Earth has mass ' $M_1$ ' radius ' $R_1$ ' and for moon mass ' $\mathrm{M}_2$ ' and radius ' $\mathrm{R}_2$ '…
Earth has mass ' $M_1$ ' radius ' $R_1$ ' and for moon mass ' $\mathrm{M}_2$ ' and radius ' $\mathrm{R}_2$ '. Distance between their centres is ' $r$ '. A body of mass ' $M$ ' is placed on the line joining them at a distance $\frac{r}{3}$ from the centre of the earth. To project a mass ' M ' to escape to infinity, the minimum speed required is
The binding energy of the body is given by
$\begin{aligned}
\text { B.E. } & =\frac{\mathrm{GM}_1 \mathrm{M}}{\frac{\mathrm{r}}{3}}+\frac{\mathrm{GM}_2 \mathrm{M}}{\frac{2 \mathrm{r}}{3}}=\frac{3 \mathrm{GM}_1 \mathrm{M}}{\mathrm{r}}+\frac{3 \mathrm{GM}_2 \mathrm{M}}{2 \mathrm{r}} \\
& =\frac{3 \mathrm{GM}}{\mathrm{r}}\left[\mathrm{M}_1+\frac{\mathrm{M}_2}{2}\right]
\end{aligned}$ If $v$ is the velocity given to the body, then
$\begin{aligned}
& \frac{1}{2} M v^2=\frac{3 G M}{r}\left[M_1+\frac{M_2}{2}\right] \\
\therefore \quad & v=\left[\frac{6 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{\frac{1}{2}}
\end{aligned}$
^