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Với những đề thi và lời giải của những năm về trước. Đây là một tài liệu thiết thực để các bạn học sinh giỏi có thể ôn luyện tham gia học sinh giỏi cấp Quốc Gia, Khu vực, Quốc Tế... Mỗi bài thi IMO bao gồm 6 bài toán, mỗi bài tương đương tối đa là 7 điểm, có nghĩa là thí sinh có thể đạt tối đa 42 điểm cho 6 bài. 6 bài toán này sẽ được giải trong 2 ngày liên tiếp, mỗi ngày thí sinh giải 3 bài trong thời gian 270 phút. Các bài toán được lựa chọn trong các vấn đề toán học sơ cấp, bao gồm 4 lĩnh vực hình học, số học, đại số và tổ hợp. Bắt đầu từ tháng 3 hàng năm, các nước tham gia thi được đề nghị gửi các đề thi mà họ lựa chọn đến nước chủ nhà, sau đó một ban lựa chọn đề thi của nước chủ nhà sẽ lập ra một danh sách các bài toán rút gọn bao gồm những bài hay nhất, không trùng lặp đề thi IMO các năm trước hoặc kì thi quốc gia của các nước tham gia, không đòi hỏi kiến thức toán cao cấp, không quá khó hoặc quá dễ nhưng yêu cầu được thí sinh phải vận dụng hết khả năng suy luận và kiến thức toán được học. Một vài ngày trước kì thi, các trưởng đoàn sẽ bỏ phiếu lựa chọn 6 bài chính thức, chính họ cũng sẽ là người dịch đề thi sang tiếng nước mình để thí sinh có thể giải toán bằng tiếng mẹ đẻ, sau đó các vị trưởng đoàn sẽ được cách ly hoàn toàn với các thí sinh để tránh gian lận. Bài thi của thí sinh sẽ được ban giám khảo và trưởng đoàn của thí sinh đó chấm song song, sau đó hai bên sẽ hội ý để đưa ra kết quả cuối cùng. Giám khảo và trưởng đoàn đều có thể phản biện cách chấm của nhau để điểm bài thi đạt được là chính xác nhất. Nếu hai bên không thể đi tới đồng thuận thì người quyết định sẽ là trưởng ban giám khảo và giải pháp cuối cùng là tất cả các trưởng đoàn bỏ phiếu. Riêng bài thi của thí sinh nước chủ nhà sẽ do giám khảo đến từ các nước có đề thi được chọn chấm.
International Physics Olympiad (IPhO) 1967-2010 Problems and Solutions Compiled & edited by scroungehound Visitors Appendix to the Statutes of the International Physics Olympiads General a The extensive use of the calculus (differentiation and integration) and the use of complex numbers or solving differential equations should not be required to solve the theoretical and practical problems b Questions may contain concepts and phenomena not contained in the Syllabus but sufficient information must be given in the questions so that candidates without previous knowledge of these topics would not be at a disadvantage c Sophisticated practical equipment likely to be unfamiliar to the candidates should not dominate a problem If such devices are used then careful instructions must be given to the candidates d The original texts of the problems have to be set in the SI units A Theoretical Part The first column contains the main entries while the second column contains comments and remarks if necessary Mechanics a) Foundation of kinematics of a point mass Vector description of the position of the point mass, velocity and acceleration as vectors b) Newton's laws, inertial systems Problems may be set on changing mass c) Closed and open systems, momentum and energy, work, power d) Conservation of energy, conservation of linear momentum, impulse e) Elastic forces, frictional forces, the law of gravitation, potential energy and work in a gravitational field Hooke's law, coefficient of friction (F/R = const), frictional forces, static and kinetic, choice of zero of potential energy f) Centripetal acceleration, Kepler's laws Mechanics of Rigid Bodies a) Statics, center of mass, torque Couples, conditions of equilibrium of bodies b) Motion of rigid bodies, translation, rotation, Conservation of angular momentum about angular velocity, angular acceleration, fixed axis only conservation of angular momentum c) External and internal forces, equation of motion of a rigid body around the fixed axis, moment of inertia, kinetic energy of a rotating body Parallel axes theorem (Steiner's theorem), additivity of the moment of inertia d) Accelerated reference systems, inertial forces Knowledge of the Coriolis force formula is not required Hydromechanics No specific questions will be set on this but students would be expected to know the elementary concepts of pressure, buoyancy and the continuity law Thermodynamics and Molecular Physics a) Internal energy, work and heat, first and second laws of thermodynamics Thermal equilibrium, quantities depending on state and quantities depending on process b) Model of a perfect gas, pressure and molecular kinetic energy, Avogadro's number, equation of state of a perfect gas, absolute temperature Also molecular approach to such simple phenomena in liquids and solids as boiling, melting etc c) Work done by an expanding