Question 1

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 $v = \sqrt{\frac{2GM}{r}}$, where $G$ is the gravitational constant and $M$ is the mass of the Earth. The escape speed from sea level ($r = R$, the radius of the Earth) is $v_1 = \sqrt{\frac{2GM}{R}}$, and from Mt. Everest ($r = R + h$, where $h$ is the height of Mt. Everest) is $v_2 = \sqrt{\frac{2GM}{R + h}}$. The ratio of the escape speeds is $\frac{v_1}{v_2} = \sqrt{\frac{R + h}{R}}$. Substituting $R = 6.38 \times 10^6 m$ and $h = 8.85 \times 10^3 m$, the ratio is very close to 1, with a slight increase due to the additional altitude of Mt. Everest, making the correct answer 1.0007.

Question 2

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 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^-11 N • m²/kg²)

Accepted Answer

Explanation:Using energy conservation, we have:$$K_i + U_i = K_f + U_f$$$$ \frac{1}{2}mv_i^2 - \frac{Gm_1m_2}{r_i} = \frac{1}{2}mv_f^2 - \frac{Gm_1m_2}{r_f}$$$$ \frac{1}{2}v_i^2 - \frac{G m_2}{r_i} = \frac{1}{2}v_f^2 - \frac{G m_2}{r_f}$$$$ \frac{1}{2}(136.0 	ext{ m/s})^2 - \frac{(6.67 	imes 10^{-11} 	ext{ N} \cdot 	ext{m}^2/	ext{kg}^2) m_2}{7514 	ext{ km}} = \frac{1}{2}(392.0 	ext{ m/s})^2 - \frac{(6.67 	imes 10^{-11} 	ext{ N} \cdot 	ext{m}^2/	ext{kg}^2) m_2}{2823 	ext{ km}}$$$$ m_2 = 4.582 	imes 10^{21} 	ext{ kg}$$

Question 4

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.

Accepted Answer

The acceleration due to gravity at a distance $r$ from the center of the Earth is given by $g' = g\left(\frac{R}{r}\right)^2$, where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity at the surface of the Earth. Given $g' = 0.9200g$, we can solve for $r$ and then find the height above the Earth's surface by subtracting the Earth's radius. Solving for $r$ gives a height of approximately 272 km above the Earth's surface.

Question 5

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)

Accepted Answer

The correct answer is A because the mass of the planet can be calculated using the formula for centripetal force, which is F = mv^2/r. In this case, the force is provided by the gravitational force between the planet and the spaceship, which is F = G * m1 * m2 / r^2. Setting these two forces equal to each other and solving for m1, the mass of the planet, gives m1 = v^2 * r / G * m2. Plugging in the given values, we get m1 = (80 m/s)^2 * 421 km / (6.67 × 10^-11 N • m^2/kg^2) * m2. Since the mass of the spaceship is negligible compared to the mass of the planet, we can simplify this to m1 = (80 m/s)^2 * 421 km / (6.67 × 10^-11 N • m^2/kg^2). Evaluating this expression gives m1 = 4.0 × 10^19 kg.

Question 6

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)

Accepted Answer

The projectile will reach its maximum height when its final velocity becomes zero. We can use conservation of energy to calculate this maximum height:

$Initial\,Energy = Kinetic\,Energy\,+\,Potential\,Energy$
$mv^2/2 = mgh$
where m is the mass of the projectile, v is the initial velocity, h is the maximum height.

From this equation, we can calculate the maximum height to be 20404.08 km. Since the observation satellite orbits at a height of 1000 km, we know that the projectile will pass the satellite on its way up and on its way down.

Using conservation of energy again, but now at the height of the satellite, we can calculate the velocity of the projectile as it passes the satellite on its way up:

$Total\,Energy = Kinetic\,Energy\,+\,Potential\,Energy$
$mv^2/2 - GmM_p/(r_p+h_s) = mv_s^2/2$
where Mp is the mass of the planet, rp is the radius of the planet, and hs is the height of the satellite.

Plugging in the values, we get:

$v = \sqrt{2(v_s^2 + G\,M_p/(r_p+h_s)) - v_i^2} = 1500\,m/s$

Therefore, the best choice is A) 1500 m/s.

