To determine if the satellite has a stable orbit, we must check if its orbital velocity is sufficient to balance the gravitational force at the given height. Here's how to analyze this step-by-step:

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1. Gravitational Force and Required Orbital Velocity

The velocity required for a stable circular orbit is given by:

$v_\text{orbit} = \sqrt{\frac{GM}{r}}$

Where:

• $G$ is the gravitational constant, $6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}$,
• $M$ is the mass of the Earth, $5.97 \times 10^{24} \, \text{kg}$,
• $r$ is the distance from the center of the Earth to the satellite.

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2. Calculate $r$ (Distance from the Center of the Earth)

The satellite is released at a height of $35,000 \, \text{km}$ above the Earth's surface. Adding the Earth's radius ($R_\text{Earth} = 6,371 \, \text{km}$):

$r = R_\text{Earth} + h = 6,371 \, \text{km} + 35,000 \, \text{km} = 41,371 \, \text{km} = 41,371,000 \, \text{m}$

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3. Calculate the Required Orbital Velocity

Substitute $G$, $M$, and $r$ into the orbital velocity formula:

$v_\text{orbit} = \sqrt{\frac{(6.674 \times 10^{-11})(5.97 \times 10^{24})}{41,371,000}}$

Simplify step by step:

$v_\text{orbit} = \sqrt{\frac{3.986 \times 10^{14}}{41,371,000}} = \sqrt{9.64 \times 10^6} \, \text{m/s}$

$v_\text{orbit} \approx 3105 \, \text{m/s}$

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4. Compare Given Velocity to Required Velocity

The satellite is released at $3100 \, \text{m/s}$, which is slightly less than $3105 \, \text{m/s}$. Since the velocity is insufficient:

• The satellite will not maintain a stable orbit.
• Instead, it will spiral downward unless corrective action is taken to increase its velocity.

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Conclusion

The satellite does not have a stable orbit at $3100 \, \text{m/s}$, as the required orbital velocity at $35,000 \, \text{km}$ altitude is $3105 \, \text{m/s}$. A small adjustment to the satellite's velocity would stabilize its orbit.

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Would you like detailed calculations for energy considerations or further explanations?
Here are 5 follow-up questions for deeper understanding:

1. What factors influence the stability of an orbit besides velocity?
2. How does elliptical motion differ from circular orbital motion?
3. What happens if the satellite's speed is greater than $v_\text{orbit}$?
4. How does Earth's atmosphere affect low-altitude satellites?
5. How would the satellite's orbit change if it were launched at a different altitude?

Tip: Always ensure to consider both altitude and velocity when evaluating orbital stability.