Transformation of an angle

A straight line passing through the origin O' of S' forms an angle alpha' with the x'-axis. a) Find a relation between alpha and alpha' b) What is the value of alpha when V tends to c?

Speed of light in a moving medium

Light moves more slowly through a material medium than through a vacuum, its phase velocity v being c/n, where n is the index of refraction of the medium. If now the medium itself moves at a velocity V << c with respect to the laboratory, show that the phase velocity of the light with respect to the laboratory is approx. c/n + V(1- 1/n^2)

Lorentz contraction for a length

A one-meter ruler moves at a speed c/2. In its own reference frame it forms an angle of 45 degrees with its velocity. What is its length as measured by a fixed observer?

On his 25th birthday, Peter leaves his twin Paul behind on the earth and goes off in a straight line for 7 years of his time at a speed of 0.96c.

On his 25th birthday, Peter leaves his twin Paul behind on the earth and goes off in a straight line for 7 years of his time at a speed of 0.96c. Peter then reverses direction and returns at the same speed. What are the ages of Peter and Paul at the moment of reunion?

Question of Lagrangian

Question of Lagrangian. See attached file for full problem description.