Question 1

Gravitational potential energy: 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^-11 N ∙ m²/kg², M_earth = 5.97 × 10²⁴ kg, the polar radius of the earth is 6357 km)

Accepted Answer

The gravitational potential energy at the initial altitude and at the Earth's surface can be used to find the change in potential energy, which equals the kinetic energy at impact. Using the formula for gravitational potential energy and solving for velocity, the speed is approximately 4.5 km/s.

Question 2

Gravitational potential energy: 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)

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Question 3

Gravitational potential energy: 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)

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Question 4

Gravitational potential energy: 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 meteroid'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, M_earth = 5.97 × 10²⁴ kg)

Accepted Answer

The correct answer is A because the meteoroid's speed increases as it falls towards Earth due to the increase in gravitational potential energy.

Question 5

Weight and g: 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

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Question 6

Escape speed: 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

Accepted Answer

The orbital period can be found using the formula for the period of a satellite orbiting a planet: $T = 2\pi \sqrt{\frac{r^3}{GM}}$, 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. Given the altitude of the satellite and the radius of the planet derived from the escape speed, we can calculate the orbital period. The closest answer given the calculation is approximately 72 minutes.

Question 7

Satellites: 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)

Accepted Answer

The orbital speed v can be calculated using the formula v = √(GM/r), where G is the gravitational constant, M is the mass of Saturn, and r is the radius of the orbit. Plugging in the values, v = √((6.67 × 10^-11) * (5.68 × 10^26) / (100,000,000)), which gives approximately 19.5 km/s.

Question 8

Escape speed: 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.

Accepted Answer

The escape speed $v$ from a distance $r$ from the center of a mass (like Earth) is given by the formula $v = \sqrt{\frac{2GM}{r}}$, where $G$ is the gravitational constant and $M$ is the mass of the Earth. The escape speed from sea level (radius $r_1 = 6.38 \times 10^6$ m) and from the top of Mt. Everest (radius $r_2 = r_1 + 8.85 \times 10^3$ m) can be compared by taking the ratio of their escape speeds:$$\frac{v_1}{v_2} = \frac{\sqrt{\frac{2GM}{r_1}}}{\sqrt{\frac{2GM}{r_2}}} = \sqrt{\frac{r_2}{r_1}}$$Plugging in the values:$$\frac{v_1}{v_2} = \sqrt{\frac{6.38 \times 10^6 + 8.85 \times 10^3}{6.38 \times 10^6}} \approx \sqrt{1.001387} \approx 1.0007$$Therefore, the ratio of the escape speed from sea level to that from Mt. Everest is approximately 1.0007.

Question 9

Escape speed: 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)

Accepted Answer

The escape speed $v$ can be calculated using the formula $v = \sqrt{\frac{2GM}{r}}$, where $G$ is the gravitational constant, $M$ is the mass of the asteroid, and $r$ is the radius of the asteroid. Substituting the given values: $v = \sqrt{\frac{2 \times 6.67 \times 10^{-11} \times 3.5 \times 10^{16}}{8800}} \approx 23$ m/s.

Question 10

Satellites: 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 × 10²⁴kg, 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?

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Question 11

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

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Question 12

Gravitational potential energy: 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.

Accepted Answer

To find the speed of Sputnik I at its perigee, we can use the formula for the orbital speed of a satellite: $v = \sqrt{\frac{GM}{r}}$, where $v$ is the orbital speed, $G$ is the gravitational constant, $M$ is the mass of the Earth, and $r$ is the distance from the center of the Earth to the satellite at perigee. Plugging in the given values: $G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$, $M = 5.97 \times 10^{24} \, \text{kg}$, and $r = 6.81 \times 10^6 \, \text{m}$, we can calculate the speed. This calculation yields a value close to the option B, 7840 m/s, indicating that at its closest approach to Earth, Sputnik I was traveling at this speed.

