Question 1

The speed of a comet in an elliptical orbit about the sun:

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Question 2

Two planets are orbiting a star in a distant galaxy. The first has a semimajor axis of 150 *10⁶ km, an eccentricity of 0.20, and a period of 1.0 Earth years. The second has a semimajor axis of 250 *10⁶ km, an eccentricity of 0.30, and a period of:

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Question 3

A spaceship is returning to Earth with its engine turned off. Consider only the gravitational field of Earth and let M be the mass of Earth, m be the mass of the spaceship, and R be the distance from the center of Earth. In moving from position 1 to position 2 the kinetic energy of the spaceship increases by:

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Question 4

A planet is in circular orbit around the Sun. Its distance from the Sun is four times the average distance of Earth from the Sun. The period of this planet, in Earth years, is:

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According to Kepler's third law, the square of the period of a planet is proportional to the cube of its average distance from the Sun. Let's use Earth's average distance from the Sun as 1 astronomical unit (AU). Therefore, the distance of the planet from the Sun is 4 AU.

Using Kepler's third law:

(T planet)^2 / (a planet)^3 = (T earth)^2 / (a earth)^3

(T planet)^2 / (4)^3 = (1)^2 / (1)^3

(T planet)^2 / 64 = 1

(T planet)^2 = 64

T planet = 8 Earth years.

Therefore, the answer is B) 8.

Question 5

An artificial Earth satellite is moved from a circular orbit with radius R to a circular orbit with radius 2R. During this move:

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Question 6

Assume that Earth is in circular orbit around the Sun with kinetic energy K and potential energy U, taken to be zero for infinite separation. Then, the relationship between K and U:

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In a circular orbit, the kinetic energy (K) is equal to half the magnitude of the potential energy (U), but with a positive sign, making the relationship K = -U/2. This is a result of the conservation of energy and the specifics of gravitational force, where the total energy (which is the sum of kinetic and potential energy) is negative, indicating a bound system.

Question 7

To measure the mass of a planet with the same radius as Earth, an astronaut drops an object from rest (relative to the planet) from an altitude of one radius above the surface. When the object hits its speed is 4 times what it would be if the same experiment were carried out for Earth. In units of Earth masses, the mass of the planet is:

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Using conservation of energy, we know that the initial gravitational potential energy of the object is equal to its final kinetic energy when it hits the surface.

For Earth, the initial potential energy is:
$$U_i = \frac{Gm_eM_e}{2R_e}$$
where $m_e$ is the mass of the object, $M_e$ is the mass of Earth, and $R_e$ is the radius of Earth.

The final kinetic energy is:
$$K_f = \frac{1}{2}m_ev_e^2$$
where $v_e$ is the final velocity of the object when it hits Earth's surface.

Setting $U_i = K_f$, we can solve for $v_e$:
$$v_e = \sqrt{\frac{2G M_e}{R_e}}$$

For the planet with the same radius as Earth, we can set up the same equations with $M_p$ as the mass of the planet:
$$U_i = \frac{Gm_pM_p}{2R_e}$$
and
$$K_f = \frac{1}{2}m_ev_p^2$$
where $v_p$ is the final velocity of the object when it hits the planet's surface.

Setting $U_i = K_f$ again, and using the fact that $v_p = 4v_e$, we get:
$$\frac{Gm_pM_p}{2R_e} = \frac{1}{2}m_e(4v_e)^2$$
Simplifying and solving for $M_p$, we get:
$$M_p = 16\frac{m_e}{M_e}$$
Since we are measuring the mass of the planet in units of Earth masses, we can substitute $m_e$ for the mass of the object in Earth masses (which is 1):
$$M_p = 16 M_\oplus$$
where $M_\oplus$ is the mass of Earth. Therefore, the answer is D) 16.

Question 8

An artificial satellite of Earth nears the end of its life due to air resistance. While still in orbit:

Accepted Answer

As an artificial satellite's orbit lowers due to air resistance, it moves faster due to conservation of angular momentum and the gravitational pull becoming stronger as it gets closer to Earth.

Question 9

A planet in another solar system orbits a star with a mass of 4.0 *10³⁰ kg. At one point in its orbit it is 250 * 10⁶ km from the star and is moving at 35 km/s. Take the universal gravitational constant to be 6.67*10^-¹¹ m²/s² . kg and calculate the semimajor axis of the planet's orbit. The result is:

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Question 10

The orbit of a certain a satellite has a semimajor axis of 1.5 * 10⁷ m and an eccentricity of 0.20. Its perigee (minimum distance) and apogee (maximum distance) are respectively:

Accepted Answer

The semimajor axis is the average distance between the satellite and the center of the Earth. The eccentricity is a measure of how elliptical the orbit is, with 0 being a perfect circle and 1 being a parabola. The perigee is the closest point in the orbit to the center of the Earth, and the apogee is the farthest point.The perigee is given by:$$r_p = a(1 - e)$$where a is the semimajor axis and e is the eccentricity.The apogee is given by:$$r_a = a(1 + e)$$Plugging in the given values, we get:$$r_p = 1.5 	imes 10^7 m (1 - 0.20) = 1.2 	imes 10^7 m$$$$r_a = 1.5 	imes 10^7 m (1 + 0.20) = 1.8 	imes 10^7 m$$Therefore, the perigee is 1.2 x 10^7 m and the apogee is 1.8 x 10^7 m.

Question 11

A planet travels in an elliptical orbit about a star as shown. At what pair of points is the speed of the planet the same?

