Earth orbiting satellite will escape if

Options

A. its speed is increased by 41 %
B. its KE is doubled
C. Both (a) and (b) are correct
D. Both (a) and (b) are wrong

Explanation

To determine the conditions under which an Earth-orbiting satellite will escape from its orbit, we need to understand the concepts of escape velocity and orbital velocity.

### Step-by-Step Solution:

1. **Understanding Orbital Velocity**:
The orbital velocity (\(v_o\)) of a satellite is the speed required to maintain a stable orbit around the Earth. It is given by the formula:
\[ v_o = \sqrt{\frac{GM}{R}} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the distance from the center of the Earth to the satellite.

2. **Understanding Escape Velocity**:
The escape velocity (\(v_e\)) is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. It is given by:
\[ v_e = \sqrt{2GM/R} \]
This means that the escape velocity is \( \sqrt{2} \) times the orbital velocity:
\[ v_e = \sqrt{2} \cdot v_o \]

3. **Condition for Escape**:
For a satellite to escape the gravitational pull of the Earth, its velocity must be equal to or greater than the escape velocity. This can be expressed as:
\[ v \geq v_e \]
where \(v\) is the current velocity of the satellite.

4. **Increasing Orbital Velocity**:
To escape, the satellite must increase its orbital velocity. The increase required to reach escape velocity can be calculated as:
\[ \Delta v = v_e - v_o = \sqrt{2} \cdot v_o - v_o = ( \sqrt{2} - 1 ) \cdot v_o \]
This implies that the satellite's speed must be increased by approximately 41% (since \( \sqrt{2} \approx 1.414 \)).

5. **Kinetic Energy Consideration**:
The kinetic energy (\(KE\)) of the satellite is given by:
\[ KE = \frac{1}{2} mv^2 \]
If the velocity increases to escape velocity, the kinetic energy will increase by a factor of 2, since:
\[ KE_{escape} = \frac{1}{2} m v_e^2 = \frac{1}{2} m (2 v_o^2) = 2 \cdot KE_{orbital} \]

6. **Conclusion**:
Therefore, an Earth-orbiting satellite will escape if its speed is increased by 41% (to reach escape velocity) and this will result in the kinetic energy increasing by a factor of 2.

### Final Answer:
An Earth-orbiting satellite will escape if its speed is increased by approximately 41% to reach escape velocity, resulting in a doubling of its kinetic energy.