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27 Nov 2019

A spacecraft is on a journey to the moon. At what point, as measuredfrom the center of the earth, does the gravitational force exerted onthe spacecraft by the earth balance that exerted by the moon? Thispoint lies on a line between the centers of the earth and the moon. Thedistance between the earth and the moon is 3.85 × 108 m, and the massof the earth is 81.4 times as great as that of the moon.

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

At a certain instant, the earth, the moon, and a stationary 1220 kgspacecraft lie at the vertices of an equilateral triangle whosesides are 3.84 \times 10^{5}\:{\rm km} in length.

Find the magnitude of the net gravitational force exerted on thespacecraft by the earth and moon.
Take the gravitational constant to be 6.673 \times 10^{-11}\:{\rmN} \cdot {\rm m}^{2}/{\rm kg}^{2}, the mass of the Earth to be 5.97\times 10^{24}\:{\rm kg}, the mass of the Moon to be 7.35 \times10^{22}\:{\rm kg}.

Find the direction of the net gravitational force exerted on thespacecraft by the earth and moon. State the direction as an anglemeasured from a line connecting the earth and the spacecraft.

What is the minimum amount of work that you would have to do tomove the spacecraft to a point far from the earth and moon? You canignore any gravitational effects due to the other planets or thesun.

In the example 12.5 (Section 12.3) in the textbook we ignored thegravitational effects of the moon on a spacecraft en route from theearth to the moon. In fact, we must include the gravitationalpotential energy due to the moon as well. For this problem, you canignore the motion of the earth and moon.

Part A
If the moon of mass m_{\rm M} has radius R_{\rm M} and the distancebetween the centers of the earth and the moon is R_{\rm EM}, findthe total gravitational potential energy of the particle-earth andparticle-moon systems when a particle with mass m is between theearth and the moon, and a distance r from the center of the earth.Take the gravitational potential energy to be zero when the objectsare far from each other. Take the mass of earth as m_{\rm E}.

Part B
There is a point along a line between the earth and the moon wherethe net gravitational force is zero. Use the expression derived inpart (a) to find the distance of this point from the center of theearth.

Part C
With what speed must a spacecraft be launched from the surface ofthe earth just barely to reach this point?

Part D
If a spacecraft were launched from the earth's surface toward themoon with an initial speed of 11.2 {\rm km/s}, with what speedwould it impact the moon?

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