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OPhO 2020, Теория — Задача 9. Piston Gun

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Всички материали от OPhO 2020

2020 Online Physics Olympiad: Invitational Contest v1.4 · 40 т.

Внимание: Бележка към темата

The v1.4 question paper, native Partial Solutions and separately attributed organizer clarifications are retained. The key is incomplete: Problem 1 is only marked TBD, Problem 8 has no key for parts f–h or an explicit critical power in e, and Problem 10 has no key for Part B f–h. Problems 2 and 9 also have source gaps noted below. Printed mathematical defects and differences between repeated questions in the key and the question original are not silently repaired. Problem2(e) gives a time-dependent spin result but no explicit answer in orbital angle; key2(c) refers to an absent diagram. Problem9(a) gives its ODE and initial conditions without an explicit speed, and 9(b) does not state whether the pressure increases or decreases. Native source typography is retained, including the Planck-constant value, missing electron-charge unit, missing kappa in key3(a), mismatched labels in key4, inconsistent theta indices and ring factors in key6, the negative probability/energy signs and r_B/K_B/k variants in key7, radiation expressions and fusion-rate/proton-rate difference in key8, and image/flux signs and field factors in key10. All these are printed-source issues, not repaired answers. Question2 omits the word define while the repeated key question includes it; the repeated key question7 adds For simplicity, ignore spin. The 4-page organizer clarification contradicts question1(d)’s frictionless table and also contains both only one final submission and multiple submissions instructions. The native original and the separately attributed clarifications remain visible.

Original credits

2020 Online Physics Olympiad (OPhO): Invitational Contest v1.4

OPhO Lead Organizers and Writers: Adithya Balachandran, Eddie Chen, Ashmit Dutta, Ethan Hu, Viraj Jayam, Rishab Parthasarathy, Kushal Thaman, Max Wang, Qilin Xue

OPhO Website Developer: Ben Oommen

Advisors: Dr. Branislav Kisačanin, Kye Shi, Kevin Zhou

July 3 - 6, 2020

Указания

If you wish to request a clarification, please use this form. To see all clarifications, see this document.

This test contains 10 long answer questions.
The total base score for the exam is 300 points; a factor incorporating the number of teams who solved the question will be added in the marking scheme. The final scores of all the teams will be available a few days after the contest ends.
The team leader should submit their final solution document in this here. We don’t anticipate a tie, but in the rare circumstance that there is one, the time you submit will be used to break it.
You can write on paper, type it up online, or a mix of both. If you wish to use a pre-made LaTeX template, we made one which you can choose to use here.
Since this is a long answer response, you will be judged on the quality of your work. To receive full points, you need to show your work, including deriving equations. As a general rule of thumb, any common equations (such as the ones in this formula sheet) can be cited without proof. Remember to state any approximations made, which system of equations were solved after every step. Explicitly showing every step of algebra is not necessary.
You may leave all final answers in symbolic form (in terms of variables) unless otherwise specified.2
Be sure to state all assumptions.

Sponsors

WOLFRAM · AwesomeMath (making x,y,z as easy as a,b,c) · Physics Olympiad · AoPS.com Art of Problem Solving

List of Constants

Proton mass, $m_p=1.67\cdot10^{-27}\ \mathrm{kg}$
Neutron mass, $m_n=1.67\cdot10^{-27}\ \mathrm{kg}$
Electron mass, $m_e=9.11\cdot10^{-31}\ \mathrm{kg}$
Avogadro’s constant, $N_0=6.02\cdot10^{23}\ \mathrm{mol^{-1}}$
Universal gas constant, $R=8.31\ \mathrm{J/(mol\cdot K)}$
Boltzmann’s constant, $k_B=1.38\cdot10^{-23}\ \mathrm{J/K}$
Electron charge magnitude, $e=1.60\cdot10^{-19}$
1 electron volt, $1\ \mathrm{eV}=1.60\cdot10^{-19}\ \mathrm J$
Speed of light, $c=3.00\cdot10^8\ \mathrm{m/s}$
Universal Gravitational constant, $G=6.67\cdot10^{-11}\ \mathrm{(N\cdot m^2)/kg^2}$
Acceleration due to gravity, $g=9.81\ \mathrm{m/s^2}$
1 unified atomic mass unit, $1\ \mathrm u=1.66\cdot10^{-27}\ \mathrm{kg}=931\ \mathrm{MeV}/c^2$
Planck’s constant, $h=6.63\cdot10^{-34}\ \mathrm{J\cdot s}=4.41\cdot10^{-15}\ \mathrm{eV\cdot s}$
Permittivity of free space, $\epsilon_0=8.85\cdot10^{-12}\ \mathrm{C^2/(N\cdot m^2)}$
Coulomb’s law constant, $k=\frac1{4\pi\epsilon_0}=8.99\cdot10^9\ \mathrm{(N\cdot m^2)/C^2}$
Permeability of free space, $\mu_0=4\pi\cdot10^{-7}\ \mathrm{T\cdot m/A}$
Magnetic constant, $\frac{\mu_0}{4\pi}=1\cdot10^{-7}\ \mathrm{(T\cdot m)/A}$
1 atmospheric pressure, $1\ \mathrm{atm}=1.01\cdot10^5\ \mathrm{N/m^2}=1.01\cdot10^5\ \mathrm{Pa}$
Wien’s displacement constant, $b=2.9\cdot10^{-3}\ \mathrm{m\cdot K}$
Stefan-Boltzmann constant, $\sigma=5.67\cdot10^{-8}\ \mathrm{W/m^2/K^4}$

Native key title

2020 Online Physics Olympiad (OPhO): Invitational Contest Partial Solutions

The same organizer, writer, developer and advisor credits appear on this key cover.