gas limited to isothermal and adiabatic processes Proof of the equation of the adiabatic process is not required d) The Carnot cycle, thermodynamic efficiency, reversible and irreversible processes, entropy (statistical approach), Boltzmann factor Entropy as a path independent function, entropy changes and reversibility, quasistatic processes Oscillations and waves a) Harmonic oscillations, equation of harmonic Solution of the equation for harmonic motion, oscillation attenuation and resonance -qualitatively b) Harmonic waves, propagation of waves, transverse and longitudinal waves, linear polarization, the classical Doppler effect, sound waves Displacement in a progressive wave and understanding of graphical representation of the wave, measurements of velocity of sound and light, Doppler effect in one dimension only, propagation of waves in homogeneous and isotropic media, reflection and refraction, Fermat's principle c) Superposition of harmonic waves, coherent Realization that intensity of wave is waves, interference, beats, standing waves proportional to the square of its amplitude Fourier analysis is not required but candidates should have some understanding that complex waves can be made from addition of simple sinusoidal waves of different frequencies Interference due to thin films and other simple systems (final formulae are not required), superposition of waves from secondary sources (diffraction) Electric Charge and Electric Field a) Conservation of charge, Coulomb's law b) Electric field, potential, Gauss' law Gauss' law confined to simple symmetric systems like sphere, cylinder, plate etc., electric dipole moment c) Capacitors, capacitance, dielectric constant, energy density of electric field Current and Magnetic Field a) Current, resistance, internal resistance of source, Ohm's law, Kirchhoff's laws, work and power of direct and alternating currents, Joule's law Simple cases of circuits containing non-ohmic devices with known V-I characteristics b) Magnetic field (B) of a current, current in a magnetic field, Lorentz force Particles in a magnetic field, simple applications like cyclotron, magnetic dipole moment c) Ampere's law Magnetic field of simple symmetric systems like straight wire, circular loop and long solenoid d) Law of electromagnetic induction, magnetic flux, Lenz's law, self-induction, inductance, permeability, energy density of magnetic field e) Alternating current, resistors, inductors and Simple AC-circuits, time constants, final formulae for parameters of concrete resonance circuits are not required capacitors in AC-circuits, voltage and current (parallel and series) resonances Electromagnetic waves a) Oscillatory circuit, frequency of oscillations, generation by feedback and resonance b) Wave optics, diffraction from one and two slits, diffraction grating,resolving power of a grating, Bragg reflection, c) Dispersion and diffraction spectra, line spectra of gases d) Electromagnetic waves as transverse waves, polarization by reflection, polarizers Superposition of polarized waves e) Resolving power of imaging systems f) Black body, Stefan-Boltzmanns law Planck's formula is not required Quantum Physics a) Photoelectric effect, energy and impulse of the photon Einstein's formula is required b) De Broglie wavelength, Heisenberg's uncertainty principle 10 Relativity a) Principle of relativity, addition of velocities, relativistic Doppler effect b) Relativistic equation of motion, momentum, energy, relation between energy and mass, conservation of energy and momentum 11 Matter a) Simple applications of the Bragg equation b) Energy levels of atoms and molecules (qualitatively), emission, absorption, spectrum of hydrogen like atoms c) Energy levels of nuclei (qualitatively), alpha- , beta- and gamma-decays, absorption of radiation, halflife and exponential decay, components of nuclei, mass defect, nuclear reactions B Practical Part The Theoretical Part of the Syllabus provides the basis for all the experimental problems The experimental problems given in the experimental contest should contain measurements Additional requirements: Candidates must be aware that instruments affect measurements Knowledge of the most common experimental techniques for measuring physical quantities mentioned in Part A Knowledge of commonly used simple laboratory instruments and devices such as calipers, thermometers, simple volt-, ohm- and ammeters, potentiometers, diodes, transistors, simple optical devices and so on Ability to use, with the help of proper instruction, some sophisticated instruments and devices such as double-beam oscilloscope, counter, ratemeter, signal and function generators, analog-to-digital converter