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 semi axis of its orbit.
equal energies. d, such that a parallel incident beam is col-
i) (2 points) The missile is launched as de-
i) (1.5 points) Find the speed of the electron lected perfectly into the focus (has aspheric scribed above in an arbitrary direction. What
v_e in the lab frame. surfaces). What is the minimal possible thick- is the period of the missile on its elliptic or-
ness of such a lens if the coefficient of refrac- bit?
ii) (0.5 points) Find the energy E_γ of photon tion of the glass is n=1.5?
A in the lab frame. ii) (2 points) According to the Kepler's I law,
iii) (2 points) If a short light pulse falls the missile's orbit is an ellipse, with one of the
iii) (1 point) Find the momentum p_γ of onto a lens, the behaviour of the holographic foci being at S. The position of the other fo-
photon A in the lab frame. lens is very different from that of the glass cus F depends on the launching angle. While
lens. Indeed, if we ignore the dispersion the launching angle is changed, the focus F
iv) (2 points) Find the angle α between the of the glass (the dependence of n on the moves along a certain curve; what curve is
z-axis and the momentum of photon A in the wavelength), the pulse duration at the fo- The rotations of the engine are transferred to it? Also, determine the geometric dimensions
lab frame. cus of the glass lens is the same as that of the wheels through a transmission gearbox. of that curve.
the incident beam. Meanwhile, in the case When the car is in the first gear, then the
v) (1 point) Why is it not possible for a of a short pulse (τ=3×10^-14 s), the pulse gear ratio from the engine to the wheels is iii) (1 point) Let us consider such a point
collision process to produce just one single duration at the focus of the holographic lens 14:1; in the second gear, the gear ratio is 7:1. Q on the missile's trajectory, the distance of
photon? becomes significantly dilated. Sketch qual- which from the star equals to |SQ|=r. De-
itatively the light intensity I at the focus of i) (2 points) At what speed of the car should termine the distance |QF|.
The rest mass of electron m_e=511 keV/c^2. the holographic lens as a function of time. A the transmission be shifted from the first
You can express your results in electron volts scale is required for time t, but not for I. The gear into the second, in order for the aver- iv) (1 point) The position of point Q (as
and related units. speed of light c=3×10^8 m/s. age acceleration of the car to be maximum? defined above) depends on the launching
angle of the missile, and so does the distance
2. HOLOGRAPHIC LENS (7 points) (by iv) (1 point) Even if the light source is an ii) (1 point) What is the acceleration of |PQ|. Among all the possible positions of the
J. Kalda) Monochromatic light can be focused ideal laser providing a perfectly monochro- the car immediately before and immediately point Q, what is the largest possible distance
into a point using an holographic lens (bet- matic radiation, the short pulse is no longer after the gear change? |PQ|?
ter known as a Fresnel zone plate). This is strictly monochromatic. Estimate the width Neglect any drag force. The mass of the
a thin film which has a system of opaque of the range of wavelengths in the pulse of car m=1400 kg, the diameter of a wheel is v) (2 points) For questions iii and iv, we as-
and transparent concentric rings: the opaque duration τ. d=60 cm. sumed that the distance r had a fixed value.
rings obstruct those light waves which would Now, if we consider r as a free parameter, the
arrive at the focus in the opposite phase (as 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
compared with the light from the transpar- 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
ent rings). In what follows, let us assume Assume that in the glass, the light wave can be hit with an interplanetary ballistic boundary of that region of space to which the
that the diameter and focal length of such group velocity v_g depends on the wavelength missile. The launching point P is motion- point Q can belong?
a lens d=f=10 cm; the wavelength of the at the rate dv_g/dλ = 0.02×v_g/λ and that for the gravitationally less in the inertial frame of reference of a
monochromatic incident light radiation is attracting star centred at S.
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 << l^2.
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 = (λ − λ_0)/p, where p is the heat flux g=9.8 m/s^2. ii) (2 points) Find the initial angular speed netic field at the centre of a solenoid sepa-
from the aluminium profile p = λA(dT/dx); ω in terms of v, α and R. rately is B = μ_0NI/l, where μ_0 is the per-
length; T and T_0 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- 2nd ball [A_1, A_2 << (x−l)^2, x^2]. 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 mm^2. g L. Franti) A conducting ring with radius R
Hint: to describe the stationary temper- P carries a large current I. The ring lies sta- ii) (4 points) Find the electromotive forces
ature distribution along the profile we can throwing tionary in the xy-plane with its center point E_1 and E_2 generated on the coils when one is
derive a one-dimensional Helmholtz equa- point at (0,0,0). An observer approaches the ring pulled out with a speed v.
tion T''(x) = (h/(kA))(T(x)−T_0), the general solution parallel to the z-axis with velocity v and
of which is 10 m measures the electric field at distance r from 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))), the axis. Assume that v << c and r >> R. one coil outwards.
where C_1 and C_2 are integration constants. 7. BOUNCY BALL (5 points) (by J. Toots) i) (2 points) Find the direction and mag- 10. VAPOUR PRESSURE (8 points) (by
A homogeneous elastic ball with radius R nitude of the magnetic field B(z) on z axis in J. Kalda, M. Heidelberg, E. Uustalu) Meas-
Equipment: Aluminium profile, wire (for bounces from a vertical wall in such a way stationary reference frame. ure the room temperature saturated vapour
heating), DC power source, infrared thermo- that it rebounds back exactly in the same pressure of the unknown fluid in the syringe.
meter, ruler. direction it came from. The ball's speed be- ii) (4 points) Estimate the magnitude and The volume of the bottle is V=0.50 ℓ, the in-
Keep the temperature of the wire be- fore the bounce is v at an angle α to the ver- direction of electric field E(z,r) measured by ner diameter of the pipe is d=6.0 mm. Cur-
low 150°C as both the wire insulation tical; the ball also has an angular speed ω. the observer when he is at a distance z from rent air pressure p_0 and room temperature
and the table surface will start to emit There was no slipping at the contact point, the ring's plane. T_0 are written on the blackboard. There is
smoke after that point! but do not assume that ωR = v cos α. The no need to repeat the measurement after a
bounce was completely elastic, ie no kinetic 9. SOLENOIDS (8 points) (by S. Ainsaar). successful attempt as getting the bottle free
6. TWO BALLS (5 points) (by J. Kalda) energy was lost; also, the horizontal compon- A tightly-wound rigid solenoid coil has been from vapours is time consuming. Estimate
The following snapshot (a larger version is on ents of incident and rebound velocities have partly inserted into another. They are con- the uncertainty of the result.
an extra sheet) depicts two balls that were the same magnitude. The moment of inertia nected to a constant current source so that
for a homogeneous ball is I = 2mR^2/5. the same current I flows through them; both Equipment: syringe with an unknown
generate a magnetic field in the same direc- fluid, syringe with water, bottle, flexible pipe,
tion. The solenoids have N turns both, their ruler, plug, stand, tape.