Question 1

A satellite is in circular orbit at an altitude of 2300 km above the surface of a nonrotating asteroid with an orbital speed of 5.9 km/s. The minimum speed needed to escape from the surface of the asteroid is 14.6 km/s, and G = 6.67 × 10^-11 N ∙ m²/kg². The mass of the asteroid is closest to A) 1.8 × 10²⁴ kg. B) 8.9 × 10²³ kg. C) 1.3 × 10²⁴ kg. D) 2.7 × 10²⁴ kg. E) 3.6 × 10²⁴ kg.

Accepted Answer

The correct answer is A because the mass of the asteroid can be calculated using the formula for the escape velocity: v = sqrt(2GM/r), where v is the escape velocity, G is the gravitational constant, M is the mass of the asteroid, and r is the radius of the asteroid. Plugging in the given values, we get: 14.6 km/s = sqrt(2 * 6.67 × 10^-11 N ∙ m^2/kg^2 * M / (6371 km + 2300 km)). Solving for M, we get: M = 1.8 × 10^24 kg.

Question 2

What is the ratio of the escape speed of a rocket launched from sea level to the escape speed of one launched from Mt. Everest (an altitude of 8.85 km)? The radius of the earth is 6.38 × 10⁶ m. A) 1.0007 B) 1.0014 C) 1.0001 D) 0.9986 E) 0.9993

Accepted Answer

The escape speed from a planet's surface is given by:
Ve = sqrt((2GM)/r)
Where G is the gravitational constant, M is the mass of the planet and r is the distance from the planet's center. Since the question asks for the ratio of the escape speeds, we can cancel out these common factors and only consider the distance from the center of the Earth:
Ve1/Ve2 = sqrt(r2/r1)
Where Ve1 is the escape speed from sea level and Ve2 is the escape speed from the altitude of Mt. Everest. Plugging in the values given, we get:
Ve1/Ve2 = sqrt((r + h)/r) = sqrt((6.38×10^6 + 8.85×10^3)/6.38×10^6) = 1.0007 (rounded to four decimal places)

Therefore, the ratio of the escape speed of a rocket launched from sea level to the escape speed of one launched from Mt. Everest is 1.0007, which is closest to choice A.

Question 3

Beings on spherical asteroid Π have observed that a large rock is approaching their asteroid in a collision course. At 7514 km from the center of the asteroid, the rock has a speed of 136.0 m/s and later at 2823 km it has a speed of 392.0 m/s. Use energy conservation to find the mass of asteroid Π. You can neglect any effects due to the parent star of the asteroid. (G = 6.67 × 10^-11N ∙ m²/kg²) A) 4.582 × 10²¹ kg B) 9.164 × 10²⁷ kg C) 6.112 × 10²⁸ kg D) 4.582 × 10²⁰ kg

Accepted Answer

Using conservation of mechanical energy, the change in kinetic energy is equal to the change in gravitational potential energy. Solving for the mass of the asteroid using the given speeds and distances gives the result.

Question 4

An astronaut is standing on the surface of a planetary satellite that has a radius of 1.74 × 10⁶ m and a mass of 7.35 × 10²² kg. An experiment is planned where a projectile needs to be launched straight up from the surface. What must be the minimum initial speed of the projectile so it will reach a height of 2.55 × 10⁶ m above this satellite's surface? (G = 6.67 × 10^-11 N ∙ m²/kg²)

Accepted Answer

The answer of An astronaut is standing on the surface...

Question 5

Find the orbital speed of an ice cube in the rings of Saturn. The mass of Saturn is 5.68 × 1026 kg and the rings have an average radius of 100,000 km. (G = 6.67 × 10^-11 N ∙ m²/kg²) A) 19.5 km/s B) 27.5 km/s C) 13.8 km/s D) 1.95 km/s

Accepted Answer

The orbital speed of an object is given by:$$v = \sqrt{\frac{GM}{r}}$$where:* v is the orbital speed in m/s* G is the gravitational constant in N ∙ m2/kg2* M is the mass of the object in kg* r is the radius of the orbit in mIn this case, we have:* G = 6.67 × 10-11 N ∙ m2/kg2* M = 5.68 × 1026 kg* r = 100,000 km = 100,000,000 mPlugging these values into the equation, we get:$$v = \sqrt{\frac{(6.67 	imes 10^{-11} N \cdot m^2/kg^2)(5.68 	imes 10^{26} kg)}{100,000,000 m}} = 19.5 km/s$$Therefore, the orbital speed of an ice cube in the rings of Saturn is 19.5 km/s.

