Questions & Answers
Browse all 18 questions from the
Cosmic Algebra study set below.
Each question shows the correct answer — select a study format above to practice interactively.
1
If the orbital period P of a planet in years is related to its semi-major axis a in AU by Kepler's Third Law (P^2 = a^3), calculate the semi-major axis of a planet with an orbital period of 64 years.
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A
4 AU
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B
8 AU
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C
16 AU
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D
32 AU
2
The luminosity L of a star is proportional to the square of its radius R and the fourth power of its surface temperature T (L = kR^2T^4). If the temperature doubles and the radius remains constant, by what factor does the luminosity increase?
3
The escape velocity v of a planet is given by v = sqrt(2GM/r). If a planet's mass M is multiplied by 9 and its radius r is multiplied by 4, by what factor does the escape velocity change?
4
Using the inverse-square law for light (I = P / 4πd^2), if the distance d to a star is tripled, by what factor does the observed intensity I decrease?
5
The Schwarzschild radius R_s of a black hole is calculated as 2GM/c^2. If you double the mass M of a black hole, how does the Schwarzschild radius change?
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A
It remains the same
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B
It doubles
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C
It quadruples
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D
It halves
6
A satellite orbits Earth at a velocity v proportional to 1/sqrt(r). If the orbital radius r is increased by a factor of 4, what is the new velocity relative to the original?
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A
1/2 v
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B
2 v
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C
1/4 v
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D
4 v
7
The gravitational force F between two objects is F = G(m1*m2)/r^2. If the mass of both objects is doubled and the distance between them is halved, by what factor does the force increase?
8
The apparent magnitude m of a star is related to its distance d in parsecs by m = M + 5log10(d) - 5. If the distance d is 100 parsecs, what is the value of 5log10(d)?
9
The period of a simple pendulum on a planet is T = 2π*sqrt(L/g). If gravity g on the planet is 4 times that of Earth, what is the ratio of the period on that planet to the period on Earth?
10
Hubble's Law states v = H_0 * d. If a galaxy is 500 Megaparsecs away and H_0 is 70 km/s/Mpc, what is its recessional velocity v?
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A
350 km/s
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B
3,500 km/s
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C
35,000 km/s
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D
350,000 km/s
11
The surface area of a spherical planet is A = 4πr^2. If you increase the radius of the planet by a factor of 10, how does the surface area change?
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A
10 times
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B
20 times
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C
100 times
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D
400 times
12
The volume of a gas cloud is V = (4/3)πr^3. If the radius of the cloud triples, what is the ratio of the new volume to the original volume?
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A
3:1
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B
9:1
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C
27:1
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D
81:1
13
In the equation E = mc^2, if mass m is converted to energy entirely and the speed of light c is approximated as 3x10^8 m/s, what is the factor by which mass is multiplied to find energy?
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A
9x10^16
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B
3x10^8
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C
6x10^16
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D
9x10^8
14
The flux of radiation from a star is F = σT^4. If the temperature T of the star increases from 5,000K to 10,000K, the flux increases by what factor?
15
The density ρ of a planet is M / ((4/3)πr^3). If you double the radius while keeping the mass constant, what is the new density relative to the original?
16
Centripetal force is F = mv^2/r. If a moon's velocity v is doubled and its orbit radius r remains constant, by what factor does the centripetal force increase?
17
The Stefan-Boltzmann law relates total power P to area A and temperature T as P = AσT^4. If a star's radius is halved and its temperature is doubled, what happens to the power P?
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A
P is halved
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B
P is doubled
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C
P is quadrupled
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D
P remains the same
18
The orbital energy of a satellite is E = -GMm/2r. If the distance r is doubled, what happens to the total orbital energy E?
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A
It becomes half as negative
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B
It becomes twice as negative
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C
It becomes four times as negative
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D
It remains the same
