Answer:
The two objects should be kept at a distance of [tex]3.535\times 10^{4}[/tex] kilometers.
Explanation:
From Newton's Law of Gravitation, gravitational force ([tex]F[/tex]), measured in newtons, between two objects is inversely proportional to the square of distance ([tex]r[/tex]), measured in meters. That is:
[tex]F \propto \frac{1}{r^{2}}[/tex] (1)
[tex]F = \frac{k}{r^{2}}[/tex] (2)
Where [tex]k[/tex] is the proportionality ratio, measured in newtons-square meter.
Now we eliminate the proportionality by constructing the following relationship:
[tex]\frac{F_{2}}{F_{1}} = \left(\frac{r_{1}}{r_{2}}\right)^{2}[/tex] (3)
If we know that [tex]\frac{F_{2}}{F_{1}} = \frac{1}{2}[/tex] and [tex]r_{1} = 2.5\times 10^{7}\,m[/tex], then the distance between the two objects so that gravitational force becomes half is:
[tex]\frac{1}{2} = \frac{(2.5\times 10^{7}\,m)^{2}}{r_{2}^{2}}[/tex]
[tex]r_{2}^{2} = 1.25\times 10^{15}\,m^{2}[/tex]
[tex]r_{2} = 3.535\times 10^{7}\,m[/tex]
[tex]r_{2} = 3.535\times 10^{4}\,km[/tex]
The two objects should be kept at a distance of [tex]3.535\times 10^{4}[/tex] kilometers.
