Certainly! Let's solve this problem step-by-step using the law of universal gravitation.
1. Identify the given values:
- Gravitational constant, [tex]\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex]
- Mass of the first asteroid, [tex]\( m_1 = 5 \times 10^8 \, \text{kg} \)[/tex]
- Distance between the asteroids, [tex]\( r = 50,000 \, \text{m} \)[/tex]
- Gravitational force between them, [tex]\( F = 8.67 \times 10^{-2} \, \text{N} \)[/tex]
2. Recall the formula for gravitational force:
[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]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two objects,
- [tex]\( r \)[/tex] is the distance between them.
3. Rearrange the formula to solve for the mass of the second asteroid [tex]\( m_2 \)[/tex]:
[tex]\[
m_2 = \frac{F \cdot r^2}{G \cdot m_1}
\][/tex]
4. Substitute the given values into the formula:
[tex]\[
m_2 = \frac{8.67 \times 10^{-2} \, \text{N} \cdot (50,000 \, \text{m})^2}{6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \cdot 5 \times 10^8 \, \text{kg}}
\][/tex]
5. Calculate the numerator:
- Distance squared: [tex]\( (50,000)^2 = 2.5 \times 10^9 \, \text{m}^2 \)[/tex]
- Force times distance squared: [tex]\( 8.67 \times 10^{-2} \times 2.5 \times 10^9 = 2.1675 \times 10^8 \)[/tex]
6. Calculate the denominator:
[tex]\[
6.67430 \times 10^{-11} \times 5 \times 10^8 = 3.33715 \times 10^{-2}
\][/tex]
7. Divide the numerator by the denominator to find [tex]\( m_2 \)[/tex]:
[tex]\[
m_2 = \frac{2.1675 \times 10^8}{3.33715 \times 10^{-2}} = 6.49506315269017 \times 10^9
\][/tex]
8. Express the final result with the correct significant figures:
[tex]\[
m_2 \approx 6.5 \times 10^9 \, \text{kg}
\][/tex]
Therefore, the correct answer is:
A. [tex]\( 6.5 \times 10^9 \, \text{kg} \)[/tex]