Question 7

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^-¹¹N • m²/kg².

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 8

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 meteoroid's final speed can be calculated using the conservation of energy principle. The initial kinetic energy plus the initial gravitational potential energy equals the final kinetic energy plus the final gravitational potential energy. The increase in speed is due to the conversion of gravitational potential energy into kinetic energy as the meteoroid falls towards Earth.

Question 9

Suppose we want a satellite to revolve around the earth 5 times a day.What should be the radius of its orbit? (The mass of the earth is 5.97 × 10²⁴ kg,G = 6.67 × 10^-11 N • m²/kg²,and you can neglect the presence of the moon.)

Accepted Answer

To find the radius of the orbit for a satellite to revolve around the Earth 5 times a day, we use the formula for the orbital period $T$ of a satellite: $T = 2\pi\sqrt{\frac{r^3}{GM}}$, where $r$ is the radius of the orbit, $G$ is the gravitational constant, and $M$ is the mass of the Earth. Rearranging the formula to solve for $r$, and knowing that the satellite revolves around the Earth 5 times a day (which means $T = \frac{24}{5}$ hours or $T = \frac{24 \times 3600}{5}$ seconds for one revolution), we can plug in the values for $G$ and $M$ to find $r$. The calculation involves solving for $r$ in the equation given the values for $T$, $G$, and $M$, leading to the correct radius for the desired orbital period. The exact calculation requires substituting the numerical values and solving for $r$, which results in the radius closest to the option A) $1.44 \times 10^{7} m$.

Question 10

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 11

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

Accepted Answer

The answer of Ekapluto is an unknown planet that has...

Question 12

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 answer of A 910-kg object is released from rest...

Question 13

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 12.0 hours and an orbital radius 6.00 × 10⁷ m. The second planet has a period 16.0 hours. What is the orbital radius of the second satellite?

Accepted Answer

The answer of A planet has two small satellites in...

Question 14

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

The answer of Sputnik I was launched into orbit around...

Question 15

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 18.0 hours and an orbital radius 2.00 × 10⁷ m. The second planet has an orbital radius 3.00 × 10⁷ m. What is the period of the second satellite?

Accepted Answer

We can use Kepler's third law to solve this problem, which states:

$$\frac{T_1^2}{R_1^3} = \frac{T_2^2}{R_2^3}$$

where $T$ is the period of the satellite and $R$ is its orbital radius.

Plugging in the values given in the problem, we get:

$$\frac{(18.0\ \mathrm{hours})^2}{(2.00	imes10^{17}\ \mathrm{m})^3} = \frac{T_2^2}{(3.00	imes10^{17}\ \mathrm{m})^3}$$

Solving for $T_2$, we get:

$$T_2 = \sqrt{\frac{(18.0\ \mathrm{hours})^2	imes(3.00	imes10^{17}\ \mathrm{m})^3}{(2.00	imes10^{17}\ \mathrm{m})^3}} \approx 33.1\ \mathrm{hours}$$

Therefore, the answer is C.

Question 16

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 velocity can be calculated using the formula $v = \sqrt{\frac{2GM}{r}}$, where $G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$, $M = 3.5 \times 10^{16} \, \text{kg}$, and $r = 8.8 \times 10^3 \, \text{m}$. Plugging in these values gives an escape velocity of approximately 23 m/s.

Question 17

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 found using the formula $v = \sqrt{\frac{GM}{r}}$, where $G$ is the gravitational constant $6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$, $M$ is the mass of Saturn $5.68 \times 10^{26} \, \text{kg}$, and $r$ is the average radius of the rings $100,000 \, \text{km} = 100,000,000 \, \text{m}$. Plugging in the values, we get $v = \sqrt{\frac{6.67 \times 10^{-11} \times 5.68 \times 10^{26}}{100,000,000}} \approx 19.5 \, \text{km/s}$.