connected to a computer, amplifier, integrator, differentiator, power supply, universal (analog and digital) volt-, ohm- and ammeters Proper identification of error sources and estimation of their influence on the final result(s) Absolute and relative errors, accuracy of measuring instruments, error of a single measurement, error of a series of measurements, error of a quantity given as a function of measured quantities Transformation of a dependence to the linear form by appropriate choice of variables and fitting a straight line to experimental points Proper use of the graph paper with different scales (for example polar and logarithmic papers) Correct rounding off and expressing the final result(s) and error(s) with correct number of significant digits 10 Standard knowledge of safety in laboratory work (Nevertheless, if the experimental set-up contains any safety hazards the appropriate warnings should be included into the text of the problem.) Problems of the 1st International Physics Olympiad (Warsaw, 1967) Waldemar Gorzkowski Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Abstract The article contains the competition problems given at he 1st International Physics Olympiad (Warsaw, 1967) and their solutions Additionally it contains comments of historical character Introduction One of the most important points when preparing the students to the International Physics Olympiads is solving and analysis of the competition problems given in the past Unfortunately, it is very difficult to find appropriate materials The proceedings of the subsequent Olympiads are published starting from the XV IPhO in Sigtuna (Sweden, 1984) It is true that some of very old problems were published (not always in English) in different books or articles, but they are practically unavailable Moreover, sometimes they are more or less substantially changed The original English versions of the problems of the 1st IPhO have not been conserved The permanent Secretariat of the IPhOs was created in 1983 Until this year the Olympic materials were collected by different persons in their private archives These archives as a rule were of amateur character and practically no one of them was complete This article is based on the books by R Kunfalvi [1], Tadeusz Pniewski [2] and Waldemar Gorzkowski [3] Tadeusz Pniewski was one of the members of the Organizing Committee of the Polish Physics Olympiad when the 1st IPhO took place, while R Kunfalvi was one of the members of the International Board at the 1st IPhO For that it seems that credibility of these materials is very high The differences between versions presented by R Kunfalvi and T Pniewski are rather very small (although the book by Pniewski is richer, especially with respect to the solution to the experimental problem) As regards the competition problems given in Sigtuna (1984) or later, they are available, in principle, in appropriate proceedings “In principle” as the proceedings usually were published in a small number of copies, not enough to satisfy present needs of people interested in our competition It is true that every year the organizers provide the permanent Secretariat with a number of copies of the proceedings for free dissemination But the needs are continually growing up and we have disseminated practically all what we had The competition problems were commonly available (at least for some time) just only from the XXVI IPhO in Canberra (Australia) as from that time the organizers started putting the problems on their home pages The Olympic home page www.jyu.fi/ipho contains the problems starting from the XXVIII IPhO in Sudbury (Canada) Unfortunately, the problems given in Canberra (XXVI IPhO) and in Oslo (XXVII IPhO) are not present there The net result is such that finding the competition problems of the Olympiads organized prior to Sudbury is very difficult It seems that the best way of improving the situation is publishing the competition problems of the older Olympiads in our journal The This is somewhat extended version of the article sent for publication in Physics Competitions in July 2003 e-mail: gorzk@ifpan.edu.pl question arises, however, who should it According to the Statutes the problems are created by the local organizing committees It is true that the texts are improved and accepted by the International Board, but always the organizers bear the main responsibility for the topics of the problems, their structure and quality On the other hand, the glory resulting of high level problems goes to them For the above it is absolutely clear to me that they should have an absolute priority with respect to any form of publication So, the best way would be to publish the problems of