When energy of a satellite - planet system is positive then satellite will

Options

A. Move around planet in circular orbit
B. Move around planet inelliptical orbit
C. Escape out with minimum speed
D. Escape out with speed greater than escape velocity

Answer

To solve the problem of determining the behavior of a satellite-planet system when the energy is positive, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy of the Satellite-Planet System**:
The total mechanical energy (E) of a satellite in orbit around a planet is given by the formula:
\[ E = K + U \]
where \( K \) is the kinetic energy and \( U \) is the gravitational potential energy.

2. **Gravitational Potential Energy**:
The gravitational potential energy (U) of a satellite of mass \( m \) at a distance \( r \) from the center of a planet of mass \( M \) is given by:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant.

3. **Kinetic Energy**:
The kinetic energy (K) of the satellite is given by:
\[ K = \frac{1}{2} mv^2 \]
where \( v \) is the orbital speed of the satellite.

4. **Total Energy**:
Therefore, the total energy of the satellite-planet system can be expressed as:
\[ E = \frac{1}{2} mv^2 - \frac{G M m}{r} \]

5. **Condition for Positive Energy**:
For the total energy \( E \) to be positive:
\[ \frac{1}{2} mv^2 - \frac{G M m}{r} > 0 \]
Rearranging gives:
\[ \frac{1}{2} mv^2 > \frac{G M m}{r} \]
Dividing by \( m \) (assuming \( m \neq 0 \)):
\[ \frac{1}{2} v^2 > \frac{G M}{r} \]
Multiplying both sides by 2:
\[ v^2 > \frac{2 G M}{r} \]
Taking the square root:
\[ v > \sqrt{\frac{2 G M}{r}} \]

6. **Conclusion**:
This means that for the energy of the satellite-planet system to be positive, the speed of the satellite must be greater than the escape velocity, which is given by:
\[ v_{escape} = \sqrt{\frac{2 G M}{r}} \]
Therefore, if the energy of the satellite-planet system is positive, the satellite will be moving with a speed greater than the escape velocity, indicating that it will escape the gravitational influence of the planet.

### Final Answer:
When the energy of a satellite-planet system is positive, the satellite will escape from the gravitational field of the planet.