Estonian-Finnish Olympiad 2015 λ=5×10^-7 m. 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 v_0, 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. GEARSHIFT (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- ii) (1.5 points) Let us consider also a glass 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 lens of the same focal length f and diameter sheet). the longer semiaxis of its orbit. equal energies. d, such that a parallel incident beam is col-
i) (1.5 points) Find the speed of the electron lected perfectly into the focus (has aspheric i) (2 points) The missile is launched as de- v_e in the lab frame. surfaces). What is the minimal possible thick- scribed above in an arbitrary direction. What
ii) (0.5 points) Find the energy E_γ of photon ness of such a lens if the coefficient of refrac- is the period of the missile on its elliptic or- A in the lab frame. tion of the glass is n=1.5? bit?
iii) (1 point) Find the momentum p_γ of iii) (2 points) If a short light pulse falls ii) (2 points) According to the Kepler's I law, photon A in the lab frame. onto a lens, the behaviour of the holographic the missile's orbit is an ellipse, with one of the
iv) (2 points) Find the angle α between the lens is very different from that of the glass foci being at S. The position of the other fo- z-axis and the momentum of photon A in the lens. Indeed, if we ignore the dispersion cus F depends on the launching angle. While lab frame. of the glass (the dependence of n on the the launching angle is changed, the focus F
The rotations of the engine are transferred to moves along a certain curve; what curve is v) (1 point) Why is it not possible for a the wheels through a transmission gearbox. it? Also, determine the geometric dimensions collision process to produce just one single wavelength), the pulse duration at the fo- photon? cus of the glass lens is the same as that of When the car is in the first gear, then the of that curve.
The rest mass of electron m_e = 511 keV/c². the incident beam. Meanwhile, in the case gear ratio from the engine to the wheels is iii) (1 point) Let us consider such a point You can express your results in electron volts of a short pulse (τ=3×10^-14 s), the pulse 14:1; in the second gear, the gear ratio is 7:1. Q on the missile's trajectory, the distance of and related units. duration at the focus of the holographic lens which from the star equals to |SQ|=r. De-
becomes significantly dilated. Sketch qual- i) (2 points) At what speed of the car should termine the distance |QF|. 2. HOLOGRAPHIC LENS (7 points) (by itatively the light intensity I at the focus of the transmission be shifted from the first J. Kalda) Monochromatic light can be focused the holographic lens as a function of time. A gear into the second, in order for the aver- iv) (1 point) The position of point Q (as into a point using an holographic lens (bet- scale is required for time t, but not for I. The age acceleration of the car to be maximum? defined above) depends on the launching ter known as a Fresnel zone plate). This is speed of light c=3×10⁸ m/s. angle of the missile, and so does the distance a thin film which has a system of opaque ii) (1 point) What is the acceleration of |PQ|. Among all the possible positions of the and transparent concentric rings: the opaque iv) (1 point) Even if the light source is an the car immediately before and immediately point Q, what is the largest possible distance rings obstruct those light waves which would ideal laser providing a perfectly monochro- after the gear change? |PQ|? arrive at the focus in the opposite phase (as matic radiation, the short pulse is no longer Neglect any drag force. The mass of the compared with the light from the transpar- strictly monochromatic. Estimate the width car m=1400 kg, the diameter of a wheel is v) (2 points) For questions iii and iv, we as- ent rings). In what follows, let us assume of the range of wavelengths in the pulse of d=60 cm. sumed that the distance r had a fixed value. that the diameter and focal length of such duration τ. Now, if we consider r as a free parameter, the a lens d = f = 10 cm; the wavelength of the v) (1 point) Estimate the duration of this 4. STAR WARS (8 points) (by J. Kalda) Here position of the point Q is a function of both monochromatic incident light radiation is light pulse in the focus of the glass lens. we'll study, in which region of space targets r and of the launching angle. What is the λ = 5×10^-7 m. Assume that in the glass, the light wave can be hit with an interplanetary ballistic boundary of that region of space to which the
group velocity v_g depends on the wavelength missile. The launching point P is motion- point Q can belong?