the older Olympiads by representatives of the organizers from different countries Poland organized the IPhOs for thee times: I IPhO (1967), VII IPhO (1974) and XX IPhO (1989) So, I have decided to give a good example and present the competition problems of these Olympiads in three subsequent articles At the same time I ask our Colleagues and Friends from other countries for doing the same with respect to the Olympiads organized in their countries prior to the XXVIII IPhO (Sudbury) I IPhO (Warsaw 1967) The problems were created by the Organizing Committee At present we are not able to recover the names of the authors of the problems Theoretical problems Problem A small ball with mass M = 0.2 kg rests on a vertical column with height h = 5m A bullet with mass m = 0.01 kg, moving with velocity v0 = 500 m/s, passes horizontally through the center of the ball (Fig 1) The ball reaches the ground at a distance s = 20 m Where does the bullet reach the ground? What part of the kinetic energy of the bullet was converted into heat when the bullet passed trough the ball? Neglect resistance of the air Assume that g = 10 m/s2 M m v0 h Fig s Solution M m v0 v – horizontal component of the velocity of the bullet after collision V – horizontal component of the velocity of the ball after collision h s d Fig We will use notation shown in Fig As no horizontal force acts on the system ball + bullet, the horizontal component of momentum of this system before collision and after collision must be the same: mv0 = mv + MV So, v = v0 − M V m From conditions described in the text of the problem it follows that v >V After collision both the ball and the bullet continue a free motion in the gravitational field with initial horizontal velocities v and V, respectively Motion of the ball and motion of the bullet are continued for the same time: t= 2h g It is time of free fall from height h The distances passed by the ball and bullet during time t are: s = Vt and d = vt , respectively Thus V =s g 2h Therefore v = v0 − M g s m 2h d = v0 2h M − s g m Finally: Numerically: d = 100 m The total kinetic energy of the system was equal to the initial kinetic energy of the bullet: mv0 E0 = Immediately after the collision the total kinetic energy of the system is equal to the sum of the kinetic energy of the bullet and the ball: Em = mv , EM = MV Their difference, converted into heat, was ∆E = E0 − ( E m + E M ) It is the following part of the initial kinetic energy of the bullet: E + EM ∆E = 1− m E0 E0 By using expressions for energies and velocities (quoted earlier) we get p= 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No OPERATION CYCLE OF A WATER-POWERED RICE-POUNDING MORTAR a) a) At the beginning there is no water in the bucket, the pestle rests on the mortar Water flows into the bucket with a small rate, but for some time the lever remains in the horizontal position b) α1 b) At some moment the amount of water is enough to lift the lever up Due to the tilt, water rushes to the farther side of the bucket, tilting the lever more quickly Water starts to flow out at α = α1 c) α=β c) As the angle α increases, water starts to flow out At some particular tilt angle, α = β , the total torque is zero d) d) α2 α e) increasing, water continues to flow out until no water remains in the bucket e) α α0 continues keeps increasing because of inertia Due to the shape of the bucket, water falls into the bucket but immediately flows out The inertial motion of the lever continues until α reaches the maximal value α f) Figure f) With no water in the bucket, the weight of the lever pulls it back to the initial horizontal position The pestle gives the mortar (with rice inside) a pound and a new cycle begins 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No C The problem Consider a water-powered rice-pounding mortar with the following parameters (Figure 3) The mass of the lever (including the pestle but without water) is M = 30 kg, The center of mass of the lever is G The lever rotates around the axis T (projected onto the point T on the figure) The moment of inertia of the lever around T is I = 12 kg ⋅ m2 When there is water in the bucket, the mass of water is denoted as m , the center of mass of the water body is denoted as N The tilt angle of the lever with respect to the horizontal axis is α The main length measurements of the mortar and the bucket are as in Figure Neglect friction at the rotation axis and the force due to water falling onto the bucket In this problem, we make an approximation that the water surface is always horizontal a =20cm Bucket T Lever L = 74 cm N h= 12 cm γ =300 Figure cm Pestle G b =15cm Mortar Design and dimensions of the rice-pounding mortar