To calculate the distance between the two asteroids given their masses and the gravitational force between them, we can use Newton's law of universal gravitation:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force between the asteroids, which is [tex]\( 1030 \)[/tex] N.
- [tex]\( G \)[/tex] is the gravitational constant, which is [tex]\( 6.67430 \times 10^{-11} \)[/tex] m[tex]\(^3\)[/tex]kg[tex]\(^{-1}\)[/tex]s[tex]\(^{-2}\)[/tex].
- [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two asteroids, each [tex]\( 1.41 \times 10^{14} \)[/tex] kg.
- [tex]\( r \)[/tex] is the distance between the centers of the two asteroids, which we need to find.
First, rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r^2 = G \frac{m_1 m_2}{F} \][/tex]
Substitute the given values into this formula:
[tex]\[ r^2 = (6.67430 \times 10^{-11}) \frac{(1.41 \times 10^{14}) (1.41 \times 10^{14})}{1030} \][/tex]
Next, compute the product [tex]\( m_1 m_2 \)[/tex]:
[tex]\[ m_1 m_2 = 1.41 \times 10^{14} \times 1.41 \times 10^{14} \][/tex]
[tex]\[ m_1 m_2 = 1.9881 \times 10^{28} \, \text{kg}^2 \][/tex]
Now, substitute back into the distance formula:
[tex]\[ r^2 = (6.67430 \times 10^{-11}) \frac{1.9881 \times 10^{28}}{1030} \][/tex]
Calculate the term inside the parentheses:
[tex]\[ G \frac{m_1 m_2}{F} = (6.67430 \times 10^{-11}) \times \frac{1.9881 \times 10^{28}}{1030} \][/tex]
[tex]\[ r^2 \approx 1.288269498058252 \times 10^{15} \, \text{m}^2 \][/tex]
To find [tex]\( r \)[/tex], take the square root of both sides:
[tex]\[ r \approx \sqrt{1.288269498058252 \times 10^{15}} \][/tex]
[tex]\[ r \approx 3.589247132837544 \times 10^{7} \, \text{m} \][/tex]
Therefore, the distance between the asteroids, written in scientific notation to three significant figures, is approximately:
[tex]\[ r \approx 3.59 \times 10^{7} \, \text{m} \][/tex]
So the distance between the asteroids is:
[tex]\[ 3.59 \times 10^{7} \, \text{m} \][/tex]
