Let's determine the distance between Pluto and Charon, given the provided values:
1. Step 1: Given Data
- Gravitational force ([tex]\( F \)[/tex]) between Pluto and Charon: [tex]\( 3.61 \times 10^{18} \, \text{N} \)[/tex]
- Mass of Pluto ([tex]\( m_1 \)[/tex]): [tex]\( 1.3 \times 10^{22} \, \text{kg} \)[/tex]
- Mass of Charon ([tex]\( m_2 \)[/tex]): [tex]\( 1.6 \times 10^{21} \, \text{kg} \)[/tex]
- Gravitational constant ([tex]\( G \)[/tex]): [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex]
2. Step 2: Formula for Gravitational Force
The gravitational force formula is given by:
[tex]\[
F = G \frac{m_1 \cdot m_2}{r^2}
\][/tex]
where [tex]\( r \)[/tex] is the distance between the centers of the two objects.
3. Step 3: Solving for [tex]\( r^2 \)[/tex]
Rearrange the above formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = G \frac{m_1 \cdot m_2}{F}
\][/tex]
4. Step 4: Substitute the Values
Substitute the given values into the rearranged formula:
[tex]\[
r^2 = \frac{(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}) \times (1.3 \times 10^{22} \, \text{kg}) \times (1.6 \times 10^{21} \, \text{kg})}{3.61 \times 10^{18} \, \text{N}}
\][/tex]
5. Step 5: Compute [tex]\( r^2 \)[/tex]
After performing the computation:
[tex]\[
r^2 = 384558005540166.2 \, \text{m}^2
\][/tex]
6. Step 6: Compute [tex]\( r \)[/tex]
Take the square root of [tex]\( r^2 \)[/tex] to get [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt{384558005540166.2} \approx 19610150.574132934 \, \text{m}
\][/tex]
7. Step 7: Closest Matching Option
Consider the options given:
- [tex]\( 2.0 \times 10^7 \, \text{m} \)[/tex]
- [tex]\( 2.4 \times 10^{12} \, \text{m} \)[/tex]
- [tex]\( 3.8 \times 10^{14} \, \text{m} \)[/tex]
- [tex]\( 5.8 \times 10^{24} \, \text{m} \)[/tex]
The closest value to [tex]\( 19610150.574132934 \, \text{m} \)[/tex] (which is approximately [tex]\( 2.0 \times 10^7 \, \text{m} \)[/tex]) is [tex]\( 2.0 \times 10^7 \, \text{m} \)[/tex].
So, the distance between Pluto and Charon is closest to:
[tex]\[ \boxed{2.0 \times 10^7 \, \text{m}} \][/tex]