Organizer clarification document — major notes

Separately quoted from the organizer clarification document, titled 2020 OPhO Invitational Clarifications:

MAJOR NOTES

Hi everyone! All team members are welcome to fill out this (optional) feedback form. Everyone who submits (and consents) will be entered in a draw to receive a $25 Amazon gift card.
Use this link to submit clarification requests. The problem writers may or may not deem it necessary to answer all clarification requests.
Link to the overleaf template (the link has been fixed) for answer submission.
Reminder: There is only one final submission at the end of the competition via the submission form only. Do NOT email us a copy of your submission. In case of internet connectivity issues, any team member may email us the file.
Make sure to read over the instructions. You are expected to give symbolic answers unless explicitly otherwise stated. If the numerical values for a symbol are stated, you do not need to solve the problem for a general case.
Uploading a .jpeg on the overleaf for a graph is allowed.
Please check the newest version v1.4 for revisions.
Please provide clear reasoning or derivations of any obscure results. Directly citing them would not necessarily fetch full points.
We will be awarding a special prize for whichever team produces the most creative and/or elegant solution to certain problems. We will not be disclosing which problems until after the competition has concluded.

Note on submission of solutions:

It is highly recommended that the solutions to the invitational round be submitted in LaTeXed form. We recommend using Overleaf. It is preferred that the solutions are quite detailed; this means that while you do not need to explicitly write all your algebra, all the relevant equations must be clearly written. A brief wording on how you solve a given system of equations is also optimal. Clearly state any approximations you make. Solutions typed in .doc/.docx, and ones neatly written by hand will also be accepted. Some students have a handwriting that’s hard to read; if that’s you, please read this. You can submit solutions multiple times. Only the latest submission will count.

Условие

In this problem, we examine a model for a certain type of gun that works by using the expansion of a gas to propel a bullet. We can model the bullet as a piston. Since we are assuming atmospheric pressure is negligible, we can assume that the whole setup is in a vacuum. Also, the gun is insulated.

An ideal monatomic gas of initial temperature $T_0$ is inside a long cylindrical container of cross-section area $A$ . One side of the container is a wall, while the other side is a piston of mass $M$ that can slide freely along the container without friction. The total mass of the gas is $m$ , and it is made up of $N$ particles. Initially, the piston is at rest and a distance $L_0$ away from the opposite wall. Then, the piston is released. After a time $t$ , the piston moves at a speed $v$ . Assume that throughout the process, the particles on average move very fast.

a) (5 pts) Assume that $m$ is negligible. Find $v$ .

b) (6 pts) From now on, do not assume that $m$ is negligible.

Find the time at which the pressure at the wall opposite the piston changes. Also, does it increase or decrease? State all assumptions.

c) (14 pts) From now on, assume $t$ is much smaller than the mean free time of the particles of the gas, and $L_0$ is much smaller than the mean free path. (During this time interval $t$ , assume that all the particles still collide many, many times with the walls, but they don’t collide with each other.) Find $v$ .

d) (6 pts) Find the recoil impulse of the gun over the time $t$ .

e) (9 pts) Let $r>1$ be a dimensionless parameter. Suppose at time $t$ , the piston is a distance $rL_0$ away from the wall; then the piston is stopped, and the gas is allowed to come to equilibrium (after a time much greater than the mean free time). Find the total entropy change (throughout the whole process) of the gas in terms of $r$ , and verify the Second Law of Thermodynamics.

Published organizer clarifications (separate source)

Separately quoted from the organizer clarification document:

Problem 9:

t is arbitrary.
In c), assume that all the particles still collide many, many times with the walls (during the time interval t), but the particles don’t collide with each other (during the time interval t).

Решение

Внимание: Непълно решение

The complete available printed key is retained. Part a explicitly gives only the differential equation and initial conditions, not the requested speed; part b gives a travel time but no answer on whether pressure increases or decreases.

Solution 9:

(a) Since $m\ll M$ , the expansion of the gas can be treated as a reversible process, because pressure, density, etc. will be uniformly distributed. The process is also adiabatic because the container is insulated. If $x$ is the position of the piston, we have $px^\gamma$ is constant, where $p$ is the pressure and $\gamma=\frac{C_p}{C_v}=\frac53$ for monatomic gas. Also, Newton’s 2nd Law gives $pA=M\ddot x$ . We have $\ddot x=\frac{p_0AL_0^\gamma}{Mx^\gamma}$ , where $p_0=\frac{NkT_0}{AL_0}$ . To find $v$ , we need to solve this differential equation and plug in the boundary conditions of $x(0)=L_0$ and $\dot x(0)=0$ . Since the final answer is very complicated, we have decided to award (almost) full credit for those who end up with correct differential equation and boundary conditions (and have accurate, thorough reasoning).