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Question 12

For a planet in orbit around a star the perihelion distance is r_p and its speed at perihelion is v_p. The aphelion distance is r_a and its speed at aphelion is v_a. Which of following is true?

Accepted Answer

The conservation of angular momentum for a planet in orbit states that the product of the speed and the distance from the star is constant at any point in the orbit, so v_a r_a = v_p r_p.

Question 13

Given the perihelion distance, aphelion distance, and speed at perihelion of a planet, which of the following CANNOT be calculated?

Accepted Answer

The mass of the planet cannot be calculated directly from the given information (perihelion distance, aphelion distance, and speed at perihelion) because these parameters are primarily influenced by the mass of the star around which the planet orbits, not the mass of the planet itself. The gravitational force (and thus the orbital characteristics) that a planet experiences is predominantly due to the star's mass according to Kepler's laws and Newton's law of universal gravitation.

Question 14

A small satellite is in elliptical orbit around Earth as shown. If L denotes the magnitude of its angular momentum and K denotes kinetic energy:

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Question 15

Planet 1 and planet 2 are both in circular orbits around the same central star. The orbit of planet 2 has a radius that is much larger than the radius of the orbit of planet 1. This means that:

Accepted Answer

According to Kepler's third law of planetary motion, the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This means that a planet with a larger orbital radius (like planet 2) will have a longer period than a planet with a smaller orbital radius (like planet 1). Additionally, in circular orbits, the orbital speed is inversely proportional to the square root of the orbital radius, meaning that planet 1, with its smaller orbit, moves faster than planet 2.

Quiz 13: Gravitation

The speed of a comet in an elliptical orbit about the sun:

Two planets are orbiting a star in a distant galaxy. The first has a semimajor axis of 150 10⁶ km, an eccentricity of 0.20, and a period of 1.0 Earth years. The second has a semimajor axis of 250 10⁶ km, an eccentricity of 0.30, and a period of:

A planet is in circular orbit around the Sun. Its distance from the Sun is four times the average distance of Earth from the Sun. The period of this planet, in Earth years, is:

An artificial Earth satellite is moved from a circular orbit with radius R to a circular orbit with radius 2R. During this move:

Assume that Earth is in circular orbit around the Sun with kinetic energy K and potential energy U, taken to be zero for infinite separation. Then, the relationship between K and U:

An artificial satellite of Earth nears the end of its life due to air resistance. While still in orbit:

The orbit of a certain a satellite has a semimajor axis of 1.5 * 10⁷ m and an eccentricity of 0.20. Its perigee (minimum distance) and apogee (maximum distance) are respectively:

A planet travels in an elliptical orbit about a star as shown. At what pair of points is the speed of the planet the same?

For a planet in orbit around a star the perihelion distance is r_p and its speed at perihelion is v_p. The aphelion distance is r_a and its speed at aphelion is v_a. Which of following is true?

Given the perihelion distance, aphelion distance, and speed at perihelion of a planet, which of the following CANNOT be calculated?

A small satellite is in elliptical orbit around Earth as shown. If L denotes the magnitude of its angular momentum and K denotes kinetic energy:

Planet 1 and planet 2 are both in circular orbits around the same central star. The orbit of planet 2 has a radius that is much larger than the radius of the orbit of planet 1. This means that:

Quiz 13: Gravitation

The speed of a comet in an elliptical orbit about the sun:

Two planets are orbiting a star in a distant galaxy. The first has a semimajor axis of 150 *106 km, an eccentricity of 0.20, and a period of 1.0 Earth years. The second has a semimajor axis of 250 *106 km, an eccentricity of 0.30, and a period of:

A planet is in circular orbit around the Sun. Its distance from the Sun is four times the average distance of Earth from the Sun. The period of this planet, in Earth years, is:

An artificial Earth satellite is moved from a circular orbit with radius R to a circular orbit with radius 2R. During this move:

Assume that Earth is in circular orbit around the Sun with kinetic energy K and potential energy U, taken to be zero for infinite separation. Then, the relationship between K and U:

An artificial satellite of Earth nears the end of its life due to air resistance. While still in orbit:

The orbit of a certain a satellite has a semimajor axis of 1.5 * 107 m and an eccentricity of 0.20. Its perigee (minimum distance) and apogee (maximum distance) are respectively:

A planet travels in an elliptical orbit about a star as shown. At what pair of points is the speed of the planet the same?

For a planet in orbit around a star the perihelion distance is rp and its speed at perihelion is vp. The aphelion distance is ra and its speed at aphelion is va. Which of following is true?

Given the perihelion distance, aphelion distance, and speed at perihelion of a planet, which of the following CANNOT be calculated?

A small satellite is in elliptical orbit around Earth as shown. If L denotes the magnitude of its angular momentum and K denotes kinetic energy:

Planet 1 and planet 2 are both in circular orbits around the same central star. The orbit of planet 2 has a radius that is much larger than the radius of the orbit of planet 1. This means that:

Two planets are orbiting a star in a distant galaxy. The first has a semimajor axis of 150 10⁶ km, an eccentricity of 0.20, and a period of 1.0 Earth years. The second has a semimajor axis of 250 10⁶ km, an eccentricity of 0.30, and a period of:

The orbit of a certain a satellite has a semimajor axis of 1.5 * 10⁷ m and an eccentricity of 0.20. Its perigee (minimum distance) and apogee (maximum distance) are respectively:

For a planet in orbit around a star the perihelion distance is r_p and its speed at perihelion is v_p. The aphelion distance is r_a and its speed at aphelion is v_a. Which of following is true?