91Ó°ÊÓ

Mass of the astronauts = $65kg$, distance between them = $1m$.

Here,

Mass of the astronauts $\mathrm{m}1=\mathrm{m}2=65\mathrm{kg}.$

Separation between astronauts $r12=1.0\mathrm{m}.$

Mass of earth localid="1648480144458" $\mathrm{Me}=5.97×{10}^{24}\mathrm{kg}.$

Therefore, the force between the astronauts :$\mathrm{F}1=G×m1×\mathrm{m}2×\mathrm{r}12×2.$

Consider the distance of the astronauts from the centre of earth is $\mathrm{r}.$

So, the gravitational force between earth and astronaut is :

$\mathrm{F}2=\mathrm{GM}2\mathrm{m}1\mathrm{r}2.$

Now equate the above two equations and solve for $r.$

$$\begin{gathered} \mathrm{Gm}1\mathrm{m}2\mathrm{r}122=G\mathrm{Me}m1\mathrm{r}2 \\ ⇒\mathrm{m}2\mathrm{r}122=\mathrm{Me}r2 \\ ⇒\mathrm{r}2=\mathrm{Me}m2\mathrm{r}122 \\ ⇒\mathrm{r}=\mathrm{Me}m2\mathrm{r}12. \end{gathered}$$

Now substitute the values of $Me,m2$and $r12$to calculate$r:$

localid="1648480176345" $$\begin{gathered} r=(5.97×{10}^{24}\mathrm{kg})(65\mathrm{kg})(1.0\mathrm{m}) \\ =3.03×{10}^{11}\mathrm{m}. \end{gathered}$$

The astronauts are $3.03×{10}^{11}m$away from the center of the earth when the gravitational force between them is as strong as the gravitational force of the earth on one of the astronauts.