# The masses and radii of the earth and moon are

Question:

The masses and radii of the earth and moon are $\left(\mathrm{M}_{1}, \mathrm{R}_{1}\right)$ and $\left(\mathrm{M}_{2}, \mathrm{R}_{2}\right)$ respectively. Their centres are at a distance ' $r$ ' apart. Find the minimum escape velocity for a particle of mass ' $m$ ' to be projected from the middle of these two masses:

1. $V=\frac{1}{2} \sqrt{\frac{4 G\left(M_{1}+M_{2}\right)}{r}}$

2. $\mathrm{V}=\sqrt{\frac{4 \mathrm{G}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)}{\mathrm{r}}}$

3. $\mathrm{V}=\frac{1}{2} \sqrt{\frac{2 \mathrm{G}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)}{\mathrm{r}}}$

4. $\mathrm{V}=\frac{\sqrt{2 \mathrm{G}}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)}{\mathrm{r}}$

Correct Option: , 2

Solution:

$\frac{1}{2} \mathrm{mV}^{2}-\frac{\mathrm{GM}_{1} \mathrm{~m}}{\mathrm{r} / 2}-\frac{\mathrm{GM}_{2} \mathrm{~m}}{\mathrm{r} / 2}=0$

$\frac{1}{2} \mathrm{mV}^{2}=\frac{2 \mathrm{Gm}}{\mathrm{r}}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)$

$V=\sqrt{\frac{4 G\left(M_{1}+M_{2}\right)}{r}}$

Option (2)