October 9, 2003
If any of these answers seems obscure, please ask us questions until we clear it up for you. Correct answers are indicated in bold type and filled squares (g).
1)
Let President
Jackson be observer #1, and you be observer #2. The distance and time interval you see between the two events
are Ģx2 =
5000 cm (the length of the car) and Ģt2 = 2 x 10-6 s. The
distance between the events as seen by President Jackson (not to
be confused with the length of the car as it appears to him!) is
Ģx1 = 8.83 x 104 cm. We
are looking for Ģt1; from the Formulas page at the
back of the test, we get a useful rearrangement of the Absolute Interval
formula:
2) The length of your car appears to President Jackson to be
g 3.00 x 103 cm.
c 8.83 x 104 cm.
c 3.00 x 107 cm.
c 8.83 x 109 cm.
c 3.00 x 1010 cm.
3) Match the pictures to the names listed below.
|
|
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|
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A |
B |
C |
D |
E |
_C_ David Wells
_E_ Bernie Williams
_D_ Alfonso Soriano
_B_ Hideki Matsui
_A_ Derek Jeter
4) Calculate the mass of an object whose horizon circumference is 0.8 times as large as the Moon's circumference.
c
4.7 x 1032 g
g 9.4 x 1032 kg
c
5.9 x 1025 g
c
1.9 x 1028 kg
c
9.4 x 1032 g
5) Calculate the horizon circumference for an object with 0.84 times the mass of the moon.
c 0.23 cm
g 0.057 cm
c 0.014 cm
c 0.114 m
c 0.114 cm
6)
What is the difference between an inertial reference frame and a non-inertial
reference frame?
In an inertial frame one experiences no external forces, and the special theory of relativity applies. In a non-inertial frame one experiences forces, and can feel accelerations; the special theory of relativity generally does not apply.
7) What is the difference between an inertial reference frame and a freely-falling frame?
In a freely-falling frame the force of gravity is experienced, and in an inertial frame no force is present. They are otherwise equivalent; the special theory of relativity applies to both.
8) You and I are both in inertial reference frames. Our motion can be described as follows:
c We must accelerate with respect to each other.
c We must fall freely under the influence of gravity.
c We must be at rest with respect to each other.
g We must move at a constant speed with respect to each other.
c Any state of relative motion is allowed, as long as the speed of light is not exceeded.
9) Which of the following quantities were considered relative in classical physics, but were treated as absolute by Einstein? Check all that apply.
c Distance.
c Time.
c Velocity.
c Simultaneity.
g The speed of light.
10) The brightest star in the northern hemisphere of the sky besides the Sun is Vega, in the constellation Lyra. Its mass is 5.0 x 1033 grams. The mass of Vega, in units of solar mass, is equal to
c 8.3 x 105
g 2.5
c 1 x 1067
c 2.5 x 103
c 1.0 x 1023
11) The distance from Vega to Earth is 2.37 x 1019 cm. In light years, that is
c 2.24 x 1037 ly.
c 2.437 ´ 1017 ly.
c 237 ly.
g 25 ly.
12) One of the first tests of general relativity was
c The description of the orbit of the moon.
c The determination of the speed of light to be constant.
c The change in the mass of a particle moving at high speeds.
c The demonstration of a hammer and a feather falling at the same rate on the moon.
g The determination of the rate of advance of the perihelion of Mercury’s orbit.
13) Another important test of Einstein’s theory of relativity was the deflection of starlight by the gravitational field of the sun. Consider the picture below taken during a solar eclipse. The arrows mark the expected and the observed position of a particular star. Please indicate which position corresponds to the expected position of the star, and which corresponds to the observed position.
Briefly explain the effect observed here.
The light from a distant star is deflected
by the gravitational field of the sun, and as a result of this deflection,
the apparent position of the star (determined on the basis of the
observed position in the sky) is different from the real position. These two position are indicated in the
picture below, taken during a solar eclipse.
The position of a star in the sky is
determined by the direction from which the light is arriving at the
telescope. Since the light is deflected by the sun,
the apparent position is different from the real position. Since the gravitational force is always
attractive, the light will be deflected as indicated in the Figure
below.
As you can see from the Figure, the
apparent position of the start is further from the sun than the real
position.
14) A right triangle, shown in the diagram below as it would look at rest, moves with respect to you at a speed of 198,294 km/sec in the x direction.
