Question

The masses of the earth and moon are $\mathrm{M}$ and ${\displaystyle \frac{M}{n}}$ respectively. Their centers are separated by a distance 2 r. The minimum speed with which a particle of mass $m$ should be projected from a point midway the two centers so as to escape to infinity is
a) $\sqrt{{\displaystyle \frac{GM(n+1)}{nr}}}$
b)$\sqrt{{\displaystyle \frac{2GM(n+1)}{nr}}}$
c) $\sqrt{{\displaystyle \frac{GM(n+1)}{2nr}}}$
d) $\sqrt{{\displaystyle \frac{2(\mathrm{GMn})}{(\mathrm{n}+1)\mathrm{r}}}}$

Answer 1

Hint: The energy conservation law states that an isolated system's total energy remains constant; it is said to be conserved over time. This means that, unless it's added from the outside, a system always has the same amount of energy. Apply law of conservation of energy and Equate Initial total energy to the final total energy .

Formula used:
$$\mathrm{\Delta }K.E=-\mathrm{\Delta }P.E$$

Complete answer:
The principles of conservation tell us that, through change, some quantity, quality, or aspect remains constant. In ancient and mediaeval natural philosophy, such principles are already emerging.
from the law of conservation of energy,
$$\mathrm{\Delta }K.E=-\mathrm{\Delta }P.E$$
here, change in kinetic energy, $$\mathrm{\Delta }K.E={\displaystyle \frac{1}{2}}m{v}^2$$, where v is the minimum speed of particle of mass m should be projected from a point midway the two centres so as to escape to infinity.
On conserving mechanical energy
 Initial total energy = final total energy
${\displaystyle \frac{-\text{G}{\text{M}}_2\text{m}}{{\displaystyle \frac{d}{2}}}}-{\displaystyle \frac{\text{G}{\text{M}}_1\text{m}}{{\displaystyle \frac{d}{2}}}}={\displaystyle \frac{1}{2}}\text{m}{\text{V}}^2$
$\Rightarrow {\mathrm{V}}^2={\displaystyle \frac{4\mathrm{G}}{\mathrm{d}}}({\mathrm{M}}_1+{\mathrm{M}}_2)$
$\Rightarrow \mathrm{V}=\sqrt{{\displaystyle \frac{4\mathrm{G}({\mathrm{M}}_1+{\mathrm{M}}_2)}{\mathrm{d}}}}$

Note:
The rotation of the celestial orbs is eternal and immutable within one significant strand of Greek cosmology. The conservation of charge means that equal numbers of positive and negative particles are always created in reactions that generate charged particles, keeping the net amount of charge unchanged. Conservation is the understanding that something, although its appearance changes, remains the same in quantity. The ability to understand that redistributing material does not affect its mass, number, volume or length is more technical conservation.