Gravitational potential energy (general form) Transitional Reasoning Download PDF

Questions

1. 1.

   A telemetry satellite of mass $m$ travels in a stable circular orbit around a planet of mass $M$ with an orbital speed $v$. In this initial orbit, the gravitational potential energy of the planet-satellite system is $U$, defined to be zero at infinite separation. Mission controllers perform maneuvers to transfer the satellite into a new, higher circular orbit where its orbital speed is reduced to $\frac{v}{2}$. In terms of the initial gravitational potential energy $U$, what is the gravitational potential energy of the system in the new orbit?

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

   A deep-space scientific probe monitors a meteoroid of mass $m$ that is momentarily at rest at a radial distance of $4R$ from the center of an isolated, spherical planet of mass $M$ and radius $R$. Under the influence of the planet's gravitational pull alone, the meteoroid accelerates along a straight radial path toward the planet. Assuming no atmospheric drag and taking the gravitational potential energy to be zero at infinity, what is the impact speed of the meteoroid when it strikes the planet's surface?

   • A) $\sqrt{\frac{3GM}{4R}}$
   • B) $\sqrt{\frac{3GM}{2R}}$
   • C) $\sqrt{\frac{2GM}{R}}$
   • D) $\sqrt{\frac{GM}{2R}}$
3. 3.

   An orbital service spacecraft of mass $m$ maintains a circular orbit of radius $r$ around a spherical planet of mass $M$. Taking the gravitational potential energy of the planet-spacecraft system to be zero at infinite separation, the potential energy of the system in this orbit is $U$. What is the minimum additional work that thrusters must perform on the spacecraft to allow it to completely escape the planet's gravitational field?

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

   A research team on an airless spherical dwarf planet of mass $M$ and radius $R$ launches a vertical probe of mass $m$ directly upward from the surface. The probe is launched with an initial kinetic energy equal to half the magnitude of the dwarf planet-probe gravitational potential energy at the surface. Assuming the gravitational potential energy is defined to be zero at infinity, what is the maximum radial distance from the center of the dwarf planet reached by the probe before it momentarily comes to rest?

   • A) $1.5R$
   • B) $\sqrt{2}R$
   • C) $2R$
   • D) $4R$
5. 5.

   An uncrewed space probe of mass $m$ is transferred from an initial circular parking orbit of radius $2R$ to a higher monitoring orbit of radius $6R$ around a spherical planet of mass $M$. The mission controllers need to evaluate the change in gravitational potential energy of the planet-probe system to determine the gravitational work involved. Assuming gravitational potential energy is defined to be zero at an infinite distance, what is the change in gravitational potential energy, $\Delta U = U_f - U_i$, of the system?

   • A) $+\frac{GMm}{6R}$
   • B) $+\frac{GMm}{3R}$
   • C) $-\frac{GMm}{3R}$
   • D) $+\frac{2GMm}{3R}$