Question 6

Spaceman Speff orbits spherical asteroid X with his spaceship. To remain in a circular orbit at 421 km from the asteroid's center, he should maintain a speed of 80 m/s. What is the mass of planet X? (G = 6.67 × 10^-11 N ∙ m²/kg²) A) 4.0 × 10¹⁹ kg B) 5.1 × 10¹⁷ kg C) 4.0 × 10¹⁶ kg D) 5.1 × 10¹⁴ kg

Accepted Answer

The correct answer is A because the mass of the planet is directly proportional to the square of the orbital speed and the cube of the orbital radius.

Question 7

A satellite that weighs 4900 N on the launchpad travels around the earth's equator in a circular orbit with a period of 1.667 h. The earth's mass is 5.97 × 10²⁴ kg, its equatorial radius is 6.3 × 10⁶ m, and G = 6.67 × 10^-11 N ∙ m²/kg². (a) Calculate the magnitude of the earth's gravitational force on the satellite. (b) Determine the altitude of the satellite above the Earth's SURFACE.

Accepted Answer

The answer of A satellite that weighs 4900 N on...

Question 8

From what height above the surface of the earth should an object be dropped to initially experience an acceleration of 0.9200g? The radius of the earth is 6.38 × 10⁶ m. A) 272 km B) 260 km C) 554 km D) 510 km

Accepted Answer

The acceleration due to gravity decreases with height according to the formula g' = g * (R / (R + h))^2, where g is the acceleration due to gravity at the surface, R is the radius of the Earth, and h is the height above the surface. Solving for h when g' = 0.9200g gives approximately 272 km.

Question 9

Ekapluto is an unknown planet that has two spherical moons in circular orbits. The table summarizes the hypothetical data about the moons. Both moons have low axial spin rates. (G = 6.67 × 10^-11 N ∙ m²/kg²) The acceleration due to gravity at the surface of Moon B is A) 0.10 m/s². B) 0.15 m/s². C) 0.20 m/s². D) 0.25 m/s². E) 0.30 m/s².

Accepted Answer

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Question 10

The moons of Mars, Phobos (Fear) and Deimos (Terror), are very close to the planet compared to Earth's Moon. Their orbital radii are 9,378 km and 23,459 km respectively. What is the ratio of the orbital speed of Phobos to that of Deimos?
A) 0.2528
B) 0.3998
C) 1.582
D) 2.858
E) 3.956

Accepted Answer

The ratio of the orbital speed is given by the inverse ratio of the square roots of the radii.

$$\frac{v_p}{v_d} = \sqrt{\frac{R_d}{R_p}} = \sqrt{\frac{23,459}{9,378}} \approx 1.582$$

Therefore, the correct answer is (C).

Question 11

A meteoroid, heading straight for Earth, has a speed of 14.8 km/s relative to the center of Earth as it crosses our moon's orbit, a distance of 3.84 × 10⁸ m from the earth's center. What is the meteoroid's speed as it hits the earth? You can neglect the effects of the moon, Earth's atmosphere, and any motion of the earth. (G = 6.67 × 10^-11 N ∙ m²/kg², Mearth = 5.97 × 10²⁴ kg) A) 18.5 km/s B) 21.5 km/s C) 32.4 km/s D) 87.3 km/s

Accepted Answer

The correct answer is A because the meteoroid's speed increases as it falls toward Earth due to the Earth's gravitational pull.

Question 12

A certain spherical asteroid has a mass of 3.5 × 10¹⁶ kg and a radius of 8.8 km. What is the minimum speed needed to escape from the surface of this asteroid? (G = 6.67 × 10^-11 N ∙ m²/kg²) A) 23 m/s B) 16 m/s C) 520 m/s D) 730 m/s

Accepted Answer

The escape velocity $ v $ is calculated using the formula $ v = \sqrt{\frac{2GM}{R}} $. Substituting the given values: $ G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 $, $ M = 3.5 \times 10^{16} \, \text{kg} $, and $ R = 8.8 \times 10^3 \, \text{m} $, we find $ v \approx 23 \, \text{m/s} $.

Question 13

What is the period (in hours) of a satellite circling Mars 100 km above the planet's surface? The mass of Mars is 6.42 × 10²³ kg, its radius is 3.40 × 10⁶ m, and G = 6.67 × 10^-11 N ∙ m²/kg². A) 1.75 h B) 1.25 h C) 1.15 h D) 1.00 h E) 1.45 h

Accepted Answer

The period of a satellite circling a planet is given by:$$T = 2\pi \sqrt{\frac{r^3}{GM}}$$where r is the distance from the center of the planet to the satellite, G is the gravitational constant, and M is the mass of the planet.Plugging in the given values, we get:$$T = 2\pi \sqrt{\frac{(3.40 	imes 10^6 m + 100 km)^3}{(6.67 	imes 10^{-11} N \cdot m^2/kg^2)(6.42 	imes 10^{23} kg)}}$$$$T = 1.75 h$$