In an instant, you measure the length of the side that lies along the x direction, and you get
c 5 meters.
c 4 meters.
g 3 meters.
c 2 meters.
c 1 meter.
15) Still watching that triangle, you measure the length of the side that lies along the y direction, and you get
c 5 meters.
c 4 meters.
g 3 meters.
c 2 meters.
c 1 meter.
16) At a distance corresponding to a circle with circumference 1.0001 times the horizon circumference from a black hole with a mass ten times as large as the mass of the sun, which of the following effects would one experience? Check all answers that apply.
g The tidal forces would be extremely large and probably rip one to bits.
g A vertical descent was necessary in order to reach this point; there are no stable orbits this close to the black hole.
c The sky looks much like it does near the dark side of a very large planet.
c Distant clocks (far away from the black hole) appear to be running very slowly.
c Light from distant stars reaches you at a speed much greater than the normal speed of light, owing to the strong warping of space-time near the horizon.
17) At a distance corresponding to a circle with circumference 1.0001 times the horizon circumference from a black hole with a mass 15 x 1012 times as large as the mass of the sun, which of the following effects would one experience? Check all answers that apply.
c The tidal forces would be extremely large and probably rip one to bits.
g A vertical descent was necessary in order to reach this point; there are no stable orbits this close to the black hole.
g The sky is compressed into a small circle directly overhead.
c Distant clocks (far away from the black hole) appear to be running very slowly.
g A laser used to signal a distant observer looks normal to those near the horizon, but the laser light appears to that distant observer to have much longer wavelength than normal.
18) Suppose you have two spaceships, the Discovery and the Enterprise. You are on the Discovery and are at rest. The Enterprise flies past you at a speed 0.73c. Pretend you have X-ray vision so that you can see what is happening on the Enterprise as it passes you. You see Captain Picard fire a cannon in the same direction the Enterprise is moving. Captain Picard thinks the cannon ball is going 12,030 km/s. What is the speed of the cannon ball as measured by you?
c 231,000 km/s
c 207,000 km/s
c 228,000 km/s
g 224,000 km/s
c 219,000 km/s
19) Which of the following statements are true? Check all that apply.
c
A sphere flies past you at 0.90 times the
speed of light. Because of its roundness and its speed, it still looks
like a sphere.
g
A meter stick, laying perpendicular to its direction of motion and
moving 0.8 times the speed of light, would NOT appear shorter due to
Lorentz length contraction.
g
A meter stick, laying along its direction of motion and moving at 0.99
times the speed of light, would look much shorter than 1 meter to an
observer who is at rest.
g
A meter stick, laying at an angle of 45 degrees to its direction of
motion and moving 0.999 times the speed of light would appear to be
very nearly perpendicular to its direction of motion to an observer
who is at rest.
c
Two 100-meter long spaceships can be seen;
each is moving at 0.98 times the speed of light, with one of them approaching
you, and one receding from you. The one approaching you looks much
longer than 100 meters while the one receding from you looks much shorter.
20)
An astronaut travels with a speed of 0.67c to a distant
object in the universe. If
the astronaut thinks the trip took 40 years, how many years have passed
for a stationary observer on Earth?
c 61 years
c 70 years
c 43 years
g 54 years
c 27 years
21) What happened to the Red Sox? Check all that apply.
g According to Prof. Wolfs’ son, they fell into a black hole.
c The team is called the Red Sox since the light they emit is red shifted by the gravitational field of their black hole.
g Who cares! These questions have nothing to do with Astronomy 102.
c Their close encounter with a black hole made them stronger; so strong that they beat the Yankees 5 to 2 last night.
g Is this a joke?
22) In your own words, describe the meaning of the following equation (note: you do NOT need to derive this equation or solve it).
This is Einstein’s general theory
of relativity. It tells
you that “Space-time, with its curvature, tells masses how
to move; masses tell space-time how to curve”.
23) You are at rest near the horizon of a black hole, and a friend is a large distance away, in a part of space far from any strong gravitational forces. You get a good look at each other’s clock.
c You see your friend’s clock ticking more slowly than yours, and she sees your clock ticking faster than hers.
g You see your friend’s clock ticking faster than yours, and she sees your clock ticking more slowly than hers.
c You each see the other’s clock ticking more slowly than your own.
c You each see the other’s clock ticking faster than your own.
