Astronomy 102, Exam #1

Astronomy 102, Midterm Exam #1

Tuesday October 11, 2005

2.00 pm – 3.15 pm

Do not turn the pages of the exam until you are instructed to do so.

You are responsible for reading the following rules carefully before beginning.

Exam rules: You may use only a writing instrument and a calculator while taking this test. You may not consult any computers, books, notes – neither on paper nor stored in a calculator – nor each other. All of your work must be written on the attached pages, using the reverse sides if necessary. Important equations, numbers and conversion factors, used in the problems, are found in the last pages of the exam, in the form of the Useful Equations sheet and the How Big Is That sheet. The final answers must be indicated clearly. Exams are due at 3:15, and will be available to be reclaimed in recitations next week.

The questions are each worth five (5) points. Partial credit is available for those questions involving essays, short answers, drawings or explicit calculations, and for multiple-choice questions indicated possibly to have more than one correct answer (e.g. “check all answers that apply”).

Name: ___________________________________

ID number: _______________________________

Recitation: ________________________________

1._ What is the difference between an inertial reference frame and a non-inertial reference frame?

______________________________________________________________________________

2._ What is the difference between an inertial reference frame and a freely-falling frame?

______________________________________________________________________________

3._ The difference between the special theory of relativity and the general theory of relativity is that the former only applies to ___________________________________________ reference frames.

4. 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.

c The speed of light.

5. Which of the following quantities were considered absolute in classical physics, but were treated as relative by Einstein? Check all that apply.

c Distance.

c Time.

c Velocity.

c Simultaneity.

c The speed of light.

6. 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.

c The determination of the rate of advance of the perihelion of Mercury’s orbit.

7. You are at rest in a part of space far from any strong gravitational fields, and a friend flies past at 0.99 times the speed of light. You get a good look at each other’s clock as she passes by.

c 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.

8. 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.

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.

9. Match the pictures to the names listed below.


A	B	C	D	E

___ Robinson Cano

___ Jorge Posada

___ Alex Rodriguez

___ Jason Giambi

___ Mariano Rivera

10. We are running together in the same direction, at a speed 0.999 times the speed of light. I hold up a mirror facing you, and in my mirror you see

c Nothing.

c Yourself, but moving slowly.

c Yourself, with the colors in the mirror bluer than normal.

c Yourself, but contracted along the direction of your motion.

c All of above.

c None of above.

AppleMark

11. A right triangle, shown in the diagram 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 meter.

c 4 meter.

c 3 meter.

c 2 meter.

c 1 meter.

12. Still watching that triangle, you measure the length of the side that lies along the y direction, and you get

c 5 meter.

c 4 meter.

c 3 meter.

c 2 meter.

c 1 meter.

13. It happens about once in a lifetime: the Red Sox winning the World Series. They won in 1918. They won in 2004. When will they win the World Series next?

c Never again.

c 2090.

c Who cares! These questions have nothing to do with Astronomy 102.

c Two data points are really not sufficient to make such a prediction, unless I know that the time intervals are constant. Most likely their rate of winning the World Series will slow down in the same way as the information coming from Arnold slowed down when he approached the black hole.

c 2005.

14. A meter stick flies past you at 0.99 times the speed of light, laying along its direction of motion. In an instant, you measure its length as it flies past, and your result is

c 7.1 meter.

c 1 meter.

c 0.99 meter.

c 0.14 meter.

15. 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 predicts that it will take you four years by his clock to get there, but you predict that it will take much less time than that by your clock, because

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.

c All of above.

c None of above.

16. 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:

c Both look much shorter than 100 meters.

c The approaching ship looks much shorter than 100 meters; the receding one looks much longer than 100 meters.

c The approaching ship looks much longer than 100 meters; the receding one looks much shorter, than 100 meters.

c Both look much longer than 100 meters.

17. The colors of the spaceships in problem 16 appear to you as follows:

c Both ships look bluer (shorter wavelength light) than their natural colors.

c The approaching ship looks bluer than its natural color; the receding ship looks redder (longer wavelength light) than its natural color.

c The approaching ship looks redder than its natural color; the receding ship looks bluer than its natural color.

c Both ships look redder than their natural colors.

c The colors of both ships are unchanged.

