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Chapter 7: Problem 59

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.

Short Answer

Expert verified
The final velocity of the rock as it enters Earth's atmosphere is obtained by solving the equation derived from conserving energy between the two states of the rock pointed out in step 5. The necessary parameters will need to be plugged into the equation obtained from step 5 and solved for \( v \).

Step by step solution

01

Determine the Initial Kinetic Energy

The rock is launched at twice the lunar escape velocity, hence its initial kinetic energy just after being launched would be \( KE_i = \frac{1}{2}m(v_{launch})^2 \), where \( v_{launch} \) is 2 times the lunar escape speed.
02

Calculate the Total Initial Energy

The total initial energy is the sum of kinetic and potential energy. However, because we are considering the instance when the rock has just left the Moon's surface, there is no potential energy. So, the total initial energy is equal to the kinetic energy.
03

Calculate the Final Potential Energy

The final potential energy of the rock is given by \( PE_f = -\frac{GM_Em}{R_E} \), where \( G \) is the gravitational constant, \( M_E \) is the Earth's mass, and \( R_E \) is the Earth's radius.
04

Calculate the Total Final Energy

At the point where the rock is entering Earth's atmosphere, we assume it is at the edge of Earth, so its kinetic energy would be the final total energy minus the potential energy.
05

Solve for Final Velocity

The final velocity of the rock as it enters Earth's atmosphere can be determined by setting its kinetic energy \( KE_f = \frac{1}{2}mv^2 \) equal to the final total energy calculated in step 4, and solving for \( v \).

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