24) You are at rest in a part of space far from any strong gravitational fields, and a friend of yours flies past at nearly the speed of light, with a constant acceleration equal to one g. You get a good look at each other’s clock as she passes by.
g You see your friend’s clock ticking more slowly than yours, and she sees your clock ticking faster than hers.
c You see your friend’s clock ticking faster than yours, and she sees your clock ticking more slowly than hers.
c You each see the other’s clock ticking more slowly than your own.
c You each see the other’s clock ticking faster than your own.
25)
Which of the following statements are true? Check
all that apply.
g
Space-time warpage is almost entirely due to the density of mass in
the warped area.
c
Due to geodesics of space-time, the sum of
the angles internal to a triangle always equals 180 degrees, even near
very massive objects.
c
The special theory of relativity does not
apply to a reference frame falling freely under the influence of gravity.
g
The Michelson-Morley experiment and subsequent ones can be taken to
have proven that light and the substance called the aether have nothing
to do with one another.
c
Einstein spent his first article on relativity
tackling the complicated form of the equations of electricity and magnetism
(Maxwell's equations) for moving reference frames specifically because
their disagreement with experiment was a fatal problem for physicists.
g During a solar eclipse the ocean tides should be slightly higher and lower than usual.
26) Which of the following statements are true? Check all that apply.
c
As your velocity approaches the speed of
light, your mass increases, and so you become harder to decelerate.
Eventually it becomes impossible for you to slow back down to rest.
g
If you are standing still relative to two observers, one traveling
at one half the speed of light relative to you and the other three
fourths the speed of light relative to you, the two observers will
disagree about how massive you are.
g
As your velocity approaches the speed of light your mass increases,
so you become harder to accelerate. This is what prevents you from
moving faster than the speed of light.
c
As your velocity approaches that of light
your mass increases. This increase of mass comes in the form of fat,
and may be reduced by dieting.
c
As your velocity approaches that of light
relative to a stationary observer, your mass as measured by that observer
decreases to nothing.
27) Two identical, 100 meter long, spaceships are moving with a speed 0.99 times the speed of light in your reference frame. One of the spaceships is approaching you, and the other one is receding from you. Their lengths appear to you as follows:
g Both look much shorter than 100 meters.
c The approaching ship looks much shorter, the receding one much longer, than 100 meters.
c The approaching ship looks much longer, the receding one much shorter, than 100 meters.
c Both look much longer than 100 meters.
c Both look like 100 meters.
28) The colors of the spaceships in the previous problems appear to you as follows:
c Both look bluer (shorter wavelength light) than their natural colors.
g The approaching ship looks bluer, the receding one redder (longer wavelength light), than their natural colors.
c The approaching ship looks redder, the receding one bluer, than their natural colors.
c Both look redder than their natural colors.
c Their colors are unchanged.
c None of the above.
29) You are in a spaceship heading toward the Sun’s nearest stellar neighbor, Alpha Centauri, 4 light years away, at a speed near that of the speed of light as seen from the Earth, and accelerating continuously while doing so. An observer on Earth has predicts that it will take you four years by his clock to get there, but you predict that it will take much less time as measured by your clock. Why (check all that apply)?
c The distance you see to the star is much shorter that 4 light years due to relativistic length contraction.
c Your clock runs slower than his, due to relativistic time dilation.
c Your spaceship is in a high-speed, non-inertial frame of reference.
g All of the above.
c None of the above.
30) Which of the following observable properties are useful in detecting the presence of a black hole from a great distance? Check all answers that apply.
c A black patch in the sky.
g A hot disk of material with two high-speed (close to the speed of light) jets of matter protruding from its poles.
g X-ray emission.
g Stars and gas clouds moving in orbits at speeds close to that of light.
c Radio-wave emission.
31) You drop a video camera into a black hole and have it view your spaceship as it falls. It transmits a radio signal to you that you can tune into with a TV and see what the camera sees. Which of the following observations do you expect to make? Check all answers that apply.
g The transmission lasts forever.
g You continuously have to adjust the TV to receive lower and lower frequencies to stay tuned in as the camera falls.
g After a long time, the image of the sky transmitted to you looks simply like a point in the middle of a black screen
g After a while, the image of the sky is a circle of ever-decreasing diameter.
g The spaceship looks bluer in the transmitted images than it does naturally.
g The camera never appears to cross the black hole’s event horizon.
c The camera never actually crosses the black hole’s event horizon.
c When the camera crosses the event horizon, the image you see is the tear in the horizon that was left when the camera punched through it.
g You never see what the camera sees as it falls past the horizon.
c The camera will continue to broadcast when it emerges from the other side of the black hole.
