Question #1 (25%) - required
In a distant future, a one-man starship breaks down far away
from any star or
planet (so that it does not feel any gravity from any celestial body).
An astronaut, trying to fix it, gets outside with a wrench.
Suddenly, an accident causes him to float away from his
starship. Explain, how an astronaut can return to his ship. He does not
have a portable rocket on his spacesuit. Make sure that you consider
(qualitatively) all of the possibilities.
Question #2 (15%) - required
In recent years astronomers discovered that galaxies are not distributed
in the universe uniformly, but are grouped into clusters of galaxies.
Very recently it became clear that the clusters of galaxies are also
not distributed uniformly, but grouped into ``superclusters''. How can
existence of these structures be consistent with a homogeneous universe?
Question #3 (25%) - optional, non-math
Suppose that a train robber decides to stop a train inside a tunnel. The
proper length of the train is 60 yards, while the proper length of the tunnel
is 50 yards. The train is traveling at 4/5 the speed of light. According to
proper lengths, the train would not fit inside the tunnel, but the robber
plans to use relativity to his advantage. The length of the moving train in
the rest frame of the tunnel, and of the robber, is 36 yards. The robber
computes this and decides to trap the train inside the tunnel, since, in
his frame, the train should fit. From the point of view of the train's
engineer, however, the tunnel is only 30 yards long, just half the length
of the train. The engineer knows that his 60 yards train will not fit
completely into the tunnel. The robber thinks that the train
will fit, whereas the engineer is sure it will not. But either the train
will fit, or it will not - it cannot do both. Who is correct? Explain
your reasoning. (You may wish
to draw a space-time diagram to guide you. This is an adaptation of review
question 13 in H&H, pp 198-199, with further hints there.)
Question #4 (20%) - optional, non-math
On a space-time diagram below, draw a world line of a spaceship,
that traveled from the Earth (x=0)
along the x-axis with the speed 2/3c
for 6 years as measured on Earth,
then turned around and traveled
in the opposite direction with the speed 1/2c
until it came back to Earth.
Question #5 (15%) - optional, non-math
In what way can time be called the fourth dimension? How does it
differ from the other three? Give at least two main differences.
Question #6 (20%) - optional, math
In 2099 a space warship of an alien race was on its way to destroy the
Earth. At the outskirts of the solar system it destroyed an unmanned
space probe, and encouraged by the first success, aliens organized
an on-board party. Every member of the crew attended, and the warship
remained without control for some time. Suddenly, the commander of the ship
realized that the warship was going with the speed 0.99c
(the boost factor 7.1) straight
into Mars. At this moment the ship was only 30 million km from Mars
(as measured by an observer at the Mars surface).
Everyone on board of the ship was so startled (and, perhaps, drunk),
that they just froze, and the commander started to scream, making one
scream every second. How many screams did he make? Explain your reasoning.