Escape speed Transitional Reasoning Download PDF

Questions

1. 1.

   A scientific probe is positioned at the midpoint between two identical, stationary spherical asteroids, each of mass $M$, whose centers are separated by a distance $2d$. Although the net gravitational force on the probe at this point is zero, escaping the gravitational field of the two asteroids requires a sufficient initial speed. If the probe is launched along the perpendicular bisector of the line connecting the centers of the two asteroids, what is the minimum launch speed required for it to escape to an infinite distance?

   • A) 0
   • B) $\sqrt{\frac{GM}{d}}$
   • C) $\sqrt{\frac{2GM}{d}}$
   • D) $2\sqrt{\frac{GM}{d}}$
2. 2.

   Planetary scientists investigating two airless, spherical rocky exoplanets determine that Exoplanet 1 has a radius of $R_1$ and an average mass density of $\rho_1$, resulting in a surface escape speed of $v_{\text{esc}}$. Exoplanet 2 has twice the radius of Exoplanet 1 ($R_2 = 2R_1$) and four times its average mass density ($\rho_2 = 4\rho_1$). Neglecting any atmospheric or rotational effects, what is the escape speed from the surface of Exoplanet 2 in terms of $v_{\text{esc}}$?

   • A) $2v_{\text{esc}}$
   • B) $4v_{\text{esc}}$
   • C) $8v_{\text{esc}}$
   • D) $16v_{\text{esc}}$
3. 3.

   A scientific satellite orbits an airless, spherical planet of mass $M$ and radius $R$ in a stable circular orbit at an altitude equal to the planet's radius ($h = R$). The satellite travels at a constant orbital speed $v_{\text{orb}}$. In terms of $v_{\text{orb}}$, what is the escape speed from the surface of this planet?

   • A) $\sqrt{2}\,v_{\text{orb}}$
   • B) $\sqrt{3}\,v_{\text{orb}}$
   • C) $2v_{\text{orb}}$
   • D) $4v_{\text{orb}}$
4. 4.

   A mission team investigates the launch trajectories of an unpowered scientific probe departing from the surface of an airless, non-rotating spherical asteroid. When launched radially outward with the asteroid's surface escape speed $v_{\text{esc}}$, the probe completely escapes the asteroid's gravity and reaches an infinite distance with zero residual speed. If the probe is instead launched from the surface with the same speed $v_{\text{esc}}$ at an angle of 60° above the local horizontal, which of the following correctly describes its subsequent motion?

   • A) It is captured into a closed elliptical orbit around the asteroid because of its non-zero tangential velocity component.
   • B) It reaches a finite maximum altitude and falls back to the surface because its radial velocity is less than the escape speed.
   • C) It successfully escapes the asteroid's gravitational field and reaches an infinite distance with zero residual speed.
   • D) It escapes the asteroid's gravitational field but retains a constant non-zero residual speed at infinity due to angular momentum conservation.
5. 5.

   An uncrewed probe is deployed from a research station located at an altitude of $3R$ above the surface of an airless, spherical planet of radius $R$. The planet is non-rotating and has uniform mass distribution. If the escape speed from the surface of the planet is $v_{\text{esc}}$, what is the minimum launch speed required for the probe to completely escape the gravitational field of the planet from this station?

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

   A research team is designing an electromagnetic catapult to launch an unpowered scientific container of mass $m$ into deep space from the surface of an airless, spherical moon. The moon has a radius $R$ and a uniform surface gravitational acceleration of magnitude $g$. Neglecting any rotational effects of the moon, what is the minimum work that the catapult must perform on the container so that it reaches escape speed from the surface?

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

   A deep-space probe is launched radially outward from the surface of an airless, spherical planet with an initial launch speed of $v_0 = \frac{5}{4}v_{\text{esc}}$, where $v_{\text{esc}}$ is the planet's surface escape speed. Assuming that the planet's gravitational pull is the only force acting on the vehicle, what will be the probe's residual speed when it has traveled infinitely far from the planet?

   • A) $\frac{1}{4}v_{\text{esc}}$
   • B) $\frac{1}{2}v_{\text{esc}}$
   • C) $\frac{3}{4}v_{\text{esc}}$
   • D) $v_{\text{esc}}$
8. 8.

   An uncrewed surface probe on an airless, spherical dwarf planet of radius $R$ launches a scientific sensor vertically upward with an initial launch speed of $v_0 = v_{\text{esc}}/\sqrt{2}$, where $v_{\text{esc}}$ represents the dwarf planet's surface escape speed. Assuming the dwarf planet is spherically symmetric and non-rotating, what maximum altitude $h$ above the surface does the sensor reach before momentarily coming to rest?

   • A) $h = \frac{R}{2}$
   • B) $h = R$
   • C) $h = 2R$
   • D) $h = 3R$
9. 9.

   A scientific probe of mass $m$ travels in a circular parking orbit of radius $r$ around an airless moon of mass $M$ with an orbital speed $v_{\text{orb}}$. Mission control intends to execute an instantaneous thruster burn directed tangentially along the probe's velocity vector so that the probe escapes the moon's gravitational pull with zero residual speed at infinity. In terms of $v_{\text{orb}}$, what minimum additional speed ($\Delta v$) must this burn provide to place the probe on an escape trajectory?

   • A) $(\sqrt{2} - 1)v_{\text{orb}}$
   • B) $\sqrt{2}v_{\text{orb}}$
   • C) $v_{\text{orb}}$
   • D) $(2 - \sqrt{2})v_{\text{orb}}$
10. 10.

    An aerospace engineering team is evaluating the minimum launch speed required for an unpowered scientific probe of mass $m$ to completely escape the gravitational field of an airless exoplanet with mass $M$ and radius $R$, which is determined to be $v_{\text{esc}}$. For a subsequent mission, a heavier probe of mass $2m$ is to be launched from the surface of a second airless exoplanet having mass $4M$ and radius $2R$. Assuming negligible atmospheric drag and that both planets are uniform spheres, what is the required escape speed from the second exoplanet in terms of $v_{\text{esc}}$?

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