Estonian-Finnish Olympiad 2015 λ = 5×10⁻⁷m. at the rate dv_g/dλ = 0.02×v_g/λ and that for the gravitationally attracting star centred at S.
Editors: S. Ainsaar, M. Heidelberg wavelength λ, group speed is equal to the The launching speed of the missile is fixed
i) (1.5 points) Starting enumeration from the phase speed. throughout this problem and equal to v₀, the (opaque) centre of the lens, what is the radius same applies to the distance to launching
1. ANNIHILATION (6 points) (by L. Franti) of the m-th transparent ring (more precisely, 3. GEAR SHIFT (3 points) (by K.A. Saar) point |SP| = R. The mass of the star is M, An electron with kinetic energy 1 MeV the circle at the middle of the ring)? The maximum torque of a car's engine de- and the gravity constant is G. Hint: the total travels along the z-axis and collides with a pends on the engine's rotation speed (see the energy of a missile of mass m moving on an positron at rest. The particles annihilate pro- following figure, a larger copy is on an extra elliptical orbit is E = −GMm/(2a), where a is ducing a pair of two photons A and B with sheet). the longer semiaxis of its orbit.
ii) (1.5 points) Let us consider also a glass equal energies. lens of the same focal length f and diameter i) (2 points) The missile is launched as de-
i) (1.5 points) Find the speed of the electron d, such that a parallel incident beam is col- scribed above in an arbitrary direction. What v_e in the lab frame. lected perfectly into the focus (has aspheric is the period of the missile on its elliptic or-
ii) (0.5 points) Find the energy E_γ of photon surfaces). What is the minimal possible thick- bit? A in the lab frame. ness of such a lens if the coefficient of refrac-
iii) (1 point) Find the momentum p_γ of tion of the glass is n = 1.5? ii) (2 points) According to the Kepler's I law, photon A in the lab frame. the missile's orbit is an ellipse, with one of the
iii) (2 points) If a short light pulse falls foci being at S. The position of the other fo-
iv) (2 points) Find the angle α between the onto a lens, the behaviour of the holographic cus F depends on the launching angle. While z-axis and the momentum of photon A in the lens is very different from that of the glass the launching angle is changed, the focus F lab frame. lens. Indeed, if we ignore the dispersion moves along a certain curve; what curve is
The rotations of the engine are transferred to of the glass (the dependence of n on the it? Also, determine the geometric dimensions v) (1 point) Why is it not possible for a the wheels through a transmission gearbox. wavelength), the pulse duration at the fo- of that curve. collision process to produce just one single cus of the glass lens is the same as that of
When the car is in the first gear, then the photon? the incident beam. Meanwhile, in the case iii) (1 point) Let us consider such a point gear ratio from the engine to the wheels is of a short pulse (τ = 3×10⁻¹⁴s), the pulse Q on the missile's trajectory, the distance of
The rest mass of electron m_e = 511 keV/c². 14:1; in the second gear, the gear ratio is 7:1. duration at the focus of the holographic lens which from the star equals to |SQ| = r. De-You can express your results in electron volts becomes significantly dilated. Sketch qual- termine the distance |QF|. and related units. itatively the light intensity I at the focus of
i) (2 points) At what speed of the car should the holographic lens as a function of time. A iv) (1 point) The position of point Q (as
2. HOLOGRAPHIC LENS (7 points) (by the transmission be shifted from the first scale is required for time t, but not for I. The defined above) depends on the launching J. Kalda) Monochromatic light can be focused gear into the second, in order for the aver- speed of light c = 3×10⁸ m/s. angle of the missile, and so does the distance into a point using an holographic lens (bet- age acceleration of the car to be maximum? |PQ|. Among all the possible positions of the ter known as a Fresnel zone plate). This is
iv) (1 point) Even if the light source is an point Q, what is the largest possible distance a thin film which has a system of opaque ii) (1 point) What is the acceleration of ideal laser providing a perfectly monochro- |PQ|? and transparent concentric rings: the opaque the car immediately before and immediately matic radiation, the short pulse is no longer rings obstruct those light waves which would after the gear change? strictly monochromatic. Estimate the width v) (2 points) For questions iii and iv, we as-arrive at the focus in the opposite phase (as Neglect any drag force. The mass of the of the range of wavelengths in the pulse of sumed that the distance r had a fixed value. compared with the light from the transpar- car m = 1400 kg, the diameter of a wheel is duration τ. Now, if we consider r as a free parameter, the ent rings). In what follows, let us assume d = 60 cm. position of the point Q is a function of both that the diameter and focal length of such v) (1 point) Estimate the duration of this r and of the launching angle. What is the a lens d = f = 10 cm; the wavelength of the light pulse in the focus of the glass lens. 4. STAR WARS (8 points) (by J. Kalda) Here boundary of that region of space to which the monochromatic incident light radiation is Assume that in the glass, the light wave we'll study, in which region of space targets point Q can belong? λ = 5×10⁻⁷m. group velocity v_g depends on the wavelength can be hit with an interplanetary ballistic missile. The launching point P is motion-
