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. $2 \sqrt{\frac{2 G}{d}\left(M_{1}+M_{2}\right)}$
• C. $2 \sqrt{\frac{G m}{d}\left(M_{1}+M_{2}\right)}$
• D. $2 \sqrt{\frac{G m\left(M_{1}+M_{2}\right)}{d\left(R_{1}+R_{2}\right)}}$

Answer 1

Answer: (A)

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