Certainly! Let's solve this problem step-by-step.
We are asked to find the distance between two men, each with a mass of 90 kg, given the gravitational force between them is [tex]\(8.64 \times 10^{-8}\)[/tex] N. The gravitational constant [tex]\(G\)[/tex] is [tex]\(6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right)\)[/tex].
We start by recalling the formula for the gravitational force between two masses:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/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,
- [tex]\(r\)[/tex] is the distance between the centers of the two masses.
Given:
- [tex]\(m_1 = 90 \text{ kg}\)[/tex],
- [tex]\(m_2 = 90 \text{ kg}\)[/tex],
- [tex]\(F = 8.64 \times 10^{-8} \text{ N}\)[/tex],
- [tex]\(G = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right)\)[/tex].
We need to solve for [tex]\(r\)[/tex]. Rearranging the formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r^2 = G \cdot \frac{m_1 \cdot m_2}{F} \][/tex]
Substituting the given values into the equation:
[tex]\[ r^2 = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right) \cdot \frac{90 \text{ kg} \cdot 90 \text{ kg}}{8.64 \times 10^{-8} \text{ N}} \][/tex]
First, let's compute the numerator and the denominator separately:
[tex]\[ \text{Numerator} = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right) \cdot 8100 \text{ kg}^2 \][/tex]
[tex]\[ \text{Numerator} = 6.67 \times 10^{-11} \cdot 8100 \text{ m}^2 \][/tex]
[tex]\[ \text{Numerator} = 5.4027 \times 10^{-7} \text{ m}^2 \][/tex]
[tex]\[ \text{Denominator} = 8.64 \times 10^{-8} \text{ N} \][/tex]
Now, divide the two results to find [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = \frac{5.4027 \times 10^{-7} \text{ m}^2}{8.64 \times 10^{-8} \text{ N}} \][/tex]
[tex]\[ r^2 \approx 6.25 \text{ m}^2 \][/tex]
Finally, take the square root of both sides to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt{6.25 \text{ m}^2} \][/tex]
[tex]\[ r \approx 2.50 \text{ m} \][/tex]
Thus, the distance between the two men is approximately 2.5 meters.
The correct answer is:
D. 2.5 m
