Question

The masses and radii of the earth and moon are $M_1,R_1$ and $M_2,R_2$, respectively. Their centres are distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centres so that it escapes to infinity is

• A. $2\sqrt{\frac{G}{d}\left(M_1+M_2\right)}$
• B. $2\sqrt{\frac{2G}{d}\left(M_1+M_2\right)}$
• C. $2\sqrt{\frac{Gm}{d}\left(M_1+M_2\right)}$
• D. $2\sqrt{\frac{Gm\left(M_1+M_2\right)}{d\left(R_1+R_2\right)}}$

Answer 1

Answer: (A)

Gravitational potential at mid point
$V=\frac{-G{M}_1}{d/2}+\frac{-G{M}_2}{d/2}$
Now, $PE=m\times V=\frac{-2Gm}{d}\left(M_1+M_2\right)$
$[m=$ mass of particle $]$
So, for projecting particle from mid point to infinity
$KE=|PE|$
$\Rightarrow \frac{1}{2}m{v}^2=\frac{2Gm}{d}\left(M_1+M_2\right)$
$\Rightarrow v=2\sqrt{\frac{G\left(M_1+M_2\right)}{d}}$