Escape speed Knowledge-Based Download PDF

Questions

1. 1.

   An object is launched from the surface of an airless, non-rotating spherical planet at an angle of $45^\circ$ above the horizontal. If the escape speed for a vertical launch from this surface is $v_e$, what is the minimum launch speed required for the object to escape the planet's gravitational field?

   • A) $v_e$
   • B) $\sqrt{2}\,v_e$
   • C) $\frac{\sqrt{2}}{2}\,v_e$
   • D) $2v_e$
2. 2.

   A spacecraft is positioned at a distance of $4R$ from the center of an isolated, spherical planet of radius $R$. If the escape speed from the planet's surface is $v_e$, what is the escape speed from this spacecraft's position?

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

   Two spherical, airless planets, X and Y, have the same uniform mass density, but the radius of planet X is three times the radius of planet Y. If the escape speed from the surface of planet Y is $v$, what is the escape speed from the surface of planet X?

   • A) $3v$
   • B) $\sqrt{3}v$
   • C) $9v$
   • D) $3\sqrt{3}v$
4. 4.

   A projectile is launched from the surface of an airless, spherical planet with an initial speed of $\sqrt{2}\,v_e$, where $v_e$ is the escape speed from the planet's surface. What is the speed of the projectile when it reaches a distance infinitely far from the planet?

   • A) 0
   • B) $(\sqrt{2} - 1)\,v_e$
   • C) $v_e$
   • D) $\sqrt{2}\,v_e$
5. 5.

   A projectile of mass $m$ is launched from the surface of a stationary, airless spherical planet with an initial speed equal to the planet's escape speed. If the gravitational potential energy is taken to be zero at an infinite separation, what is the total mechanical energy of the projectile–planet system throughout the trajectory?

   • A) Exactly zero
   • B) Strictly positive and constant
   • C) Strictly negative and constant
   • D) Increasing from negative toward zero
6. 6.

   A spherical planet of radius $R$ has an acceleration due to gravity of magnitude $g$ at its surface. Neglecting atmospheric drag and planetary rotation, what is the escape speed from the surface of this planet?

   • A) $\frac{1}{2}gR$
   • B) $\sqrt{gR}$
   • C) $2\sqrt{gR}$
   • D) $\sqrt{2gR}$
7. 7.

   An object of mass $m$ launched from the surface of an airless spherical planet requires a minimum speed $v$ to escape the planet's gravitational field. What is the escape speed from the same surface for an object of mass $4m$?

   • A) $\frac{v}{2}$
   • B) $2v$
   • C) $v$
   • D) $4v$
8. 8.

   A satellite of mass $m$ is launched from the surface of a stationary, airless spherical planet of mass $M$ and radius $R$. What is the minimum kinetic energy required at launch for the satellite to escape the planet's gravitational influence completely?

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

   What is the relationship between the escape speed $v_e$ from the surface of a spherical planet and the speed $v_o$ of an object in a circular orbit immediately above its surface?

   • A) $v_e = \sqrt{2}\,v_o$
   • B) $v_e = 2\,v_o$
   • C) $v_e = \frac{v_o}{\sqrt{2}}$
   • D) $v_e = 4\,v_o$
10. 10.

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

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