USAPhO 1997, Полуфинал — Задача 2. A2

United States Physics Team — 1997 Exam 2 · 20 т.

Указания

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.

Possibly useful information:

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

Gravitational constant: G=6.67×10−11 N-m2/kg2G = 6.67\times10^{-11}\ \mathrm{N\text{-}m^2 / kg^2}

Coulomb constant: k=1/4πϵo=8.99×109 N-m2/C2k = 1 / 4\pi\epsilon_o = 8.99\times10^{9}\ \mathrm{N\text{-}m^2 / C^2}

Speed of light in vacuum: c=3.00×108 m/sc = 3.00\times10^{8}\ \mathrm{m/s}

Permeability of empty space: μo\mu_o, where μoϵo=1/c2\mu_o\epsilon_o = 1/c^2

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

Boltzmann’s constant: kB=1.38×10−23 J/Kk_B = 1.38\times10^{-23}\ \mathrm{J/K}

Avagadro’s number: NA=6.02×1023N_A = 6.02\times10^{23}

1 electron volt = 1 eV=1.60×10−19 J1\ \mathrm{eV} = 1.60\times10^{-19}\ \mathrm{J}

Planck’s constant: h=6.63×10−34 J-s=4.14×10−15 eV-sh = 6.63\times10^{-34}\ \mathrm{J\text{-}s} = 4.14\times10^{-15}\ \mathrm{eV\text{-}s}

(1−x)−1=1+x+x2+x3+…(1 - x)^{-1} = 1 + x + x^2 + x^3 + \ldots for ∣x∣<1|x| < 1 \qquad cos⁡θ≈1−θ2/2\cos\theta \approx 1 - \theta^2/2 for θ≪1\theta \ll 1

(1+x)n=1+nx+(1/2!)n(n−1)x2+…(1 + x)^n = 1 + nx + (1/2!)n(n-1)x^2 + \ldots for ∣x∣<1|x| < 1 \qquad sin⁡θ≈θ\sin\theta \approx \theta for θ≪1\theta \ll 1

Условие

A ball is tossed onto the floor, where it makes a succession of bounces as illustrated in the figure below. Assume that because of internal elasticity and friction with the floor, at each bounce the magnitude of the vertical velocity component is reduced by a factor ϵy\epsilon_y and the horizontal component is reduced by a factor ϵx\epsilon_x. That is, if voy,n+1v_{oy,n+1} denotes the yy-component of the velocity as the ball emerges from the (n+1)(n+1)st bounce, then voy,n+1=ϵyvoy,nv_{oy,n+1} = \epsilon_y v_{oy,n}, and similarly for the xx-component. Note that ϵy\epsilon_y and ϵx\epsilon_x are <1< 1. Thus after each bounce, the ball moves slower and hops a shorter distance than it did after the preceding bounce.

Let the ball's succession of bounces traverse a total horizontal distance L∗L^* which takes the time t∗t^*. (As a practical matter, we measure L∗L^* and t∗t^* as the length and time where the bounces become imperceptible; mathematically, the number of bounces goes to infinity.) Find θ1\theta_1, the angle the ball's velocity makes with the horizontal immediately after the first bounce, written in terms of L∗L^*, t∗t^*, ϵy\epsilon_y, ϵx\epsilon_x, and needed constants. Neglect air resistance.

A ball's trajectory arriving from the upper left and bouncing along a horizontal floor with ever smaller arcs; bounce points labelled 1, 2, 3, 4, 5 (row label Bounce); angle θ1 marked between the floor and the dotted tangent after bounce 1; dotted verticals mark the hop intervals L1, t1 and L2, t2, and the whole distance L*, t* from bounce 1 to where the bounces die out.

Отговори

Решение

Оригинал в Архива: Semifinal 1997.pdf · официални решения: Semifinal 1997 Solutions.pdf