The structure of the mortar At the beginning, the bucket is empty, and the lever lies horizontally Then water flows into the bucket until the lever starts rotating The amount of water in the bucket at this moment is m = 1.0 kg 1.1 Determine the distance from the center of mass G of the lever to the rotation axis T It is known that GT is horizontal when the bucket is empty 1.2 Water starts flowing out of the bucket when the angle between the lever and the horizontal axis reaches α1 The bucket is completely empty when this angle is α Determine α1 and α 1.3 Let μ (α ) be the total torque (relative to the axis T) which comes from the 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No weight of the lever and the water in the bucket μ (α ) is zero when α = β Determine β and the mass m1 of water in the bucket at this instant Parameters of the working mode Let water flow into the bucket with a flow rate Φ which is constant and small The amount of water flowing into the bucket when the lever is in motion is negligible In this part, neglect the change of the moment of inertia during the working cycle 2.1 Sketch a graph of the torque μ as a function of the angle α , μ (α ) , during one operation cycle Write down explicitly the values of μ (α ) at angle α1, α2, and α = 2.2 From the graph found in section 2.1., discuss and give the geometric interpretation of the value of the total energy Wtotal produced by μ (α ) and the work Wpounding that is transferred from the pestle to the rice 2.3 From the graph representing μ versus α , estimate α and Wpounding (assume the kinetic energy of water flowing into the bucket and out of the bucket is negligible.) You may replace curve lines by zigzag lines, if it simplifies the calculation The rest mode Let water flow into the bucket with a constant rate Φ , but one cannot neglect the amount of water flowing into the bucket during the motion of the lever 3.1 Assuming the bucket is always overflown with water, 3.1.1 Sketch a graph of the torque μ as a function of the angle α in the vicinity of α = β To which kind of equilibrium does the position α = β of the lever belong? 3.1.2 Find the analytic form of the torque μ (α ) as a function of Δα when α = β + Δα , and Δα is small 3.1.3 Write down the equation of motion of the lever, which moves with zero initial velocity from the position α = β + Δα ( Δα is small) Show that the motion is, with good accuracy, harmonic oscillation Compute the period τ 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No 3.2 At a given Φ , the bucket is overflown with water at all times only if the lever moves sufficiently slowly There is an upper limit on the amplitude of harmonic oscillation, which depends on Φ Determine the minimal value Φ1 of Φ (in kg/s) so that the lever can make a harmonic oscillator motion with amplitude 1o 3.3 Assume that Φ is sufficiently large so that during the free motion of the lever when the tilting angle decreases from α to α1 the bucket is always overflown with water However, if Φ is too large the mortar cannot operate Assuming that the motion of the lever is that of a harmonic oscillator, estimate the minimal flow rate Φ for the rice-pounding mortar to not work 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No CHERENKOV LIGHT AND RING IMAGING COUNTER Light propagates in vacuum with the speed c There is no particle which moves with a speed higher than c However, it is possible that in a transparent medium a particle moves with a speed v higher than the speed of the light in the same medium c , where n n is the refraction index of the medium Experiment (Cherenkov, 1934) and theory (Tamm and Frank, 1937) showed that a charged particle, moving with a speed v in a transparent medium with refractive index n such that v > c , radiates light, called n Cherenkov light, in directions forming with the trajectory an angle θ = arccos (1) βn where β = A θ B θ v c To establish this fact, consider a particle moving at constant velocity v > c on a n straight line It passes A at time and B at time t1 As the problem is symmetric with respect to rotations around AB, it is sufficient to consider light rays in a plane containing AB At any point C between A and B, the particle emits a spherical light wave, which propagates with velocity c We define the wave front at a given time t as the envelope n of all these spheres at this time 1.1 Determine the wave front at time t1 and draw its intersection with a plane containing the trajectory of the particle 1.2 Express the angle ϕ between this intersection and the trajectory of the particle in terms of n and β Let us consider a beam of particles moving with velocity v > c , such that the angle n θ is small, along