(b) Now, we cannot neglect $m$ anymore, so the pressure will not be uniform throughout the gas. In fact, a pressure wave will start at the piston and travel to the opposite wall at speed $c=\sqrt{\frac{\gamma p}{\rho}}=\sqrt{\frac{5NkT_0}{3m}}$ .

The time it takes is $\frac{L_0}{c}=\boxed{L_0\sqrt{\frac{3m}{5NkT_0}}}$ .

(c) Let $L(t)$ be the position of the piston as a function of time, where $L(0)=L_0$ . Let $v=\dot L$ . Let the x direction be along the cylinder’s axis. Consider a particle in the gas with speed $v_x(t)$ in the x direction. We will show that $Lv_x$ is a conserved quantity. Since the mean free time is large, we can assume the particles rarely collide with each other. Since the gas is monatomic, the particle makes elastic collisions with the walls. When it collides with a stationary wall, the particle’s speed stays the same. However, if it moves at speed $v_x$ and collides with the piston moving at speed $v$ , it rebounds with a new speed $v_x-2v$ , which can be seen by shifting to the piston’s frame and noting that the particle’s mass is much smaller than the piston’s. Thus, its decrease in speed is $2v$ . Since it is moving very fast, on average, it loses speed $2v$ every time it takes to go back and forth, which is $\frac{2L}{v_x}$ . We can write $\frac{dv_x}{dt}=-\frac{2v}{\frac{2L}{v_x}}$ . This simplifies to $L\dot v_x+vv_x=0$ . Integrating, we get $Lv_x=c$ where $c$ is an integration constant. Plugging in initial conditions, we have $Lv_x=L_0v_{x0}$ , where $v_{x0}$ is the initial x-velocity of the particle.

Next, we write the conservation of energy equation after time $t$ . For simplicity, we redefine $m$ as the mass of each particle.

\int_{-\infty}^\infty\frac12Nm(v_{x0}^2-v_x^2)f(v_{x0})\,dv_{x0}=\frac12Mv^2

where $f(v_{x0})=\sqrt{\frac m{2\pi kT_0}}e^{-\frac{mv_{x0}^2}{2kT_0}}$ is the Maxwell-Boltzmann distribution in the x direction. Using the fact that $Lv_x=L_0v_{x0}$ and substituting for $f$ , we get

\int_{-\infty}^\infty Nmv_{x0}^2\left(1-\frac{L_0^2}{L^2}\right)\sqrt{\frac m{2\pi kT_0}}e^{-\frac{mv_{x0}^2}{2kT_0}}\,dv_{x0}=Mv^2

Integrating and simplifying, we get

\dot L=\sqrt{\frac{NkT_0}{M}}\sqrt{1-\frac{L_0^2}{L^2}}

\frac L{\sqrt{L^2-L_0^2}}dL=\sqrt{\frac{NkT_0}{M}}dt

Integrating and plugging in initial conditions gives

\sqrt{L^2-L_0^2}=\sqrt{\frac{NkT_0}{M}}t

L=\sqrt{L_0^2+\frac{NkT_0}{M}t^2}

Finally, we have $\boxed{v=\frac{NkT_0t}{\sqrt{M^2L_0^2+NkT_0Mt^2}}}$

(d) The recoil impulse is simply the momentum of the gas + piston. The center of mass of the gas moves at half the speed of the piston, so we have the total momentum $Mv+\frac{mv}{2}=\boxed{\left(M+\frac m2\right)\frac{NkT_0t}{\sqrt{M^2L_0^2+NkT_0Mt^2}}}$

(e) After a long time, the pressure will again become uniform, and we can use the formula for entropy of an ideal gas: $Nk\ln PV_0^\gamma$ . To do this, we need to first realize that when the piston is stopped, $\frac12Mv^2$ of energy is lost, and combining this fact with the ideal gas law gives us enough information to solve for the final pressure of the gas. After calculation, we get $\boxed{\Delta S=Nk\left(\frac32\ln\left(\frac23+\frac1{3r^2}\right)+\ln r\right)}$ . Note that there are no surroundings (vacuum). Also, it is easy to see $\Delta S$ is positive if one finds that the derivative with respect to $r$ is positive for $r>1$ . Thus, the Second Law of Thermodynamics is verified.

Organizer clarification document — other notes

Separately quoted from the organizer clarification document:

Other Notes

Uploading images: View this link to find how to upload images on overleaf.
You are allowed to mix LaTeX and handwritten PDFs for your answer sheet. Note that you need to write your solution to the problem alongside your final answers.
You can use basic formulae related to topics on the IPhO syllabus without derivation but all other formulae which go beyond that must be derived.
If you find that space is running out on your LaTeX file, try to use $\verb|\newpage|$ and see if that fixes it.

Оригинал в Архива: invitational.pdf · официални решения: invitational_sol.pdf · допълнителни данни: Clarifications.pdf