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,M_earth = 5.97 × 10²⁴ kg,R_earth = 6.38 × 10⁶ m)

Accepted Answer

The time it takes for an object to drop a certain distance is related to the acceleration due to gravity on the surface of the celestial body. The acceleration due to gravity, $g$, can be calculated using the formula $g = \frac{G \cdot M}{R^2}$, where $G$ is the gravitational constant, $M$ is the mass of the celestial body, and $R$ is its radius.For Earth, $g_{\text{earth}} = \frac{6.67 \times 10^{-11} \cdot 5.97 \times 10^{24}}{(6.38 \times 10^6)^2} \approx 9.8 \, \text{m/s}^2$.For asteroid X, $g_{\text{X}} = \frac{6.67 \times 10^{-11} \cdot 1.1 \times 10^{22}}{(4.0 \times 10^5)^2}$.The time to fall a distance $h$ is given by $t = \sqrt{\frac{2h}{g}}$. Since $h$ is constant, the ratio of times on Earth and asteroid X is proportional to the square root of the inverse ratio of their gravities, i.e., $t_{\text{X}} = t_{\text{earth}} \cdot \sqrt{\frac{g_{\text{earth}}}{g_{\text{X}}}}$.Calculating $g_{\text{X}}$ and then $t_{\text{X}}$ with the given values, you find that the time on asteroid X is longer due to its smaller mass and radius, which results in a lower gravitational acceleration compared to Earth. The exact calculation would show that the correct answer is 9.2 seconds, reflecting the reduced gravity's effect on the object's fall time.

Question 19

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.)

Accepted Answer

The correct answer is A because the mass of the planet is 7.2 kg × 10^25 kg and the mass of the star is 2.9 kg × 10^30 kg.

Question 20

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 $v$ of a moon is given by $v = \sqrt{\frac{GM}{r}}$, where $G$ is the gravitational constant, $M$ is the mass of the planet, and $r$ is the orbital radius. The ratio of the orbital speeds of Phobos to Deimos is thus $\sqrt{\frac{r_{Deimos}}{r_{Phobos}}}$ = $\sqrt{\frac{23459}{9378}}$ ≈ 1.582, because the mass of Mars $M$ and the gravitational constant $G$ cancel out.

Quiz 13: Gravity

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.

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.

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)

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^-¹¹N • m²/kg².

Suppose we want a satellite to revolve around the earth 5 times a day.What should be the radius of its orbit? (The mass of the earth is 5.97 × 10²⁴ kg,G = 6.67 × 10^-11 N • m²/kg²,and you can neglect the presence of the moon.)

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 planet has two small satellites in circular orbits around the planet. The first satellite has a period 12.0 hours and an orbital radius 6.00 × 10⁷ m. The second planet has a period 16.0 hours. What is the orbital radius of the second satellite?

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 18.0 hours and an orbital radius 2.00 × 10⁷ m. The second planet has an orbital radius 3.00 × 10⁷ m. What is the period of the second satellite?

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)

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)

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 13: Gravity

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.

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.

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)

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.

Suppose we want a satellite to revolve around the earth 5 times a day.What should be the radius of its orbit? (The mass of the earth is 5.97 × 1024 kg,G = 6.67 × 10-11 N • m2/kg2,and you can neglect the presence of the moon.)

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 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,Mearth = 5.97 × 1024 kg,the polar radius of the earth is 6357 km)

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 12.0 hours and an orbital radius 6.00 × 107 m. The second planet has a period 16.0 hours. What is the orbital radius of the second satellite?

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 18.0 hours and an orbital radius 2.00 × 107 m. The second planet has an orbital radius 3.00 × 107 m. What is the period of the second satellite?

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)

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)

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?

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.

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.

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)

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^-¹¹N • m²/kg².

Suppose we want a satellite to revolve around the earth 5 times a day.What should be the radius of its orbit? (The mass of the earth is 5.97 × 10²⁴ kg,G = 6.67 × 10^-11 N • m²/kg²,and you can neglect the presence of the moon.)

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 planet has two small satellites in circular orbits around the planet. The first satellite has a period 12.0 hours and an orbital radius 6.00 × 10⁷ m. The second planet has a period 16.0 hours. What is the orbital radius of the second satellite?

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 18.0 hours and an orbital radius 2.00 × 10⁷ m. The second planet has an orbital radius 3.00 × 10⁷ m. What is the period of the second satellite?

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)

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)