a straight line IS The beam crosses a concave spherical mirror of focal length f and center C, at point S SC makes with SI a small angle α (see the figure in the Answer Sheet) The particle beam creates a ring image in the focal plane of the mirror 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No Explain why with the help of a sketch illustrating this fact Give the position of the center O and the radius r of the ring image This set up is used in ring imaging Cherenkov counters (RICH) and the medium which the particle traverses is called the radiator Note: in all questions of the present problem, terms of second order and higher in α and θ will be neglected A beam of particles of known momentum p = 10.0 GeV/c consists of three types of particles: protons, kaons and pions, with rest mass M p = 0.94 GeV / c , M κ = 0.50 GeV / c and M π = 0.14 GeV / c , respectively Remember that pc and Mc have the dimension of an energy, and eV is the energy acquired by an electron after being accelerated by a voltage V, and GeV = 109 eV, MeV = 106 eV The particle beam traverses an air medium (the radiator) under the pressure P The refraction index of air depends on the air pressure P according to the relation n = + aP where a = 2.7×10-4 atm-1 3.1 Calculate for each of the three particle types the minimal value Pmin of the air pressure such that they emit Cherenkov light 3.2 Calculate the pressure P1 such that the ring image of kaons has a radius equal to one half of that corresponding to pions Calculate the values of θ κ and θ π in this case Is it possible to observe the ring image of protons under this pressure? Assume now that the beam is not perfectly monochromatic: the particles momenta are distributed over an interval centered at 10 GeV / c having a half width at half height Δp This makes the ring image broaden, correspondingly θ distribution has a half width at half height Δθ The pressure of the radiator is P1 determined in 3.2 4.1 Calculate Δθ π Δθ κ Δθ and , the values taken by in the pions and kaons Δp Δp Δp cases 4.2 When the separation between the two ring images, θ π − θ κ , is greater than 10 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No times the half-width sum Δθ = Δθ κ + Δθ π , that is θ π − θ κ > 10 Δθ , it is possible to distinguish well the two ring images Calculate the maximal value of Δp such that the two ring images can still be well distinguished Cherenkov first discovered the effect bearing his name when he was observing a bottle of water located near a radioactive source He saw that the water in the bottle emitted light 5.1 Find out the minimal kinetic energy Tmin of a particle with a rest mass M moving in water, such that it emits Cherenkov light The index of refraction of water is n = 1.33 5.2 The radioactive source used by Cherenkov emits either α particles (i.e helium nuclei) having a rest mass M α = 3.8 GeV / c or β particles (i.e electrons) having a rest mass M e = 0.51 MeV / c Calculate the numerical values of Tmin for α particles and β particles Knowing that the kinetic energy of particles emitted by radioactive sources never exceeds a few MeV, find out which particles give rise to the radiation observed by Cherenkov In the previous sections of the problem, the dependence of the Cherenkov effect on wavelength λ has been ignored We now take into account the fact that the Cherenkov radiation of a particle has a broad continuous spectrum including the visible range (wavelengths from 0.4 µm to 0.8 µm) We know also that the index of refraction n of the radiator decreases linearly by 2% of n − when λ increases over this range 6.1 Consider a beam of pions with definite momentum of 10.0 GeV / c moving in air at pressure atm Find out the angular difference δθ associated with the two ends of the visible range 6.2 On this basis, study qualitatively the effect of the dispersion on the ring image of pions with momentum distributed over an interval centered at p = 10 GeV / c and having a half width at half height Δp = 0.3 GeV / c 6.2.1 Calculate the broadening due to dispersion (varying refraction index) and that due to achromaticity of the beam (varying momentum) 6.2.2 Describe how the color of the ring changes when going from its inner to outer edges by checking the appropriate boxes in the Answer Sheet 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No CHANGE OF AIR TEMPERATURE WITH ALTITUDE, ATMOSPHERIC STABILITY AND AIR POLLUTION Vertical motion of air governs many atmospheric processes, such as the formation of clouds and precipitation and the dispersal of air pollutants If the atmosphere is stable, vertical motion is restricted and air pollutants tend to be accumulated around the emission site rather