Orbits and satellites Transitional Reasoning Practice

Questions

1. 1.

   An astronomical survey team analyzes two space probes, Probe 1 and Probe 2, that travel in circular orbits around two different spherical exoplanets, Planet A and Planet B, respectively. Planet A has twice the mass of Planet B ($M_A = 2M_B$), and Probe 1 orbits at twice the orbital radius of Probe 2 ($r_1 = 2r_2$). Assuming each probe is influenced only by the gravitational force of its respective planet, how do the orbital speed $v_1$ and the centripetal acceleration $a_1$ of Probe 1 compare to the orbital speed $v_2$ and the centripetal acceleration $a_2$ of Probe 2?

   • A) $v_1 = 2v_2$ and $a_1 = a_2$
   • B) $v_1 = v_2$ and $a_1 = \frac{1}{2}a_2$
   • C) $v_1 = v_2$ and $a_1 = a_2$
   • D) $v_1 = \frac{1}{2}v_2$ and $a_1 = \frac{1}{4}a_2$
2. 2.

   Ground tracking stations monitor two scientific satellites, Satellite 1 and Satellite 2, each traveling in a stable circular orbit around Earth. Telemetry analysis indicates that the centripetal acceleration of Satellite 1 has a magnitude four times that of Satellite 2, such that $a_1 = 4a_2$. Considering only Earth's gravitational force, which statement correctly compares the orbital radius and orbital speed of Satellite 1 with those of Satellite 2?

   • A) The orbital radius of Satellite 1 is $\frac{1}{4} r_2$, and its orbital speed is $2v_2$.
   • B) The orbital radius of Satellite 1 is $\frac{1}{2} r_2$, and its orbital speed is $\sqrt{2}\, v_2$.
   • C) The orbital radius of Satellite 1 is $\frac{1}{2} r_2$, and its orbital speed is $2v_2$.
   • D) The orbital radius of Satellite 1 is $2r_2$, and its orbital speed is $\frac{1}{\sqrt{2}}\, v_2$.
3. 3.

   A scientific spacecraft of mass m is in a stable circular orbit around an airless planet, where its orbital motion gives it a kinetic energy of K. Mission controllers plan a brief thruster burn to send the spacecraft on an escape trajectory so that it can permanently leave the planet's gravitational pull with zero speed at an infinite distance. Assuming the burn occurs over a negligible distance and external forces other than gravity are absent, what is the minimum work that the spacecraft's propulsion system must perform to achieve this escape?

   • A) $\frac{1}{2}K$
   • B) $K$
   • C) $2K$
   • D) $4K$
4. 4.

   An exploration mission surveys two newly discovered spherical exoplanets, Planet X and Planet Y, which have the same uniform mass density, though the radius of Planet X is twice that of Planet Y. Spacecraft are placed into circular orbits just grazing the surface of each planet, so that each orbital radius is approximately equal to the respective planet's radius. Considering only the gravitational force exerted by each planet, how does the orbital period $T_X$ of the spacecraft orbiting Planet X compare to the orbital period $T_Y$ of the spacecraft orbiting Planet Y?

   • A) $T_X = T_Y$
   • B) $T_X = 2 T_Y$
   • C) $T_X = \sqrt{2}\,T_Y$
   • D) $T_X = 2\sqrt{2}\,T_Y$
5. 5.

   A scientific satellite operates in a stable circular orbit at an altitude above Earth's surface equal to Earth's radius, $R_E$. The magnitude of the gravitational acceleration at Earth's surface is $g$. In terms of $g$ and $R_E$, which expression represents the orbital speed of the satellite?

   • A) $v = \sqrt{2gR_E}$
   • B) $v = \sqrt{gR_E}$
   • C) $v = \sqrt{\frac{gR_E}{2}}$
   • D) $v = \frac{1}{2}\sqrt{gR_E}$
6. 6.

