Escape speed Transitional Reasoning Practice

Questions

1. 1.

   A satellite travels in a stable circular orbit around an airless, spherical planet of radius $R$. Telemetry confirms that the orbital speed of the satellite is exactly half of the escape speed from the planet's surface. What is the altitude of the satellite above the surface of the planet?

   • A) $R$
   • B) $2R$
   • C) $3R$
   • D) $4R$
2. 2.

   Planetary astronomers discover two spherical, airless exoplanets, Planet X and Planet Y, that possess identical surface escape speeds. Detailed observations determine that Planet Y has four times the mass of Planet X. If the acceleration due to gravity at the surface of Planet X is $g$, what is the acceleration due to gravity at the surface of Planet Y?

   • A) $4g$
   • B) $2g$
   • C) $g$
   • D) $\frac{g}{4}$
3. 3.

   Two identical spherical asteroids, each of mass $M$, are fixed in deep space with their centers separated by a distance $2d$. A small space probe of mass $m$ is situated at rest at the midpoint between the centers of the two asteroids. If the probe is launched perpendicular to the line connecting the centers of the asteroids, what is the minimum launch speed required for it to completely escape the gravitational pull of both bodies?

   • A) $2\sqrt{\frac{GM}{d}}$
   • B) $\sqrt{\frac{2GM}{d}}$
   • C) $\sqrt{\frac{GM}{d}}$
   • D) $2\sqrt{\frac{2GM}{d}}$
4. 4.

   A deep-space probe stationed on the surface of an airless, spherical planet of radius $R$ launches radially outward to rendezvous with a craft coasting at a distance of $4R$ from the planet's center. The probe coasts with its propulsion off immediately after launch and momentarily comes to rest relative to the planet just as it reaches this rendezvous distance. If the escape speed from the surface of the planet is $v_{\text{esc}}$, what initial launch speed must the probe have at the surface?

   • A) $\frac{1}{4} v_{\text{esc}}$
   • B) $\frac{1}{2} v_{\text{esc}}$
   • C) $\frac{\sqrt{3}}{4} v_{\text{esc}}$
   • D) $\frac{\sqrt{3}}{2} v_{\text{esc}}$
5. 5.

   An astrophysics team analyzes two airless, spherical planets to determine the launch parameters for departure probes. Planet 1 has a surface gravitational acceleration of $g$ and a radius of $R$, resulting in an escape speed of $v$. Planet 2 has a surface gravitational acceleration of $4g$ and a radius of $0.25R$. What is the minimum launch speed required for an object to escape the gravitational influence of Planet 2 from its surface?

   • A) $0.25v$
   • B) $0.5v$
   • C) $v$
   • D) $2v$
6. 6.

   Engineers designing a sample-return craft on the surface of an airless, spherical dwarf planet evaluate two launch profiles. Trajectory 1 launches the craft straight up (radially outward), whereas Trajectory 2 launches the craft at an angle of 45° above the local horizontal terrain. Assuming no topographical obstacles and that propulsion cuts off immediately after launch, how do the minimum initial speeds required to completely escape the dwarf planet's gravitational pull compare?

   • A) Trajectory 2 requires a higher launch speed because only the radial component of the initial velocity contributes to overcoming gravitational potential energy.
   • B) Both trajectories require the identical minimum launch speed because escape depends solely on the scalar mechanical energy being non-negative.
   • C) Trajectory 1 requires a higher launch speed because gravitational acceleration opposes the craft's motion directly along its entire flight path.
   • D) Trajectory 2 cannot achieve escape because any non-zero tangential velocity component forces the craft into a bound elliptical orbit.
7. 7.

   A space agency designs two robotic return capsules, Capsule 1 with a mass of 250 kg and Capsule 2 with a mass of 750 kg, to retrieve surface samples from an airless, spherical moon of mass $M$ and radius $R$. Each capsule must be launched radially from the surface with the minimum initial speed required to completely escape the moon's gravitational influence. If the escape speed required for Capsule 1 is $v$, which statement correctly identifies the escape speed required for Capsule 2 along with the underlying physical reasoning?

   • A) $v/3$, because Capsule 2 possesses greater inertia to overcome the moon's gravitational pull.
   • B) $3v$, because the gravitational force exerted on Capsule 2 at the surface is three times greater.
   • C) $v$, because escape speed depends solely on the moon's mass and radius, making it independent of capsule mass.
   • D) $9v$, because escaping the gravitational potential well requires an initial kinetic energy proportional to mass squared.
8. 8.

   A scientific probe is launched radially outward from the surface of an airless, spherical moon with an initial launch speed of $2v_{\text{esc}}$, where $v_{\text{esc}}$ represents the moon's escape speed. After the initial launch, the probe coasts freely with its engines turned off and experiences no external gravitational influences other than the moon. In terms of $v_{\text{esc}}$, what is the speed of the probe when it reaches deep space infinitely far from the moon?

   • A) $\sqrt{3}v_{\text{esc}}$
   • B) $v_{\text{esc}}$
   • C) $\sqrt{2}v_{\text{esc}}$
   • D) $3v_{\text{esc}}$
9. 9.

   An autonomous research probe is in a stable circular parking orbit of radius $r$ around a spherical asteroid, traveling at an orbital speed $v$. Mission control plans to fire the thrusters to send the probe into deep space, completely escaping the asteroid's gravitational influence. What is the minimum speed the probe must attain at this radial distance to achieve escape?

   • A) $\frac{v}{\sqrt{2}}$
   • B) $v$
   • C) $2v$
   • D) $\sqrt{2}v$
10. 10.

    An exploration spacecraft visits two spherical exoplanets, Planet 1 and Planet 2, that have identical uniform densities. Telescopic measurements determine that the radius of Planet 2 is twice the radius of Planet 1. If the minimum launch speed required for an object to escape the gravitational field of Planet 1 from its surface is $v$, what is the escape speed from the surface of Planet 2?

    • A) $2v$
    • B) $\sqrt{2}v$
    • C) $4v$
    • D) $2\sqrt{2}v$

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