18. You are near the horizon of a massive black hole. The stars in the sky appear to be bluer than their natural colors, owing to

c The gravitational acceleration (increase in speed) of light toward the black hole.

c The gravitational Doppler shift of light toward the black hole.

c The effect of the strong tidal forces on your perception of color.

c The Lorentz length contraction of the distances to the stars.

19. 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.

c A hot disk of material with two high-speed (close to the speed of light) jets of matter protruding from its poles.

c X-ray emission.

c Stars and gas clouds moving in orbits at speeds close to that of light.

c Radio-wave emission.

20. Consider two black holes, one with a mass of 10³³ gm, and the other with a mass of 10³⁵ gm. Which of the following quantities are greater for the more massive black hole? Check all answers that apply.

c The horizon circumference.

c The force due to gravity on an object 100 light years away.

c The tidal force in an orbit with circumference twice as large as the horizon.

c The force due to gravity on an object just barely above the horizon.

c All of above.

21. 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.

c The transmission lasts forever.

c You continuously have to adjust the TV to receive lower and lower frequencies to stay tuned in as the camera falls.

c After a long time, the image of the sky transmitted to you looks simply like a point in the middle of a black screen.

c After a while, the image of the sky is a circle of ever-decreasing diameter.

c The spaceship looks bluer in the transmitted images than it does naturally.

c 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.

c 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.

22. 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.

c The tidal forces would be extremely large and probably rip one to bits.

c 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.

23. 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.

c 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 is compressed into a small circle directly overhead.

c Distant clocks (far away from the black hole) appear to be running very slowly.

c 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.

24. The region immediately surrounding the black hole in 3C 273 has a luminosity of 10¹² L_sun, larger than that of our Milky Way galaxy by a factor of

c 10000.

c 1000.

c 100.

c 10.

c 1 (They have the same luminosity).

25. The region around the black hole in 3C 273 in which this luminosity is produced is about the size of the event horizon itself, and is 6´10¹⁷ cm in diameter. The diameter of the Milky Way is a factor of

c 2.7´10⁵ larger.

c 2.7´10³ larger.

c 2.7 larger.

c 1 (They are the same size.)

c 2.7´10^-3 smaller.

26. 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.

c 2.5.

c 1 x 10⁶⁷.

c 2.5 x 10³.

c 1.0 x 10²³.

27. 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.

c 25 ly.

28. 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.

c 300,000 AU.

c 1.5 x 10¹¹ AU.

29. Minkowski’s formula for the absolute interval between two events (call them A and B) is

___ What do Dx₁, Dt₁, Dx₂, Dt₂ and c stand for in this formula?

______________________________________________________________________________

30. 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.)

______________________________________________________________________________

31. 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.)

______________________________________________________________________________

___ ___________________________________________________________________________

______________________________________________________________________________

32. 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.)

______________________________________________________________________________

33. 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 (at rest, like you).

c 0.8 times the speed of light.

c 0.98 times the speed of light.

c 1.6 times the speed of light.

34. 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.

c 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.

35. Due to the curvature of space-time by the sun, light from stars that passes near the edge of the sun will

c Be bent so that the stars appear further from the edge of the sun than if space-time was not curved.

c Be bent so that the stars appear closer to the edge of the sun than if space-time was not curved.

c Be bent so that the stars are no longer visible.

c Not be affected by the curvature of space-time.

c Be focused so that the stars appear brighter than if space-time was not curved.

36. The Schwarzschild radius of a black hole with a mass of 2 M_sun is approximately equal to

c 6 km.

c 4 km.

c 2 km.

c 12 km.

c 36 km.

37. An isolated black hole in space would be difficult to detect because

c There would be no light sources nearby.

c It would not be rotating rapidly.

c It would be stationary.

c Very little matter would be falling into it.

c There would be very few stars behind it whose light it could block out.

38. The event horizon of a black hole

c Is believed to be a singularity.

c Is the surface of the black hole, similar to the surface of the Earth.

c Has a radius equal to the Schwarzschild radius.

c Marks the boundary where in-falling material starts to emit X-rays.

39. 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.

c The curvature of space-time.

c The presence of microscopic black holes all over the universe.

c None of the above.