5. RADIATOR (8 points) (by M. Heidel- thrown simultaneously and with the same i) (1 point) Find the angular speed ω₂ of the length is l, their cross-sectional areas are A₁ berg) Measure the heat transfer coefficient h initial speed, but in different directions from ball after the bounce. and A₂. It may be assumed that A₁, A₂ ≪ l². between the environment and an aluminium point P. What was the initial speed? Use You may find it useful to know that the mag-profile. h = q/(T − T₀), where q is the heat flux g = 9.8 m/s². ii) (2 points) Find the initial angular speed netic field at the centre of one solenoid taken from the aluminium profile to air; l is the ω in terms of v, α and R. separately is B = μ₀IN/l, where μ₀ is the per-profile length; T and T₀ are the temperatures of alu- 1st ball meability of vacuum. minium and the environment, respectively. iii) (2 points) What is the minimum value
For measurements, keep the aluminium pro- for coefficient of friction μ that such bounce i) (2 points) The centres of the coils are file flat on your table surface, to limit nonlin- could happen? at a distance x < l along their common axis ear effects from turbulence. Thermal conduct- g [A₁, A₂ ≪ (x−l)², x²]. Find the total energy ivity of aluminium is k = 205 W/(K·m) and 8. ELECTRIC FIELD (6 points) (by E_m of the magnetic field in the system. the cross-section of the profile is A = 36 mm². 2nd ball L. Franti) A conducting ring with radius R
carries a large current I. The ring lies sta- ii) (4 points) Find the electromotive forces Hint: to describe the stationary temper- P tionary in the xy-plane with its center point E₁ and E₂ generated on the coils when one is ature distribution along the profile we can throwing at (0,0,0). An observer approaches the ring pulled out with a speed v. derive a one-dimensional Helmholtz equa- point parallel to the z-axis with velocity v and tion T''(x) = h/(kA)(T(x) − T₀), the general solution measures the electric field at distance r from iii) (2 points) Find the force F needed to pull of which is 10 m the axis. Assume that v ≪ c and r ≪ R. one coil outwards.
T(x) = T₀ + C₁e^(x√(h/kA)) + C₂e^(-x√(h/kA)), i) (2 points) Find the direction and mag- 10. VAPOUR PRESSURE (8 points) (by nitude of the magnetic field B(z) on z axis in J. Kalda, M. Heidelberg, E. Uustalu) Meas-where C₁ and C₂ are integration constants. 7. BOUNCY BALL (5 points) (by J. Toots) stationary reference frame. ure the room temperature saturated vapour
A homogeneous elastic ball with radius R pressure of the unknown fluid in the syringe. Equipment: Aluminium profile, wire (for bounces from a vertical wall in such a way ii) (4 points) Estimate the magnitude and The volume of the bottle is V = 0.50 ℓ, the in-heating), DC power source, infrared thermo- that it rebounds back exactly in the same direction of electric field E(z,r) measured by ner diameter of the pipe is d = 6.0 mm. Cur-meter, ruler. direction it came from. The ball's speed be- the observer when he is at a distance z from rent air pressure p₀ and room temperature
Keep the temperature of the wire be- fore the bounce is v at an angle α to the ver- the ring's plane. T₀ are written on the blackboard. There is low 150°C as both the wire insulation tical; the ball also has an angular speed ω. no need to repeat the measurement after a and the table surface will start to emit There was no slipping at the contact point, 9. SOLENOIDS (8 points) (by S. Ainsaar). successful attempt as getting the bottle free smoke after that point! but do not assume that ωR = v cos α. The A tightly-wound rigid solenoid coil has been from vapours is time consuming. Estimate
bounce was completely elastic, ie no kinetic partly inserted into another. They are con- the uncertainty of the result. 6. TWO BALLS (5 points) (by J. Kalda) energy was lost; also, the horizontal compon- nected to a constant current source so that
The following snapshot (a larger version is on ents of incident and rebound velocities have the same current I flows through them; both Equipment: syringe with an unknown an extra sheet) depicts two balls that were the same magnitude. The moment of inertia generate a magnetic field in the same direc- fluid, syringe with water, bottle, flexible pipe, for a homogeneous ball is I = 2mR²/5. tion. The solenoids have N turns both, their ruler, plug, stand, tape.