1. Minimum energy needed to never return

For a rocket launched from the surface of a spherical body, the minimum energy required to just escape is the gravitational binding energy per unit mass times the rocket mass:

$E_{\text{esc}} = \frac{GMm}{R}$

So, for the same rocket (same mass $m$), the required energy is proportional to

$E \propto \frac{GM}{R}$

Since

$M = \frac{4}{3}\pi R^3 \rho,$

we get

$\frac{GM}{R} = G\cdot \frac{\frac{4}{3}\pi R^3\rho}{R} = \frac{4}{3}\pi G\rho R^2$

Thus, for bodies of the same density,

$E \propto R^2$

2. Use the volume ratio

Given:

$V_E = 64 V_M$

But volume of a sphere is proportional to $R^3$, so

$\frac{V_E}{V_M} = \left(\frac{R_E}{R_M}\right)^3 = 64$

Hence,

$\frac{R_E}{R_M} = 4$

So,

$R_M = \frac{R_E}{4}$

3. Compare escape energies

Since $E \propto R^2$,

$\frac{E_M}{E_E} = \left(\frac{R_M}{R_E}\right)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16}$

Given that the minimum energy from Earth is $E$,

$E_M = \frac{E}{16}$

4. Check options

• A: $E/32$ ❌
• B: $E/16$ ✅
• C: $E/4$ ❌
• D: $E/64$ ❌

Therefore, the correct option is B.