40. A black hole, whose horizon has a circumference equal to that of the moon, has a mass of

c 0.94 x 10³⁶ g.

c 2.36 x 10³⁶ kg.

c 1.18 x 10³⁶ g.

c 2.36 x 10³⁶ g.

c 0.94 x 10³⁶ kg.

c 0.59 x 10³⁶ g.

c 1.18 x 10³⁶ kg.

c 0.59 x 10³⁶ kg.

Useful Equations (so far)

Length contraction, time dilation and velocity addition:

Minkowski absolute interval:

Useful rearrangements of the Absolute Interval formula:

Schwarzschild circumference:

How big is that?

Diameter of hydrogen atom	1.06 ´ 10^-8 cm
Diameter of the Moon	3.5 ´ 10³ km
Diameter of the Earth	1.3 ´ 10⁴ km
Diameter of the Sun	1.4 ´ 10⁶ km
Diameter of the Milky Way galaxy	1.7 ´ 10⁵ ly

Distance to the Moon	3.8 ´ 10⁵ km
Distance to the Sun	1.5 ´ 10⁸ km
Distance to the next nearest star	4 ly
Distance to the center of the Milky Way	2.7 ´ 10⁴ ly
Distance to the nearest galaxy	1.7 ´ 10⁵ ly

Mass of hydrogen atom	1.67 ´ 10^-24 gm
Mass of the Moon	7.4 ´ 10²⁵ gm
Mass of the Earth	6.0 ´ 10²⁷ gm
Mass of the Sun	2.0 ´ 10³³ gm (1 M_O)
Mass of the Milky Way galaxy	5 ´ 10¹⁰ M_O

Luminosity of the Sun	3.8 ´ 10³³ erg/s (1 L_O)
Luminosity of the largest stars	10⁵ L_O
Luminosity of the Milky Way galaxy	10¹⁰ L_O
Luminosity of quasar 3C 273	10¹² L_O

Earth’s rotation period	8.64 ´ 10⁴ s (1 day)
Moon’s revolution period	28 days
Earth’s revolution period	365.25 days (1 year)
Sun’s revolution period within Milky Way	2.4 ´ 10⁸ years
Age of the solar system	4.6 ´ 10⁹ years
Expected life span of the Sun	1.5 ´ 10¹⁰ years
Age of the Universe	1.3 ´ 10¹⁰ years

Earth’s equator rotation speed	0.47 km/s
Earth’s revolution speed	30 km/s
Sun’s speed within the Milky Way	220 km/s
Milky Way’s speed within the local Universe	500 km/s

Typical lengths:
Normal star diameter	10⁶ km
Distance between stars	a few ly
Normal galaxy diameter	10⁵ ly
Distance between galaxies	10⁶ ly

Typical masses:
Smallest star	0.1 M_O
Normal star	1 M_O
Giant star	10 M_O
Normal galaxy	10¹⁰ - 10¹¹ M_O
Galaxy cluster	10¹⁴ - 10¹⁵ M_O

Typical luminosities:
Normal star	1 L_O
Giant star	10³ - 10⁵ L_O
Normal galaxy	10⁹ - 10¹⁰ L_O
Quasar	10¹² - 10¹³ L_O

Typical time spans:
Planetary revolution	1 year
Galaxy rotation	10⁷ - 10⁹ years
Life of giant stars	10⁶ - 10⁹ years
Life of normal star	10¹⁰ years

Typical speeds:
Planetary orbits	10 km/s
Stellar motion in galaxy	100 km/s
Between nearby galaxies	100 km/s

Other important constants:
1 ly = 9.46 ´ 10¹² km = 9.46 ´ 10¹⁷ cm	1 Mly = 10⁶ ly
1 km = 10⁵ cm	1 erg = 1 gm cm²/s²
1 hour = 3600 seconds	1 year = 3.16 ´ 10⁷ seconds
p = 3.14159265359	Hubble’s constant: H₀ = 20 km/(sec Mly)
Speed of light: c = 2.99792458 ´ 10⁵ km/s = 2.99792458 ´ 10¹⁰ cm/s = 1 ly/year	Newton’s gravitational constant: G = 6.67 ´ 10^-8 cm³/(gm s²)

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