Kepler's laws Knowledge-Based Practice

Questions

1. 1.

   Two planets orbit the same star. Planet X travels in an elliptical orbit with a semi-major axis of $a$ and an orbital period of $T$, while Planet Y travels in a circular orbit of radius $a$. In terms of $T$, what is the orbital period of Planet Y?

   • A) $\frac{1}{2}T$
   • B) $\frac{1}{\sqrt{2}}T$
   • C) $\sqrt{2}T$
   • D) $T$
2. 2.

   A planet moves in an elliptical orbit with the Sun located at one focus. At which point or points along the orbit is the planet's instantaneous velocity vector strictly perpendicular to the position vector drawn from the Sun to the planet?

   • A) Only at perihelion and aphelion
   • B) Only at perihelion
   • C) Only at the two endpoints of the semi-minor axis
   • D) At every point along the orbit
3. 3.

   An asteroid orbits the Sun in an elliptical path with a perihelion distance of 3 AU and an aphelion distance of 15 AU. According to Kepler's laws of planetary motion, what is the orbital period of this asteroid in Earth years?

   • A) 3 years
   • B) 9 years
   • C) 18 years
   • D) 27 years
4. 4.

   A planet moves in an elliptical orbit around a star such that its distance from the star at aphelion is three times its distance at perihelion. What is the ratio of the planet's orbital speed at perihelion to its orbital speed at aphelion?

   • A) 1/9
   • B) 1/3
   • C) 1
   • D) 3
5. 5.

   A planet moves in an elliptical orbit around the Sun. If the line segment joining the planet to the Sun sweeps out an area $A$ over a duration of 10 days near perihelion, how much area does it sweep out over a duration of 10 days near aphelion?

   • A) An area less than $A$, because the orbital speed is smaller near aphelion
   • B) An area greater than $A$, because the distance from the Sun is larger near aphelion
   • C) An area that cannot be determined without knowing the eccentricity of the orbit
   • D) An area equal to $A$, because the rate at which area is swept out remains constant throughout the orbit
6. 6.

   According to Kepler's first law of planetary motion, which of the following statements correctly describes the shape of a planet's orbit and the position of the Sun?

   • A) The orbit is an ellipse with the Sun located at one of the two foci.
   • B) The orbit is an ellipse with the Sun located at the geometric center.
   • C) The orbit is a circle with the Sun located at an eccentric point off-center.
   • D) The orbit is a parabola with the Sun located at the focal point.
7. 7.

   For satellites in circular orbits around a celestial body, the ratio $T^2 / r^3$ is constant according to Kepler's third law. According to Newton's law of universal gravitation, on which of the following quantities does the value of this ratio depend?

   • A) The mass of the orbiting object
   • B) The radius of the orbiting object
   • C) The orbital speed of the object
   • D) The mass of the central body
8. 8.

   A planet moves in an elliptical orbit around the Sun. According to Kepler's laws of planetary motion, which quantity remains constant throughout the planet's entire orbit?

   • A) The rate at which the position vector from the Sun sweeps out area
   • B) The linear speed of the planet along its orbital trajectory
   • C) The magnitude of the gravitational force exerted on the planet by the Sun
   • D) The magnitude of the linear momentum of the planet
9. 9.

   A planet moves in an elliptical orbit around the Sun. According to Kepler's second law, which of the following statements correctly describes the planet's motion as it travels from aphelion to perihelion?

   • A) Its orbital speed increases, while the rate at which its position vector sweeps out area remains constant.
   • B) Both its orbital speed and the rate at which its position vector sweeps out area increase.
   • C) Its orbital speed remains constant, while the rate at which its position vector sweeps out area increases.
   • D) Its orbital speed decreases, while the rate at which its position vector sweeps out area remains constant.
10. 10.

    A satellite moves in a circular orbit of radius $R$ around a planet with an orbital period of $T$. What is the orbital period of a second satellite orbiting the same planet in a circular orbit of radius $4R$?

    • A) $2T$
    • B) $4T$
    • C) $16T$
    • D) $8T$

Answers and step-by-step solutions appear after you finish the test.