Let's solve the problem step-by-step:
1. Understanding the problem: We are given the gravitational force between two masses is 250 N at a distance of [tex]\( 2.5 \times 10^4 \)[/tex] km. We need to find the distance at which the gravitational force would be reduced to half of its original value.
2. Gravitational force relationship:
The gravitational force [tex]\( F \)[/tex] between two masses is given by:
[tex]\[
F = G \frac{m_1 m_2}{r^2}
\][/tex]
where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses, and [tex]\( r \)[/tex] is the distance between them.
3. Initial conditions:
[tex]\[
F_{\text{initial}} = 250 \text{ N}
\][/tex]
[tex]\[
r_{\text{initial}} = 2.5 \times 10^4 \text{ km}
\][/tex]
4. Target conditions: We want the force to be half of the initial force:
[tex]\[
F_{\text{target}} = \frac{F_{\text{initial}}}{2} = \frac{250}{2} = 125 \text{ N}
\][/tex]
5. Relationship between forces and distances:
Since the gravitational force [tex]\( F \)[/tex] is inversely proportional to the square of the distance [tex]\( r \)[/tex]:
[tex]\[
\frac{F_{\text{initial}}}{F_{\text{target}}} = \left(\frac{r_{\text{target}}}{r_{\text{initial}}}\right)^2
\][/tex]
6. Substitute known values:
[tex]\[
\frac{250 \text{ N}}{125 \text{ N}} = \left(\frac{r_{\text{target}}}{2.5 \times 10^4 \text{ km}}\right)^2
\][/tex]
This simplifies to:
[tex]\[
2 = \left(\frac{r_{\text{target}}}{2.5 \times 10^4 \text{ km}}\right)^2
\][/tex]
7. Solve for [tex]\( r_{\text{target}} \)[/tex]:
Take the square root of both sides:
[tex]\[
\sqrt{2} = \frac{r_{\text{target}}}{2.5 \times 10^4 \text{ km}}
\][/tex]
Therefore:
[tex]\[
r_{\text{target}} = 2.5 \times 10^4 \text{ km} \times \sqrt{2}
\][/tex]
8. Calculate [tex]\( r_{\text{target}} \)[/tex]:
[tex]\[
r_{\text{target}} = 2.5 \times 10^4 \times 1.414 \approx 35355.339 \text{ km}
\][/tex]
Thus, the distance between the two masses should be approximately [tex]\( 35355.339 \)[/tex] km to reduce the gravitational force between them by half.