Question 14

A 910-kg object is released from rest at an altitude of 1200 km above the north pole of the earth. If we ignore atmospheric friction, with what speed does the object strike the surface of the earth? (G = 6.67 × 10^-11N ∙ m²/kg², Mearth = 5.97 × 10²⁴ kg, the polar radius of the earth is 6357 km) A) 4.5 km/s B) 2.2 km/s C) 2.7 km/s D) 3.2 km/s E) 4.8 km/s

Accepted Answer

The correct answer is A because the object is released from rest at an altitude of 1200 km above the north pole of the earth, and we ignore atmospheric friction. So the object will fall freely under the gravity of the earth. The speed of the object when it strikes the surface of the earth can be calculated by using the conservation of energy. The initial energy of the object is all potential energy, and the final energy of the object is all kinetic energy. So we have:$$mgh = \frac{1}{2}mv^2$$where m is the mass of the object, g is the acceleration due to gravity, h is the altitude of the object, and v is the speed of the object.Solving for v, we get:$$v = \sqrt{2gh}$$Substituting the given values, we get:$$v = \sqrt{2 	imes 6.67 	imes 10^{-11} 	imes 5.97 	imes 10^{24} 	imes 1200000} = 4.5 	ext{ km/s}$$

Question 15

Sputnik I was launched into orbit around Earth in 1957. It had a perigee (the closest approach to Earth, measured from Earth's center) of 6.81 × 10⁶ m and an apogee (the furthest point from Earth's center) of 7.53 × 10⁶ m. What was its speed when it was at its perigee? The mass of Earth is 5.97 × 10²⁴ kg and G = 6.67 × 10^-11 N ∙ m²/kg². A) 7180 m/s B) 7840 m/s C) 8230 m/s D) 11,000 m/s E) 13,400 m/s

Accepted Answer

The speed at perigee can be calculated using the vis-viva equation: v = sqrt(GM(2/r - 1/a)), where r is the perigee distance, a is the semi-major axis, and GM is the gravitational parameter. Calculating this gives approximately 7840 m/s.

Question 16

Jupiter completes one revolution about its own axis every 9.92 hours. What is the radius of the orbit required for a satellite to revolve about Jupiter with the same period? Jupiter has a mass of 1.90 × 10²⁷ kg and G = 6.67 × 10^-11 N ∙ m²/kg². A) 1.04 × 10⁷ m B) 2.26 × 10⁹ m C) 1.60 × 10⁸ m D) 3.41 × 10⁸ m E) 7.45 × 10⁸ m

Accepted Answer

The period of revolution of a satellite around a planet can be found using Kepler's third law:

T² = (4π²a³)/(GM)

where T is the period of revolution, a is the radius of the orbit, G is the gravitational constant, and M is the mass of the planet.

In this case, we want to find the radius of the orbit (a) for a satellite to revolve around Jupiter with a period of 9.92 hours. We can rearrange Kepler's third law to solve for a:

a = [(GMT²)/(4π²)]^(1/3)

Plugging in the values:

a = [((6.67 × 10^-11) * (1.90 × 10^27) * (9.92*3600)^2)/(4π²)]^(1/3)

a ≈ 1.60 × 10^18 meters

Therefore, the best answer is (C).

Question 17

A satellite is in circular orbit at an altitude of 1500 km above the surface of a nonrotating planet with an orbital speed of 9.2 km/s. The minimum speed needed to escape from the surface of the planet is 14.9 km/s, and G = 6.67 × 10^-11 N ∙ m²/kg². The orbital period of the satellite is closest to A) 72 min. B) 65 min. C) 58 min. D) 51 min. E) 44 min.

Accepted Answer

The orbital period can be calculated using the formula T = 2πr/v, where r is the orbital radius and v is the orbital speed. Given the altitude and the escape speed, we can find the planet's radius and mass, and then calculate the orbital period. The calculations show that the period is closest to 72 minutes.

Question 18

An object drops a distance h in a time of 6.3 s on the surface of the earth (neglecting air effects). How long would it take the same object to drop the same distance on the surface of spherical asteroid X having a mass of 1.1 × 10²² kg and a radius of 4.0 × 10⁵ m? (G = 6.67 × 10^-11 N ∙ m²/kg², Mearth = 5.97 × 10²⁴ kg, Rearth = 6.38 × 10⁶ m) A) 9.2 s B) 6.3 s C) 5.2 s D) 12 s E) 3.3 s

Accepted Answer

The gravitational acceleration on asteroid X is less than that on Earth due to its smaller mass and size. This results in a longer time to fall the same distance. Calculating the gravitational acceleration on asteroid X and comparing it to Earth's, we find the time to be approximately 9.2 s.

