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

Answer 1

Answer: (A)

Condition for escaping of a particle is $ KE=PE $
or $\frac{1}{2} m v^{2}=\frac{G M_{1} m}{d / 2}+\frac{G M_{2} m}{d / 2}$
$v^{2}=\frac{4 G}{d}\left(M_{1}+M_{2}\right)$
$v=2 \sqrt{\frac{G\left(M_{1}+M_{2}\right)}{d}}$