Problem 1:

The radius of the galaxy is 3*10²⁰ m, measured in its own rest frame.
(a)  If the time it takes a spaceship to cross the entire galaxy is 300 years measured in the spaceship’s rest frame, what is the relative speed of spaceship and galaxy?
(b)  How much time elapses on Earth during this trip?

Problem 2:

A solar prominence (a large bright feature extending outwards from the sun's surface) was observed 10 minutes following a volcanic eruption on Earth.  Can those two events be related?  Why or why not?  The Earth-Sun distance is approximately 1.5 ·10⁸ km.

Problem 3:

In the year 2310 construction of the large space colony “Heavy Metal“ is finally completed.  The distance between the “East” and “West” edges of the colony is 400 km in the rest frame of the colony.  To celebrate the end of construction, two flares are lighted at midnight during a New Year’s celebration.  The light from the flares reaches an observer at the center of the colony at the same time.  A fast spacecraft flying with a constant speed of v = 0.95c “East” to “West” is directly over the “East” edge of “Heavy Metal”.  Let Event 1 be flare is lighted on “East” edge and Event 2 be flare is lighted on “West” edge.  In the reference frame of the spacecraft, does Event 1 occur before, after, or at the same time as Event 2?  Explain your reasoning!

Problem 4:

An atomic clock is taken to the North Pole, while another stays at the Equator.  How far will they be out of synchronization after a year has elapsed?

Problem 5:

(a) Observer A is stationary in a laboratory on Earth’s surface and observer B is in a laboratory moving at high velocity parallel to Earth’s surface (neglect the curvature of the earth in the subsequent considerations) and parallel to the x axis.  Assume that the observers each witness two separate lightning strikes on the surface of the earth lying along the x axis.
(i) Describe concisely a practical scheme whereby observer A can determine whether the two lighting strikes occurred simultaneously in her reference frame.
(ii) Show that if observer A concludes from her observations that the strikes were simultaneous, observer B generally must conclude from his observations of the same events that the strikes are not simultaneous, thus demonstrating that simultaneity is a relative and not absolute concept.
(b) Einstein was strongly inspired by Maxwell’s theory of electromagnetism and viewed Galilean invariance for transformations between reference frames as being seriously flawed because it was inconsistent with Maxwell’s theory.  The special theory of relativity removed this inconsistency.  Explain the meaning of the preceding two statements.

Problem 6:

For an observer on Earth, the star Sirius is 8.8 ly away.  Assume Sirius is stationary with respect to Earth.  At t = 0, a spaceship moving with velocity v = 0.95ci towards Sirius passes Earth.  Earth and the ship synchronize clocks. 
An observer on Earth tells the following story at a much later time:
At t = 0, just as the spaceship passed Earth, a flare erupted on Sirius.  The light from the flare reached Earth at t = 8.8 y.  At t = 0.9 years a comet hit Earth.  At t = 9.26 years the spaceship passed Sirius and sent us a light signal.  The signal reached us at 18.06 years.
How does an observer on the ship tell the same story?

Problem 7:

Imagine a train going through a tunnel.  The train and the tunnel both have length l in their own frame. The train moves through the tunnel with speed v.  There is a bomb at the front of the train which is designed to explode when the front of the train passes out the far end of the tunnel.  However, there is a disarming sensor located on the back of the train which will disarm the bomb just as the back of the train enters the near end of the tunnel.  Will the bomb explode?