Question 19

The International Space Station is orbiting at an altitude of about 370 km above the earth's surface. The mass of the earth is 5.97 × 10²⁴ kg, the radius of the earth is 6.38 × 10⁶ m, and G = 6.67 × 10^-11 N ∙ m²/kg². Assuming a circular orbit, (a) what is the period of the International Space Station's orbit? (b) what is the speed of the International Space Station in its orbit?

Accepted Answer

The answer of The International Space Station is orbiting at...

Question 20

A huge cannon is assembled on an airless planet having insignificant axial spin. The planet has a radius of 5.00 × 10⁶ m and a mass of 3.95 × 10²³ kg. The cannon fires a projectile straight up at 2000 m/s. An observation satellite orbits the planet at a height of 1000 km. What is the projectile's speed as it passes the satellite? (G = 6.67 × 10^-11 N ∙ m²/kg²) A) 1500 m/s B) 1380 m/s C) 1610 m/s D) 1280 m/s

Accepted Answer

The answer of A huge cannon is assembled on an...

Quiz 12: Gravitation

A satellite is in circular orbit at an altitude of 2300 km above the surface of a nonrotating asteroid with an orbital speed of 5.9 km/s. The minimum speed needed to escape from the surface of the asteroid is 14.6 km/s, and G = 6.67 × 10^-11 N ∙ m²/kg². The mass of the asteroid is closest to

What is the ratio of the escape speed of a rocket launched from sea level to the escape speed of one launched from Mt. Everest (an altitude of 8.85 km) ? The radius of the earth is 6.38 × 10⁶ m.

Find the orbital speed of an ice cube in the rings of Saturn. The mass of Saturn is 5.68 × 1026 kg and the rings have an average radius of 100,000 km. (G = 6.67 × 10^-11 N ∙ m²/kg²)

Spaceman Speff orbits spherical asteroid X with his spaceship. To remain in a circular orbit at 421 km from the asteroid's center, he should maintain a speed of 80 m/s. What is the mass of planet X? (G = 6.67 × 10^-11 N ∙ m²/kg²)

From what height above the surface of the earth should an object be dropped to initially experience an acceleration of 0.9200g? The radius of the earth is 6.38 × 10⁶ m.

Ekapluto is an unknown planet that has two spherical moons in circular orbits. The table summarizes the hypothetical data about the moons. Both moons have low axial spin rates. (G = 6.67 × 10^-11 N ∙ m²/kg²) The acceleration due to gravity at the surface of Moon B is

The moons of Mars, Phobos (Fear) and Deimos (Terror) , are very close to the planet compared to Earth's Moon. Their orbital radii are 9,378 km and 23,459 km respectively. What is the ratio of the orbital speed of Phobos to that of Deimos?

A certain spherical asteroid has a mass of 3.5 × 10¹⁶ kg and a radius of 8.8 km. What is the minimum speed needed to escape from the surface of this asteroid? (G = 6.67 × 10^-11 N ∙ m²/kg²)

What is the period (in hours) of a satellite circling Mars 100 km above the planet's surface? The mass of Mars is 6.42 × 10²³ kg, its radius is 3.40 × 10⁶ m, and G = 6.67 × 10^-11 N ∙ m²/kg².

Jupiter completes one revolution about its own axis every 9.92 hours. What is the radius of the orbit required for a satellite to revolve about Jupiter with the same period? Jupiter has a mass of 1.90 × 10²⁷ kg and G = 6.67 × 10^-11 N ∙ m²/kg².

Quiz 12: Gravitation

A satellite is in circular orbit at an altitude of 2300 km above the surface of a nonrotating asteroid with an orbital speed of 5.9 km/s. The minimum speed needed to escape from the surface of the asteroid is 14.6 km/s, and G = 6.67 × 10-11 N ∙ m2/kg2. The mass of the asteroid is closest to

What is the ratio of the escape speed of a rocket launched from sea level to the escape speed of one launched from Mt. Everest (an altitude of 8.85 km) ? The radius of the earth is 6.38 × 106 m.

Find the orbital speed of an ice cube in the rings of Saturn. The mass of Saturn is 5.68 × 1026 kg and the rings have an average radius of 100,000 km. (G = 6.67 × 10-11 N ∙ m2/kg2)

Spaceman Speff orbits spherical asteroid X with his spaceship. To remain in a circular orbit at 421 km from the asteroid's center, he should maintain a speed of 80 m/s. What is the mass of planet X? (G = 6.67 × 10-11 N ∙ m2/kg2)

From what height above the surface of the earth should an object be dropped to initially experience an acceleration of 0.9200g? The radius of the earth is 6.38 × 106 m.

Ekapluto is an unknown planet that has two spherical moons in circular orbits. The table summarizes the hypothetical data about the moons. Both moons have low axial spin rates. (G = 6.67 × 10-11 N ∙ m2/kg2) The acceleration due to gravity at the surface of Moon B is