Escape speed Knowledge-Based Practice

Questions

1. 1.

   A spacecraft of mass $m$ is launched from the surface of an airless, spherical planet of mass $M$ and radius $R$ at exactly the escape speed. Taking gravitational potential energy to be zero at infinite separation, what is the total mechanical energy of the spacecraft during its unpowered flight?

   • A) 0
   • B) $-\frac{GMm}{R}$
   • C) $\frac{GMm}{R}$
   • D) $-\frac{GMm}{2R}$
2. 2.

   Two space probes with masses $m$ and $4m$ are launched from the surface of an airless, spherical planet. What is the ratio of the escape speed of the probe of mass $4m$ to that of the probe of mass $m$?

   • A) 1/4
   • B) 1/2
   • C) 1
   • D) 2
3. 3.

   A projectile is launched from the surface of an airless, non-rotating spherical planet at an angle of $60^\circ$ above the horizontal. If the planet has a surface escape speed of $v_{\text{esc}}$, what is the minimum launch speed required for the projectile to escape the planet's gravitational field?

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

   A spacecraft is launched radially outward from the surface of an airless, spherical planet with an initial speed $v_0 = 2v_{\text{esc}}$, where $v_{\text{esc}}$ is the escape speed from the planet's surface. Neglecting all other celestial bodies, what will be the speed of the spacecraft when it reaches a distance infinitely far from the planet?

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

   A projectile is launched from the surface of an airless, spherical planet with an initial speed equal to the planet's escape speed, $v_{\text{esc}}$. In terms of $v_{\text{esc}}$, what is the speed of the projectile when it reaches an altitude equal to the radius of the planet?

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

   A spherical asteroid has a uniform mass density $\rho$ and a radius $R$. In terms of the universal gravitational constant $G$, $\rho$, and $R$, which expression represents the escape speed from the asteroid's surface?

   • A) $R\sqrt{\frac{4\pi G \rho}{3}}$
   • B) $R\sqrt{\frac{8\pi G \rho}{3}}$
   • C) $R\sqrt{\frac{2\pi G \rho}{3}}$
   • D) $R^2\sqrt{\frac{8\pi G \rho}{3}}$
7. 7.

   An object is launched from the surface of a spherical planet of radius $R$ where the acceleration due to gravity at the surface has magnitude $g$. Neglecting atmospheric resistance and planetary rotation, which expression represents the escape speed from the planet's surface?

   • A) $\sqrt{gR}$
   • B) $\sqrt{\frac{gR}{2}}$
   • C) $\sqrt{2gR}$
   • D) $2\sqrt{gR}$
8. 8.

   A projectile of mass $m$ is launched from the surface of a uniform spherical planet of mass $M$ and radius $R$. In the absence of atmospheric resistance, what is the minimum kinetic energy the projectile must have at launch to completely escape the planet's gravitational field?

   • A) $\frac{GMm}{4R}$
   • B) $\frac{GMm}{2R}$
   • C) $\frac{2GMm}{R}$
   • D) $\frac{GMm}{R}$
9. 9.

   A space probe moves in a stable circular orbit of radius $r$ around a planet with orbital speed $v_{\text{orb}}$. What is the escape speed of the probe from this orbital distance?

   • A) $\frac{1}{2} v_{\text{orb}}$
   • B) $\sqrt{2} v_{\text{orb}}$
   • C) $2 v_{\text{orb}}$
   • D) $4 v_{\text{orb}}$
10. 10.

    A hypothetical spherical planet has twice the mass and half the radius of Earth. If the escape speed from the surface of Earth is $v$, what is the escape speed from the surface of this planet?

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

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