x^2 + \ldots$ for $|x| < 1$ $\qquad$ $\sin\theta \approx \theta$ for $\theta \ll 1$