Question

$E$ is the minimum energy that is required by a rocket that is launched from Earth's surface to never return back. Find the minimum energy that the rocket should have if it is to be launched from the moon's surface instead. If this energy can be expressed as $\frac{E}{N}$ , then find the value of $N$ .
[Assume $\rho _{m}=\rho _{e}$ , Volume of Earth $=64\times $ Volume of moon]

Answer 1

The energy corresponding to escape speed is
$E=\frac{1}{2}m\left(\right.2gR\left.\right)=\frac{1}{2}m\left(2 \frac{GM}{R^{2}} R\right)$
$=\frac{1}{2}m\left(2 \frac{G M}{R}\right)=\frac{G M m}{R}$
$V_{e}=64V_{m}$
$\Rightarrow \frac{4}{3}\pi R_{e}^{3}=\frac{4}{3}\times \pi \times R_{m}^{3}\times 64$ ' $\Rightarrow R_{c}=4R_{m}$
$\because \rho _{c}=\rho _{m}$
$\frac{M_{e}}{V_{e}}=\frac{M_{m}}{V_{m}}$
$\Rightarrow \frac{M_{e}}{64 V_{m}}=\frac{M_{m}}{V_{m}}$
$\Rightarrow M_{e}=64M_{m}$
So, $\frac{E_{m}}{E_{c}}=\frac{M_{m}}{R_{m}}\times \frac{R_{r}}{M_{c}}$
$=\frac{1}{64}\times 4=\frac{1}{16}$ $\ldots .\left[\because R_{e} = 4 R_{m}\right]$
$E_{m}=\frac{E}{16}$