than dispersed and diluted Meanwhile, in an unstable atmosphere, vertical motion of air encourages the vertical dispersal of air pollutants Therefore, the pollutants’ concentrations depend not only on the strength of emission sources but also on the stability of the atmosphere We shall determine the atmospheric stability by using the concept of air parcel in meteorology and compare the temperature of the air parcel rising or sinking adiabatically in the atmosphere to that of the surrounding air We will see that in many cases an air parcel containing air pollutants and rising from the ground will come to rest at a certain altitude, called a mixing height The greater the mixing height, the lower the air pollutant concentration We will evaluate the mixing height and the concentration of carbon monoxide emitted by motorbikes in the Hanoi metropolitan area for a morning rush hour scenario, in which the vertical mixing is restricted due to a temperature inversion (air temperature increases with altitude) at elevations above 119 m Let us consider the air as an ideal diatomic gas, with molar mass μ = 29 g/mol Quasi equilibrium adiabatic transformation obey the equation pV γ = const , where γ= cp cV is the ratio between isobaric and isochoric heat capacities of the gas The student may use the following data if necessary: The universal gas constant is R = 8.31 J/(mol.K) The atmospheric pressure on ground is p0 = 101.3 kPa The acceleration due to gravity is constant, g = 9.81 m/s2 R for air The molar isochoric heat capacity is cV = R for air The molar isobaric heat capacity is c p = 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No Mathematical hints d ( A + Bx ) = ln ( A + Bx ) A + Bx B dx b The solution of the differential equation + Ax =B (with A and B constant) is dt B dx x ( t ) = x1 ( t ) + where x1 ( t ) is the solution of the differential equation + Ax =0 dt A a dx ∫ A + Bx = B ∫ ⎛ ⎝ c lim x →∞ ⎜ + x 1⎞ =e x⎟ ⎠ Change of pressure with altitude 1.1 Assume that the temperature of the atmosphere is uniform and equal to T0 Write down the expression giving the atmospheric pressure p as a function of the altitude z 1.2 Assume that the temperature of the atmosphere varies with the altitude according to the relation T ( z ) = T ( 0) − Λ z where Λ is a constant, called the temperature lapse rate of the atmosphere (the vertical gradient of temperature is - Λ ) 1.2.1 Write down the expression giving the atmospheric pressure p as a function of the altitude z 1.2.2 A process called free convection occurs when the air density increases with altitude At which values of Λ does the free convection occur? Change of the temperature of an air parcel in vertical motion Consider an air parcel moving upward and downward in the atmosphere An air parcel is a body of air of sufficient dimension, several meters across, to be treated as an independent thermodynamical entity, yet small enough for its temperature to be considered uniform The vertical motion of an air parcel can be treated as a quasi adiabatic process, i.e the exchange of heat with the surrounding air is negligible If the air parcel rises in the atmosphere, it expands and cools Conversely, if it moves downward, the increasing outside pressure will compress the air inside the parcel and its temperature will increase As the size of the parcel is not large, the atmospheric pressure at different points on 39th International Physics Olympiad - Hanoi - Vietnam - 2008 Theoretical Problem No the parcel boundary can be considered to have the same value p ( z ) , with z - the altitude of the parcel center The temperature in the parcel is uniform and equals to Tparcel ( z ) , which is generally different from the temperature of the surrounding air T ( z ) In parts 2.1 and 2.2, we not make any assumption about the form of T(z) 2.1 The change of the parcel temperature Tparcel with altitude is defined by dTparcel dz = −G Derive the expression of G (T, Tparcel) 2.2 Consider a special atmospheric condition in which at any altitude z the temperature T of the atmosphere equals to that of the parcel Tparcel , T ( z ) = Tparcel ( z ) We use Γ to denote the value of G when T = Tparcel , that is Γ = − dTparcel dz (with T = Tparcel ) Γ is called dry adiabatic lapse rate 2.2.1 Derive the expression of Γ 2.2.2 Calculate the numerical value of Γ 2.2.3 Derive the expression of the atmospheric temperature T ( z ) as a function of the altitude 2.3 Assume that the atmospheric temperature depends on altitude according to the relation T ( z ) = T ( ) − Λ z , where Λ is a constant Find the dependence of the parcel temperature Tparcel ( z ) on altitude z 2.4 Write down the approximate expression of Tparcel ( z ) when Λz