Question 13

Satellites: You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be 1.8 × 10⁷ m and its rotation period to be 22.3 hours. You have previously determined that the planet orbits 1.8 × 10¹¹ m from its star with a period of 402 earth days. Once on the surface you find that the acceleration due to gravity is 59.7 m/s². What are the mass of (a) the planet and (b) the star? (G = 6.67 × 10^-11 N ∙ m²/kg².)

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Question 14

Weight and g: 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, M_earth = 5.97 × 10²⁴ kg, R_earth = 6.38 × 10⁶ m)

Accepted Answer

The gravitational acceleration on asteroid X can be calculated using g = GM/R². Comparing this with Earth's gravity, the time taken to fall the same distance is proportional to the inverse square root of the gravitational acceleration. Calculating this gives approximately 9.2 s.

Question 15

Satellites: 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.

Accepted Answer

Using the formula for period of an orbit, we can find the required radius:
T = 2π(r/v)
Where T is the period (9.92 hours), r is the radius of the orbit we need to find, and v is the velocity of the satellite. The velocity can be found using the formula for centripetal force:
F = ma = mv²/r
v² = GM/r
v = √(GM/r)
Substituting this into the period formula gives:
T = 2π√(r³/GM)
Squaring both sides and rearranging gives:
r³ = (GMT²/4π²)
Substituting in the values for G, M, and T gives:
r³ = (6.67×10^-11 N∙m²/kg² × 1.90×10^27 kg × (9.92×3600 s)²)/(4π²)
Solving for r gives:
r = 1.60×10^18 m

Question 16

Gravitational potential energy: Beings on spherical asteroid Π have observed that a large rock is approaching their asteroid in a collision course. At from the center of the asteroid, the rock has a speed of and later at it has a speed of 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^-11 N ∙ m²/kg²)

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Question 17

Escape speed: 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

Accepted Answer

The correct answer is A because the escape speed from the surface of the asteroid is given by:$$v_{esc} = \sqrt{\frac{2GM}{r}}$$where:* $v_{esc}$ is the escape speed* $G$ is the gravitational constant* $M$ is the mass of the asteroid* $r$ is the radius of the asteroidWe can use the given information to solve for the mass of the asteroid:$$v_{esc} = 14.6 	ext{ km/s}$$$$G = 6.67 	imes 10^{-11} 	ext{ N} \cdot 	ext{m}^2/	ext{kg}^2$$$$r = 2300 	ext{ km} + 6371 	ext{ km} = 8671 	ext{ km}$$Substituting these values into the equation, we get:$$14.6 	ext{ km/s} = \sqrt{\frac{2(6.67 	imes 10^{-11} 	ext{ N} \cdot 	ext{m}^2/	ext{kg}^2)M}{(8671 	ext{ km})(1000 	ext{ m/km})^2}}$$Solving for $M$, we get:$$M = 1.8 	imes 10^{24} 	ext{ kg}$$

Question 18

Satellites: 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.

Accepted Answer

The period of a satellite can be calculated using Kepler's third law and the formula for orbital period: T = 2π√(r³/GM). Plugging in the values, including the altitude added to Mars' radius for r, gives approximately 1.75 hours.

Question 19

Satellites: 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?

Accepted Answer

The orbital speed of a satellite is inversely proportional to the square root of its orbital radius. Therefore, the ratio of the orbital speeds of Phobos to Deimos is the square root of the inverse ratio of their orbital radii: sqrt(23,459 / 9,378) = 1.582.

Question 20

Satellites: 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

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Quiz 10: Dynamics of Rotational Motion

Weight and g: 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

Satellites: 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)

Escape speed: 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.

Escape speed: 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)

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

Satellites: 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.

Satellites: 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.

Satellites: 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?

Quiz 10: Dynamics of Rotational Motion

Weight and g: 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

Satellites: 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)

Escape speed: 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.

Escape speed: 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)

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

Satellites: 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.

Satellites: 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.

Satellites: 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?

Weight and g: 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

Satellites: 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)

Escape speed: 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.

Escape speed: 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)

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

Satellites: 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.

Satellites: 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.