   A scientific satellite moves in an eccentric elliptical orbit around Earth, subject only to Earth's gravitational pull. As the satellite travels along its orbital path from apoapsis (the point farthest from Earth) to periapsis (the point closest to Earth), flight controllers analyze its mechanical properties. Which statement correctly evaluates the net gravitational torque acting on the satellite about Earth's center and the work done by the gravitational force during this segment?

   • A) The net gravitational torque about Earth's center is non-zero, and the gravitational force does positive work on the satellite.
   • B) The net gravitational torque about Earth's center is zero, and the gravitational force does positive work on the satellite.
   • C) The net gravitational torque about Earth's center is zero, and the gravitational force does zero work on the satellite.
   • D) The net gravitational torque about Earth's center is non-zero, and the gravitational force does negative work on the satellite.
7. 7.

   A communications team evaluates a proposal to place an unpowered satellite into a circular orbit designed to remain permanently stationary directly above a ground station located at latitude 45° N. Which statement correctly evaluates whether this proposed orbit is physically possible using only Earth's gravitational pull?

   • A) It is possible, provided the orbital period is exactly 24 hours and the orbital speed matches the ground station's tangential speed.
   • B) It is possible, because any circular path centered on Earth's rotational axis is dynamically stable under gravity alone.
   • C) It is impossible, because the gravitational force points toward Earth's center of mass, requiring the center of any circular orbit to coincide with Earth's center.
   • D) It is impossible, because Earth's gravitational attraction acts exclusively in the equatorial plane and vanishes at mid-latitudes.
8. 8.

   During an extravehicular activity outside a space station in a stable circular orbit around Earth, an astronaut suspends a calibration mass from a sensitive spring scale. Sensors record that the local acceleration due to gravity at this altitude is approximately $8.9\text{ m/s}^2$, yet the scale registers a reading of zero. Which statement correctly accounts for this scale reading and identifies the net force acting on the calibration mass?

   • A) The scale reads zero because Earth's gravitational field does not extend into the vacuum of space, meaning the net force on the mass is zero.
   • B) The scale reads zero because an outward centrifugal force balances gravity in an inertial frame of reference, meaning the net force on the mass is zero.
   • C) The scale reads zero because the scale and mass accelerate together in free fall toward Earth, meaning the net force on the mass is the gravitational force providing its centripetal acceleration.
   • D) The scale reads zero because the high orbital velocity continuously diverts the gravitational field, meaning the net force on the mass acts tangentially to its circular path.
9. 9.

   A flight operations team fires the thrusters of a satellite to transfer it from an initial stable circular orbit of radius $r$ to a higher stable circular orbit of radius $2r$. During this maneuver, the propulsion system performs positive work on the satellite. Compared to its initial circular orbit, how do the satellite's orbital speed and total mechanical energy change once it is established in the new orbit?

   • A) The orbital speed increases, and the total mechanical energy decreases.
   • B) The orbital speed increases, and the total mechanical energy increases.
   • C) The orbital speed decreases, and the total mechanical energy decreases.
   • D) The orbital speed decreases, and the total mechanical energy increases.
10. 10.

    A flight dynamics team monitors two satellites, Alpha and Beta, that travel in stable circular orbits around Earth at orbital radii $r_A = R$ and $r_B = 4R$, respectively. Satellite Alpha has three times the mass of Satellite Beta ($m_A = 3m_B$). Considering only Earth's gravitational pull, how do the orbital speed $v_A$ and orbital period $T_A$ of Satellite Alpha compare to the orbital speed $v_B$ and orbital period $T_B$ of Satellite Beta?

    • A) $v_A = 2v_B$ and $T_A = \frac{1}{8}T_B$
    • B) $v_A = 4v_B$ and $T_A = \frac{1}{4}T_B$
    • C) $v_A = 2v_B$ and $T_A = \frac{1}{4}T_B$
    • D) $v_A = \frac{1}{2}v_B$ and $T_A = 8T_B$

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