Question 1

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

Question 2

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 3

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 calculated using the formula for the period of a satellite in circular orbit: $T = 2\pi \sqrt{\frac{r^3}{GM}}$, 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 is 1500 km above the surface and the radius of the planet can be derived from the escape velocity formula $v_{esc} = \sqrt{\frac{2GM}{R}}$, we can find $M$ and subsequently $r$ to calculate $T$. However, without going through the detailed calculation, the correct choice is determined by understanding the relationship between orbital speed, altitude, and period, which is not directly calculable from the provided information without additional steps. The choice is based on understanding the principles of orbital mechanics rather than a direct calculation from the given data.

Question 4

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 5

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 6

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 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^-11 N ∙ m²/kg².

Accepted Answer

We can use Kepler's Third Law to calculate the period of the satellite:
$T^2 = \frac{4\pi^2}{GM}r^3$
where T is the period, G is the gravitational constant, M is the mass of Mars, and r is the distance between the satellite and the center of Mars.
Substituting the given values, we get:
$T^2 = \frac{4\pi^2}{(6.67	imes 10^{-11}	ext{ N} \cdot 	ext{m}^2/	ext{kg}^2)(6.42	imes 10^{23}	ext{ kg})}(100000+3.40	imes 10^{6})^3$
Simplifying, we get:
$T^2 \approx 93415$
$T \approx 1.75$ hours (rounded to two decimal places)
Therefore, the best choice is (A).

Question 8

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 9

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)

Accepted Answer

The answer of Beings on spherical asteroid Π have observed...

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

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 12

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 13

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 14

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 15

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.

Question 16

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 17

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 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 distance from the center of the asteroid to the surface. Plugging in the given values, we get: 14.6 km/s = sqrt(2 * 6.67 × 10^-11 N ∙ m^2/kg^2 * M / (2300 km + 6371 km)). Solving for M, we get: M = 1.8 × 10^24 kg.

Question 18

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 What is the mass of planet X? (G = 6.67 × 10^-11 N ∙ m²/kg²)

Accepted Answer

The answer of Spaceman Speff orbits spherical asteroid X with...

Question 19

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 solve for $r$, and then find the height above the surface by subtracting the Earth's radius.

Question 20

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

The answer of Jupiter completes one revolution about its own...

Quiz 13: Newtons Theory of Gravity

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.

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.

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?

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 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

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 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.

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 13: Newtons Theory of Gravity

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 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.

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)

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.

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?

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 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

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 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.

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 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.

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.

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

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 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

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 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.

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.