Satum’s moon Enceladus has a mass of 1.08 x 1020 kg and a radius of 2.52 x 106 m. What speed would be required for a 75.0 kg satellite to be in a stable orbit 165 km above the surface of this moon?
The correct answer and explanation is :
To determine the speed required for a 75.0 kg satellite to maintain a stable orbit 165 km above the surface of Saturn’s moon Enceladus, we can use the formula for orbital velocity:
[ v = \sqrt{\frac{G \times M}{r}} ]
Where:
- ( v ) is the orbital velocity in meters per second (m/s).
- ( G ) is the gravitational constant, approximately ( 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 ).
- ( M ) is the mass of Enceladus, given as ( 1.08 \times 10^{20} \, \text{kg} ).
- ( r ) is the distance from the center of Enceladus to the satellite, which is the sum of Enceladus’s radius and the altitude of the orbit:
- Radius of Enceladus: ( 2.52 \times 10^6 \, \text{m} ).
- Altitude of the orbit: 165 km = ( 165,000 \, \text{m} ).
- Therefore, ( r = 2.52 \times 10^6 \, \text{m} + 165,000 \, \text{m} = 2.685 \times 10^6 \, \text{m} ).
Plugging these values into the orbital velocity formula:
[ v = \sqrt{\frac{6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 1.08 \times 10^{20} \, \text{kg}}{2.685 \times 10^6 \, \text{m}}} ]
Calculating the numerator:
[ 6.674 \times 10^{-11} \times 1.08 \times 10^{20} = 7.210 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{kg} ]
Now, calculating the orbital velocity:
[ v = \sqrt{\frac{7.210 \times 10^9}{2.685 \times 10^6}} \approx \sqrt{2,682.5} \approx 51.8 \, \text{m/s} ]
Therefore, the required orbital speed for the satellite is approximately 51.8 meters per second (m/s).
Explanation:
To achieve a stable orbit around a celestial body, a satellite must balance the gravitational pull of the body with its centrifugal force due to its motion. This balance results in a constant orbital velocity, which depends on the mass of the celestial body and the distance from its center. By applying the orbital velocity formula, we can calculate the precise speed needed for a satellite to maintain a circular orbit at a given altitude above the body’s surface.