Step 1: Calculate Orbital Speed using Formula
The orbital speed \( v \) of a satellite at height \( h \) above Earth is determined by:
\[v = \sqrt{\frac{GM}{r}}\]
where:
- \( G \) is the gravitational constant.
- \( M \) is Earth's mass.
- \( r \) is the distance from Earth's center to the satellite, calculated as \( r = R + h \), with \( R \) being Earth's radius.
Step 2: Input Given Values and Calculate
Provided data:
- \( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)
- \( M = 6 \times 10^{24} \, \text{kg} \)
- Earth's Radius \( R = 6.4 \times 10^6 \, \text{m} \)
- Satellite Height \( h = 10^4 \, \text{km} = 10^7 \, \text{m} \)
Total distance from Earth's center:
\[r = 6.4 \times 10^6 + 10^7 = 1.64 \times 10^7 \, \text{m}\]
Substituting into the orbital speed formula:
\[v = \sqrt{\frac{6.67 \times 10^{-11} \times 6 \times 10^{24}}{1.64 \times 10^7}}\]
\[v = \sqrt{\frac{4.002 \times 10^{14}}{1.64 \times 10^7}} = \sqrt{2.44 \times 10^7} = 4.93 \times 10^3 \, \text{m/s} = 7.0 \, \text{km/s}\]
Answer: The satellite's orbital speed is \( 7.0 \, \text{km/s} \). This corresponds to option (1).