Gravitational potential energy (general form) Transitional Reasoning Practice

Questions

1. 1.

   A scientific spacecraft of mass $m$ travels in a stable circular orbit of radius $r$ around a spherical planet of mass $M$. A propulsion maneuver shifts the spacecraft into a higher stable circular orbit where its orbital speed is exactly half of its initial speed. Taking the gravitational potential energy to be zero at infinite separation ($U(\infty) = 0$), what is the resulting change in the gravitational potential energy of the planet–spacecraft system, $\Delta U = U_{\text{final}} - U_{\text{initial}}$?

   • A) $-\frac{3GMm}{4r}$
   • B) $+\frac{3GMm}{4r}$
   • C) $+\frac{GMm}{2r}$
   • D) $-\frac{GMm}{2r}$
2. 2.

   A deep-space transport vessel of mass $m$ is initially at rest at the midpoint between two identical, stationary spherical planets, each of mass $M$, whose centers are separated by a distance $2D$. The vessel's thrusters maneuver it along the line connecting the planetary centers to a new stationary position located at a distance of $\frac{1}{2}D$ from the center of one of the planets. Taking the gravitational potential energy to be zero at infinite separation, what is the net change in the gravitational potential energy of the system, $\Delta U = U_{\text{final}} - U_{\text{initial}}$?

   • A) $+\frac{2GMm}{3D}$
   • B) $-\frac{2GMm}{3D}$
   • C) $-\frac{4GMm}{3D}$
   • D) $+\frac{4GMm}{3D}$
3. 3.

   A flight-dynamics team models the ascent of a scientific payload of mass $m$ lifted from the surface of a spherical, airless planet of mass $M$ and radius $R$ to an altitude of $h = \frac{1}{3}R$. An engineer initially approximates the required energy using the uniform gravitational field formula $\Delta U_{\text{approx}} = mgh$, where $g = \frac{GM}{R^2}$ is the surface gravitational acceleration. Using the general form of gravitational potential energy, what is the ratio of the actual potential energy change, $\Delta U_{\text{exact}}$, to the engineer's approximated value, $\frac{\Delta U_{\text{exact}}}{\Delta U_{\text{approx}}}$?

   • A) $\frac{4}{3}$
   • B) $\frac{2}{3}$
   • C) $\frac{1}{4}$
   • D) $\frac{3}{4}$
4. 4.

   A deep-space observatory consists of three identical scientific probes, each of mass $m$, positioned in deep space at the vertices of an equilateral triangle of side length $L$. Thrusters are used to slowly and quasi-statically expand the formation until the probes form an equilateral triangle with side length $2L$, starting and ending at rest. Taking gravitational potential energy to be zero at infinite separation, what is the minimum work that the external thruster forces must do on the three-probe system to achieve this reconfiguration?

   • A) $-\frac{3 G m^2}{2L}$
   • B) $\frac{G m^2}{2L}$
   • C) $\frac{3 G m^2}{4L}$
   • D) $\frac{3 G m^2}{2L}$
5. 5.

   A scientific probe of mass $m$ is released from rest at an altitude of $3R$ above the surface of a spherical, airless planet of mass $M$ and radius $R$. The probe falls radially toward the planet under the influence of the planet's gravity alone. Taking the standard convention where gravitational potential energy is defined as zero at infinite separation, what is the ratio of the probe's kinetic energy when it reaches an altitude of $R$ above the surface to its kinetic energy just before impacting the surface?

   • A) $\frac{2}{3}$
   • B) $\frac{1}{2}$
   • C) $\frac{1}{4}$
   • D) $\frac{1}{3}$
6. 6.

   A scientific satellite of mass $m$ maintains a stable circular orbit of radius $r$ around a uniform spherical planet of mass $M$. With the standard reference configuration where gravitational potential energy is defined as zero at infinite separation ($U(\infty) = 0$), the gravitational potential energy of the planet–satellite system is $U$. What is the minimum additional energy that must be supplied to the satellite by its propulsion system so that it can completely escape the gravitational field of the planet?

   • A) $-U$
   • B) $-\frac{1}{4}U$
   • C) $-2U$
   • D) $-\frac{1}{2}U$
7. 7.

   An automated research projectile of mass $m$ is launched radially outward from the surface of a spherical, airless asteroid of mass $M$ and radius $R$. The launch mechanism imparts an initial speed of $v_0 = \sqrt{\frac{GM}{R}}$ to the projectile. Assuming gravity is the only force acting on the projectile and taking gravitational potential energy to be zero at infinite separation, what is the maximum distance from the center of the asteroid reached by the projectile before it momentarily stops?

   • A) $\frac{3}{2}R$
   • B) $\sqrt{2}R$
   • C) $4R$
   • D) $2R$
8. 8.

   A robotic scientific explorer of mass $m$ is initially held at rest at a radial distance $r$ from the center of an isolated, spherical dwarf planet of mass $M$. Thrusters on the craft are fired to slowly transport it outward to a new stationary position at distance $2r$ at a negligible speed. Taking the gravitational potential energy to be zero at infinite separation, what is the work done by the dwarf planet's gravitational force on the explorer during this displacement, and how does the gravitational potential energy of the system change?

   • A) The work done by gravity is $+\frac{GMm}{2r}$, and the gravitational potential energy increases.
   • B) The work done by gravity is $-\frac{GMm}{2r}$, and the gravitational potential energy decreases.
   • C) The work done by gravity is $-\frac{GMm}{2r}$, and the gravitational potential energy increases.
   • D) The work done by gravity is $+\frac{GMm}{2r}$, and the gravitational potential energy decreases.
9. 9.

   A space agency operates two research orbiters around a uniform spherical planet of mass $M$. Orbiter 1 has mass $m$ and orbits at a radius of $2R$ from the planet's center, while Orbiter 2 has mass $3m$ and orbits at a radius of $4R$. Taking the standard convention where gravitational potential energy is zero at infinite separation ($U(\infty) = 0$), what is the ratio of the gravitational potential energy of the planet–Orbiter 1 system to that of the planet–Orbiter 2 system, $\frac{U_1}{U_2}$?

   • A) $\frac{3}{2}$
   • B) $-\frac{2}{3}$
   • C) $\frac{2}{3}$
   • D) $\frac{3}{8}$
10. 10.

    An uncrewed scientific probe of mass $m$ is launched from the surface of a spherical, airless moon of mass $M$ and radius $R$. The probe is maneuvered to a circular orbit at an altitude of $2R$ above the surface. What is the change in the gravitational potential energy, $\Delta U = U_{\text{final}} - U_{\text{initial}}$, of the moon–probe system as a result of this transfer?

    • A) It decreases by $\frac{2GMm}{3R}$.
    • B) It increases by $\frac{GMm}{3R}$.
    • C) It increases by $\frac{GMm}{2R}$.
    • D) It increases by $\frac{2GMm}{3R}$.

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