Two $65kg$astronauts leave earth in a spacecraft, sitting $1.0m$apart. How far are they 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?

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" $\mathrm{Me}=5.97\times {10}^{24}\mathrm{kg}.$

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

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 $r.$

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

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

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

The astronauts are $3.03\times {10}^{11}m$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.