The gravitational force between the Sun (mass = 1.99 × 1030 kg) and Mercury (mass = 3.30 × 1023 kg) is 8.99 × 1021 N. How far is Mercury from the Sun? 6.98 × 1010 km 6.98 × 107 km 4.87 × 1013 km 4.87 × 1024 km
The Correct Answer and Explanation is:
To solve for the distance between Mercury and the Sun, we can use Newton’s Law of Universal Gravitation, which is given by:F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}F=Gr2m1m2
Where:
- FFF is the gravitational force between the two objects (in this case, the Sun and Mercury),
- GGG is the gravitational constant, which is approximately 6.674×10−11 N⋅m2/kg26.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^26.674×10−11N⋅m2/kg2,
- m1m_1m1 is the mass of the first object (the Sun),
- m2m_2m2 is the mass of the second object (Mercury),
- rrr is the distance between the two objects.
From the problem, we know the following values:
- The gravitational force, F=8.99×1021 NF = 8.99 \times 10^{21} \, \text{N}F=8.99×1021N,
- The mass of the Sun, m1=1.99×1030 kgm_1 = 1.99 \times 10^{30} \, \text{kg}m1=1.99×1030kg,
- The mass of Mercury, m2=3.30×1023 kgm_2 = 3.30 \times 10^{23} \, \text{kg}m2=3.30×1023kg.
We need to solve for rrr, the distance between Mercury and the Sun.
Rearranging the formula to solve for rrr:r=Gm1m2Fr = \sqrt{ \frac{G m_1 m_2}{F} }r=FGm1m2
Substituting the known values:r=(6.674×10−11)×(1.99×1030)×(3.30×1023)8.99×1021r = \sqrt{ \frac{(6.674 \times 10^{-11}) \times (1.99 \times 10^{30}) \times (3.30 \times 10^{23})}{8.99 \times 10^{21}} }r=8.99×1021(6.674×10−11)×(1.99×1030)×(3.30×1023)
First, calculate the numerator:(6.674×10−11)×(1.99×1030)=1.327×1020(6.674 \times 10^{-11}) \times (1.99 \times 10^{30}) = 1.327 \times 10^{20}(6.674×10−11)×(1.99×1030)=1.327×1020
Then,(1.327×1020)×(3.30×1023)=4.379×1043(1.327 \times 10^{20}) \times (3.30 \times 10^{23}) = 4.379 \times 10^{43}(1.327×1020)×(3.30×1023)=4.379×1043
Now divide by the gravitational force:4.379×10438.99×1021=4.87×1022\frac{4.379 \times 10^{43}}{8.99 \times 10^{21}} = 4.87 \times 10^{22}8.99×10214.379×1043=4.87×1022
Finally, take the square root:r=4.87×1022≈6.98×1010 mr = \sqrt{4.87 \times 10^{22}} \approx 6.98 \times 10^{10} \, \text{m}r=4.87×1022≈6.98×1010m
To convert meters to kilometers, divide by 100010001000:r≈6.98×107 kmr \approx 6.98 \times 10^7 \, \text{km}r≈6.98×107km
Thus, the distance between Mercury and the Sun is approximately 6.98 × 10^7 km.
So, the correct answer is: 6.98 × 10^7 km.
