To calculate the force of gravity between the two spaceships, we will use Newton's law of universal gravitation, which states that the gravitational force [tex]\( F \)[/tex] between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is given by:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where:
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the first spaceship, [tex]\( 300,000 \, \text{kg} \)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the second spaceship, [tex]\( 300,000 \, \text{kg} \)[/tex],
- [tex]\( r \)[/tex] is the distance between the two spaceships, [tex]\( 250 \, \text{m} \)[/tex].
Following these steps:
1. Substitute the given values into the formula:
[tex]\[
F = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times \frac{300,000 \, \text{kg} \times 300,000 \, \text{kg}}{(250 \, \text{m})^2}
\][/tex]
2. Calculate the product of the masses:
[tex]\[
300,000 \times 300,000 = 90,000,000,000 \, \text{kg}^2
\][/tex]
3. Calculate the square of the distance:
[tex]\[
(250 \, \text{m})^2 = 62,500 \, \text{m}^2
\][/tex]
4. Substitute these results back into the formula:
[tex]\[
F = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times \frac{90,000,000,000 \, \text{kg}^2}{62,500 \, \text{m}^2}
\][/tex]
5. Perform the division in the fraction:
[tex]\[
\frac{90,000,000,000}{62,500} = 1,440,000
\][/tex]
6. Multiply by the gravitational constant:
[tex]\[
F = 6.67 \times 10^{-11} \times 1,440,000 \, \text{N}
\][/tex]
7. Calculate the final result:
[tex]\[
F \approx 9.6048 \times 10^{-5} \, \text{N}
\][/tex]
Therefore, the force of gravity between the two spaceships is [tex]\( 9.6 \times 10^{-5} \)[/tex] N, which corresponds to option B.
