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.
The astronaut has to throw his wrench in the direction opposite to the direction of his motion. Since there is no net external force that acts on him and the wrench (the gravity of the ship is way too weak), the total momentum of the astronaut and the wrench is conserved. If he can give the wrench more momentum than this total, his own momentum will reverse direction, which means he is going back to the ship! But if the total momentum of the astronaut and the wrench is larger than the momentum he can give to the wrench (i.e. if he is flying away too fast), he cannot come back.
A much more risky alternative is to punch his spacesuit or to disconnect his air supply and use it to propel him back. But whether he can come back or not still depends on his initial momentum since the air supply is limited and is only able to provide a fixed amount of momentum.
You got the full credit for this problem only if you considered both possibilities (small total momentum and large total momentum).
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?
Homogeneity depends on the scale at which one is looking at things. A perfectly homogeneous sheet of paper looks very inhomogeneous on a scale of single atoms (in one place there is an atom, in the other one there is nothing). The universe is homogeneous only on large enough scale, on smaller scales it has structures. Indeed, superclusters are the largest structures in the universe - on larger scales it is observed to be homogeneous.
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? (You may wish
to draw a space-time diagram to guide you).
There are two correct answers, depending on whether you consider the train stopped or moving through the tunnel. The `moving' solution is basically the more interesting
If the train moves, then both are correct: the robber will indeed see the train go completely inside the tunnel (i.e. the front end of the train will still be inside the tunnel when the rear end enters the tunnel), and the engineer will see quite the opposite: the front end will exit the tunnel before the rear end enters it. This is because those two events - the front end exits the tunnel and the rear end enters it - are not causally connected and can happen in different order to different observers.
However, if the train stopped, the train and the tunnel will be in the same reference frame, and the train will not fit, since its length will be longer. In this case the engineer is correct.
Each of the explanations is sufficient to get the full credit.
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.
The time is called the fourth dimension because it is the fourth number that is used to label a point (event) in the space-time (in addition to the three spatial coordinates: length, width, and height).
Some of the differences are:
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?
From the point of view of a person on Mars, the spaceship will hit Mars in 30 million/300,000 = 100 seconds. Since the time on the ship goes slower, the time until the collision will be less. The boost factor simply quantifies how much less, i.e. 100/7.1 = 14 seconds. The commander will make 14 screams.