To solve the problem of finding the gravitational force between the two spaceships, we use Newton's law of universal gravitation, which is given by:
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force,
- [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] and [tex]\( m_2 \)[/tex] are the masses of the two objects, [tex]\( 300,000 \, \text{kg} \)[/tex] each,
- [tex]\( r \)[/tex] is the distance between the centers of the two objects, [tex]\( 250 \, \text{m} \)[/tex].
Let's plug these values into the formula:
1. Substitute the known values:
[tex]\[ F = 6.67 \times 10^{-11} \times \frac{300,000 \times 300,000}{250^2} \][/tex]
2. Calculate the numerator:
[tex]\[ 300,000 \times 300,000 = 90,000,000,000 \text{ kg}^2 \][/tex]
3. Calculate the denominator:
[tex]\[ 250^2 = 62,500 \text{ m}^2 \][/tex]
4. Divide the numerator by the denominator:
[tex]\[ \frac{90,000,000,000}{62,500} = 1,440,000 \][/tex]
5. Multiply by [tex]\( G \)[/tex]:
[tex]\[ F = 6.67 \times 10^{-11} \times 1,440,000 \][/tex]
6. Multiply these numbers:
[tex]\[ 6.67 \times 1,440,000 = 9,604,800 \times 10^{-11} \][/tex]
7. Convert the result to scientific notation:
[tex]\[ F = 9.6048 \times 10^{-5} \, \text{N} \][/tex]
After performing these calculations, we find that the gravitational force between the two spaceships is:
[tex]\[ 9.6048 \times 10^{-5} \, \text{N} \][/tex]
Now let's compare this result to the given choices:
A. [tex]\( 7.23 \times 10^{-6} \, \text{N} \)[/tex]
B. [tex]\( 8.00 \times 10^{-8} \, \text{N} \)[/tex]
C. [tex]\( 9.6 \times 10^{-5} \, \text{N} \)[/tex]
D. [tex]\( 2.40 \times 10^{-2} \, \text{N} \)[/tex]
Clearly, the calculated result [tex]\( 9.6048 \times 10^{-5} \, \text{N} \)[/tex] is closest to choice C, [tex]\( 9.6 \times 10^{-5} \, \text{N} \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{9.6 \times 10^{-5} \, \text{N}} \][/tex]
