To find the gravitational force between the two spaceships, we can use Newton's law of universal gravitation. The formula is given by:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force between the objects,
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.67 \times 10^{-11} \, N \cdot m^2 / kg^2 \)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the first spaceship [tex]\( 300,000 \, kg \)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the second spaceship [tex]\( 300,000 \, kg \)[/tex],
- [tex]\( r \)[/tex] is the distance between the centers of the two spaceships, [tex]\( 250 \, m \)[/tex].
Step-by-step solution:
1. Write down the given values:
- [tex]\( G = 6.67 \times 10^{-11} \, N \cdot m^2 / kg^2 \)[/tex]
- [tex]\( m_1 = 300,000 \, kg \)[/tex]
- [tex]\( m_2 = 300,000 \, kg \)[/tex]
- [tex]\( r = 250 \, m \)[/tex]
2. Substitute these values into the formula:
[tex]\[ F = 6.67 \times 10^{-11} \frac{(300,000) \times (300,000)}{(250)^2} \][/tex]
3. Calculate the masses product:
[tex]\[ m_1 \times m_2 = 300,000 \times 300,000 = 90,000,000,000 \, kg^2 \][/tex]
4. Calculate the square of the distance:
[tex]\[ r^2 = 250^2 = 62,500 \, m^2 \][/tex]
5. Substitute these into the equation to find the force:
[tex]\[ F = 6.67 \times 10^{-11} \times \frac{90,000,000,000}{62,500} \][/tex]
6. Divide the masses product by the distance squared:
[tex]\[ \frac{90,000,000,000}{62,500} = 1,440,000 \][/tex]
7. Multiply by the gravitational constant:
[tex]\[ F = 6.67 \times 10^{-11} \times 1,440,000 \][/tex]
8. Calculate the final value:
[tex]\[ F = 9.6048 \times 10^{-5} \, N \][/tex]
Therefore, the force of gravity between the two spaceships is [tex]\( 9.6048 \times 10^{-5} \, N \)[/tex], which corresponds to option:
A. [tex]$9.6 \times 10^{-5} \, N$[/tex]