5. RADIATOR (8 points) (by M. Heidel- thrown simultaneously and with the same i) (1 point) Find the angular speed ω_2 of the length is l, their cross-sectional areas are A_1 berg) Measure the heat transfer coefficient h initial speed, but in different directions from ball after the bounce. and A_2. It may be assumed that A_1, A_2 << 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. The heat flux from the aluminium g = 9.8 m/s². ii) (2 points) Find the initial angular speed netic field separately at the centre of one so- profile to the air is q = h(T−T_0), where T ω in terms of v, α and R. lenoid is B = μ_0NI/l, where μ_0 is the per- and T_0 are the temperatures of alumi- 1st ball meability of vacuum. nium 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- 2nd ball [A_1, A_2 << (x−l)², x²]. Find the total energy ivity of aluminium is k = 205 W/(K·m) and g E_m of the magnetic field in the system. the cross-section of the profile is A = 36 mm². 8. ELECTRIC FIELD (6 points) (by
Hint: to describe the stationary temper- P L. Franti) A conducting ring with radius R ii) (4 points) Find the electromotive forces ature distribution along the profile we can throwing carries a large current I. The ring lies sta- E_1 and E_2 generated on the coils when one is derive a one-dimensional Helmholtz equa- point tionary in the xy-plane with its center point pulled out with a speed v. tion T''(x) = (h/kA)(T(x)−T_0), the general solution at (0,0,0). An observer approaches the ring of which is 10 m parallel to the z-axis with velocity v and iii) (2 points) Find the force F needed to pull
T(x) = T_0 + C_1 e^(x√(h/kA)) + C_2 e^(−x√(h/kA)), measures the electric field at distance r from one coil outwards.
the axis. Assume that v << c and r << R. 10. VAPOUR PRESSURE (8 points) (by where C_1 and C_2 are integration constants. i) (2 points) Find the direction and mag- J. Kalda, M. Heidelberg, E. Uustalu) Meas-
Equipment: Aluminium profile, wire (for nitude of the magnetic field B(z) on z axis in ure the room temperature saturated vapour heating), DC power source, infrared thermo- 7. BOUNCY BALL (5 points) (by J. Toots) stationary reference frame. pressure of the unknown fluid in the syringe. meter, ruler. A homogeneous elastic ball with radius R The volume of the bottle is V = 0.50 ℓ, the in-
Keep the temperature of the wire be- bounces from a vertical wall in such a way ii) (4 points) Estimate the magnitude and ner diameter of the pipe is d = 6.0 mm. Cur- low 150°C as both the wire insulation that it rebounds back exactly in the same direction of electric field E(z,r) measured by rent air pressure p_0 and room temperature and the table surface will start to emit direction it came from. The ball's speed be- the observer when he is at a distance z from T_0 are written on the blackboard. There is smoke after that point! fore the bounce is v at an angle α to the ver- the ring's plane. no need to repeat the measurement after a
6. TWO BALLS (5 points) (by J. Kalda) tical; the ball also has an angular speed ω. 9. SOLENOIDS (8 points) (by S. Ainsaar). successful attempt as getting the bottle free The following snapshot (a larger version is on There was no slipping at the contact point, A tightly-wound rigid solenoid coil has been from vapours is time consuming. Estimate an extra sheet) depicts two balls that were but do not assume that ωR = v cos α. The partly inserted into another. They are con- the uncertainty of the result.
bounce was completely elastic, ie no kinetic nected to a constant current source so that Equipment: syringe with an unknown energy was lost; also, the horizontal compon- the same current I flows through them; both fluid, syringe with water, bottle, flexible pipe, ents of incident and rebound velocities have generate a magnetic field in the same direc- ruler, plug, stand, tape. the same magnitude. The moment of inertia tion. The solenoids have N turns both, their for a homogeneous ball is I = (2/5)mR².