   Chapter 7, Problem 59AP

Chapter
Section
Textbook Problem

In Robert Heinlein’s The Moon Is a Harsh Mistress, the colonial inhabitants of the Moon threaten to launch rocks down onto Earth if they are not given independence (or at least representation). Assuming a gun could launch a rock of mass m at twice the lunar escape speed, calculate the speed of the rock as it enters Earth’s atmosphere.

To determine
The speed of the rock as it enters the earth atmosphere.

Explanation

Given info: A gun can launch a rock of mass m onto earth with a speed twice the lunar escape speed.

Explanation:

The lunar escape speed is given by,

vescape=2GMMRM

• MM is the mass of moon
• RM is the radius of the moon
• G is the universal gravitational constant

Since the launch velocity of the rocks is twice the escape speed of moon, the launch velocity of the rocks will be,

vlaunch=22GMMRM

Apply the conservation of energy considering launch of the rocks as the initial state and the impact with the earth atmosphere as the final state,

12mvimpact2+(PE)f=12mvlaunch2+(PE)i

The impact velocity will be given by,

vimpact=vlaunch2+2m((PE)i(PE)f)

The potential energy at the time of launch will be,

(PE)i=GMMmRMGMEmr

• ME is the mass of the earth
• m is the mass of the rocks
• r is the radius of the orbit of moon around earth

The potential energy at the time of impact will be,

(PE)f=GMEmREGMMmr

Using the expression for the launch velocity and the expressions for the potential energies, the impact velocity will be,

vimpact=(8GMMRM)+2m((GMMmRMGMEmr)(GMEmREGMMmr))=2G(3MMRM+MERE(MEMM)r)

Substitute 6

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