- Home
- Standard 11
- Physics
Masses and radii of earth and moon are $M_1,\, M_2$ and $R_1,\, R_2$ respectively. The distance between their centre is $'d'$ . The minimum velocity given to mass $'M'$ from the mid point of line joining their centre so that it will escape
$\sqrt {\frac{{4G\left( {{M_1} + {M_2}} \right)}}{d}} $
$\sqrt {\frac{{4G}}{d}\frac{{{M_1}{M_2}}}{{({M_1} + {M_2})}}} $
$\sqrt {\frac{{2G}}{d}\left( {\frac{{{M_1} + {M_2}}}{{{M_1}{M_2}}}} \right)} $
$\sqrt {\frac{{2G}}{d}\left( {{M_1} + {M_2}} \right)} $
Solution
Potential energy of mass $m$ when it is midway between masses $M_{1}$ and $M_{2}$ is
$U=-\frac{G M_{1} m}{d / 2}-\frac{G M_{2} m}{d / 2}=-\frac{2 G m}{d}\left(M_{1}+M_{2}\right)$
According to law of conservation of energy $\frac{1}{2} m v_{e}^{2}=\frac{2 G m}{d}\left(M_{1}+M_{2}\right)$
Therefore, escape velocity $v_{e}=\sqrt{\frac{4 G\left(M_{1}+M_{2}\right)}{d}}$
