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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

By how much shall the temperature of two identical steel balls rise after their collision?They move in the same direction with speeds $v_{1}=0,7c$ and $v_{2}=0,9c$ where $c$ is the speed of light. Assume that the heat capacity is constant and that the balls are still solid.

What would be the world like if the speed of light was only $c=1000\;\mathrm{km}\cdot h^{-1}$ while all the other fundamental constants stayed unchanged? What would be the impact on life on Earth? Would it even be possible for people to exist in such a world?

• Any theory of quantum gravity is useful only when we deal with very small distances where the effects of gravitation are comparable to quantum effects. Gravitation is characterized by the gravitational constant, quantum mechanics by the Planck constant, and special relativity by the speed of light. Look up numerical values of these constants, and, using standard algebraic operations, combine them to obtain a quantity with the dimensions of length. This is the length scale where both quantum mechanics and gravitation are important.

• Prove that the special Lorentz transform (i.e. a change of the reference frame to one that is moving with speed $v$ in the $x¹;$ direction)

$$x^0_\;\mathrm{nov}=\frac{x^0-\frac{v}{c}x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^1_\mathrm{nov}=\frac{-\frac{v}{c}x^0 x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^2_\mathrm{nov}= x^2\,,\quad x^3_\mathrm{nov}= x^3$$ leaves the spacetime interval invariant. * Set $Δx=Δx=0$ in the definition of a spacetime interval. You should get

$$(\Delta s)^2 = -\(\Delta x^0\)^2 \(\Delta x^1\)^2$$

What is the region of the plane ( $Δx^{0},Δx¹;)$ where the spacetime interval ( $Δs)$ is positive? Where negative? What is the curve ( $Δs)=0?$

At the speeds close to speed of light you can observe contraction of objects. On the other hand the objects looks longer, then they are in reality (try to follow and compare light beams from closer and distant part of a body). Calculate, if these two effect will cancel for a sphere.

An rich space-tourist has payed for a trip to the deep space. The Racket flies from the Earth and is accelerating with constant acceleration $a$, which can be verified by dropping a small ball. He is quite bored and therefore is watching disappearing Earth through the back window. After some time (how long it will take?) he starts to see, that something is not right. The Earth is not getting much smaller, and he deduces, that the space ship is slowing down, which does not correspond with the constant acceleration of $a$. However the tourist is not as good physicist and goes to file complain with captain. What should the captain to tell him?

Assume, that the tourist see whole electromagnetic spectrum and will survive the observation.