A rocket has to be launched from earth in such a way that it never returns. If $$E$$ is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have, if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that the earth's volume is 64 times the volume of the moon.

 Step 1: Escape energy formula

Minimum energy needed to escape a planet is:

$$E=\ \frac{\ GMm}{R}$$

 So, energy depends directly on mass (M) and inversely on radius (R) of the planet.

 Step 2: Use volume relation

Given:

$$Ve​=64Vm$$

$$Since\ V∝R^3:$$

$$Re^3=64Rm^3​⇒Re​=4Rm$$

 Step 3: Use density condition

$$Same\ density\ ⇒\ \ \frac{\ M}{V}​=cons\tan t$$

$$M∝V⇒Me​=64Mm​$$

 Step 4: Compare escape energies

$$\ \frac{\ E_e}{E_m}=\ \frac{\ M_e}{M_m}\cdot\ \frac{\ R_m}{R_e}$$

$$=64\times\ \frac{\ 1}{4}=16$$

 Final Result

$$E_{m\ =\ \frac{\ E}{16}}$$