32) I am driving on a long, straight road at a speed 0.99 times the speed of light, flashing a light every second according to my clock. You stand on the sidewalk. Calculate the time (in seconds) that you would measure between flashes. (You must show your work, to receive partial credit for an incorrect answer.)
Since
you are moving with constant speed, we can use the theory of special
relativity to calculate the time you would observed between light
pulses coming from the moving clock. The
time you measure is
where V =
0.99c and Dt2 = 1 s. Using this information we can determine
the time difference you would measure, which is 7.1 s.
33) One of the flashes described in the previous problem occurs just as I pass you. Use the Minkowski absolute interval to calculate how far I am down the road when you see the next flash. (You must show your work, to receive partial credit for an incorrect answer.)
In
my frame of reference, the flashes occur at the same position, and
thus Dx2 = 0 m. In my reference frame the time difference
between the flashes is 1 s, and thus Dt2 = 1 s. In problem 32 we calculated that you
see a time difference between the flashed of 7.1 s, and thus Dt1 = 7.1 s. Using the Minkowski absolute interval,
we can now calculate the difference in position of the two flashes
in your reference frame:
Of
course, this is not a surprising answer. You
would have gotten the same result if you multiplied the velocity
of me, measured by you, with the time you measured between flashes.
34) If the light takes 8 minutes to reach the Earth from the sun and the nearest star is 4.7 light years from the sun, what is the distance from the sun to the nearest star in astronomical units (1 astronomical unit, AU, is the distance between the earth and the sun)?
c 37.7 AU
c 1.7 AU
c 214 AU
g 300,000 AU
c 1.5 x 1011 AU
35) Flying past you in a spaceship going 0.8 times the speed of light, I launch a rocket straight behind and give it a speed of 0.6 times the speed of light with respect to me. Calculate the speed you would measure for the rocket. (You must show your work, to receive partial credit for an incorrect answer.)
We solve this problem by using the formula
for velocity addition:
The velocity V is equal to the velocity of the spaceship and is
equal to +0.8c. The velocity v2 is the velocity of the rocket,
which is equal to –0.6c (note
the minus sign since the rocket id directed backwards). Using these numbers we can calculate v1:
36) Flying past you in a spaceship going 0.8 times the speed of light, I launch a rocket straight ahead and give it a speed of 0.8 times the speed of light with respect to me. You measure the speed of the rocket, and your result is
c 0.62 times the speed of light.
c 0.71 times the speed of light.
c 0.80 times the speed of light.
g 0.98 times the speed of light.
c 0.89 times the speed of light.
37) According to the general theory of relativity, gravity is caused by
c The change in the mass of a moving object.
c The constant speed of light.
g The curvature of space-time.
c The presence of microscopic black holes all over the universe.
c None of the above.
38)
You observe a friend of yours rolling bowling balls on a
moving flatbed truck. She
rolls the balls at 88 km/s west with respect to the truck, and the
truck itself moves west at 84 km/s with respect to you. According
to the special theory of relativity, what is the speed (in km/s) of
the bowling balls with respect to you?
c 4 km/s
g 172 km/s
c 184 km/s
c 86 km/s
39)
You are running at 99.9% of the speed of light. If you hold a mirror
in front of yourself, you see
c Nothing.
g Yourself.
c Yourself, with the colors in the mirror bluer than normal.
c Yourself, but contracted along the direction of your motion.
c Yourself, with the colors in the mirror redder than normal.
40)
The following attributes are required of a scientific theory before
it is considered valid (check
all that apply):
c Agreement with expert opinion.
c Understandable to first-year physics and astronomy majors.
c Disagreement with other, competing theories.
c Public support.
c Aesthetic (i.e. “artistic”) appeal.
g Mathematical and/or logical self-consistency.
g Predictions that can be tested experimentally.
g Precise agreement with experimental results.
c Approved by Joe Torre.
c Tangible benefits to society.
Last updated on Thursday, June 30, 2005 10:52
