Prove that for any relative velocity v between two observers, a beam of light sent from one to the other will approach at speed c (provided that v is less than c , of course).

Suppose an astronaut travels to a star 10 lightyears away at a significant fraction of the speed of light (for example, v=0.99c) and returns. Assume negligible amount of time is spent at the destination. Does this round-trip journey take less than, equal to, or more than 20 years as observed by an Earth-based observer? Does this round-trip journey take less than, equal to, or more than 20 years as observed by the astronaut?

The crew of an enemy spacecraft attempts to escape from your spacecraft by moving away from you at 0.283 of the speed of light. But all is not lost! You launch a space torpedo toward the foe at 0.351 of the speed of light with respect to you. (a) at what speed in kilometers per second does the enemy crew observe the torpedo approaching its spacecraft? (b) Is this more or less than the classical limit? Use the Galilean transform to prove this. (c) What if the torpedo is launched at the speed of light? At what speed in kilometers per second does the enemy crew observe the torpedo approaching its spacecraft? (Show all work.) (d) How fast would the second craft have to be going to measure the torpedoes speed as 10% greater than the classical limit. (Assume the torpedo is launched at the original speed, 0.351 of the speed of light.)

A light source G is moving, with respect to an observer O, at an angle θ=�=154∘∘ between the direction of relative motion and the line of sight from O to G. The redshift of the light emitted by G and measured by O is z=0�=0. Find the speed of G with respect to O in units of c�, the speed of light. Enter your answer to 3 decimal places.