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Calculate the phase shift $\Delta \Phi $ when an optical beam with a wavelength $\lambda _0$ goes through a glass plate with thickness $h$ and the index of refraction $n$ that is moving along the beam with constant speed $v$ relative to a case when the plate is stationary. We are interested mainly about the first nonzero term of Taylor series of $\Delta \Phi (v)$.

Before he set off on his flight towards Mars, the Starman in his Tesla Roadster arranged with Musk that once he reaches the distance $r=5 \cdot 10^{6} \mathrm{km}$ from the centre of mass of the Earth, Musk will shine a powerful green laser at him. The wavelength of the laser increases under the influence of the gravitational field of Earth. Compare this change of the wavelength to the electromagnetic Doppler effect. Study each of these effects separately. Assume that the Starman is moving away from Earth with velocity $v=4 \mathrm{km\cdot s^{-1}}$.

We are sorry. This type of task is not translated to English.

We are sorry. This type of task is not translated to English.

From what height would we need to „drop“ an object on a neutron star to make it land with a speed 0,1 $c$ (0,1 of speed of light). Our neutron star is 1.5 times heavier than our Sun and has diameter $d=10\;\mathrm{km}$. Ignore both the atmosphere of the star and its rotation. You can also ignore the correction for special relativity. However, do compare the results for a homogenous gravitational field (with the same strength as is on the star surface) and for a radial gravitational field. Bonus: Do not ignore the special relativity correction.

Superman and Flash decided to race each other. The race takes place in deep space as there is no straight beach long enough on Earth. As Flash is slower, he starts with a length lead $l$ ahead of Superman. At one moment, Flash starts with a constant speed $v_{F}$ comparable with the speed of light. At the moment Superman sees that Flash started, he starts running at a constant speed $v_{S}>v_{F}$. How long will it take Superman to catch up with Flash (from Superman's point of view)? How long will it take from Flash's point of view? Was the starting method fair? Can you devise a more fair method (keeping the length lead $l)?$

Lukas has been weightlifting and he managed to make a black hole of mass 1 kg. As he isn't too fond of quantum field theory in curved spacetime, the black hole does not radiate. Lukas drops this hole and it begins oscillating within the earth. Try to estimate how long would it take for the mass of the black hole to double. Is it safe to make black holes at home?

During his visit to Ankh-Morpork Two flower also visited a pub. It wouldn't have been a good pub if a general brawl didn't occur. A brawl during which chairs, bottles and other things fly fromone side of the pub to the other. Twoflower obviously documented everything with his camera. Now he is currently taking a picture of a ball of radius $Rthat$ is flying with a velocity $v$ (which is close to the speed of light $c)$. Even in such establishments the theory of relativity is valid and from it stems that Twoflower could have measured the Fitzgerald contraction of the ball in his rest frame in the direction of movement by a factor of

$$\\ \sqrt{1- \frac{v^2}{c^2}}$$

What radius of the ball was documented by the camera with a negligibly small exposition?The position of the camera is general.

Terka went on a trip once again. This time she was walking during equinox at twelve o'clock on the Equator. What would her velocity be relative to Ales, if he would want to (foolishly) watch her from the surface of the Sun on the Equator at a point nearest to his object of interest (Terka)? The axial tilt of the sun with respect to the plane of the ecliptic can be considered negligible.

Teresa was approaching with a relativistic speed $v$ a plane mirror. She was approaching perpendicularly to the mirror's plane. While doing so she is watching herself approach the mirror. What is the actual speed that she is approaching her image with and what is the speed she is observing?

Bonus: It isn't a plane mirror but a spherical one