What should be the percentage increase in the kinetic energy of a satellite to enable it to escape?

Explanation

To determine the percentage increase in the kinetic energy of a satellite to enable it to escape, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Kinetic Energy in Orbit**:
The kinetic energy (KE) of a satellite in orbit is given by the formula:
\[ KE_{\text{orbital}} = \frac{1}{2} m V_0^2 \]
where \( V_0 \) is the orbital velocity of the satellite.

2. **Identify the Escape Velocity**:
The escape velocity (\( V \)) is given by the formula:
\[ V = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius from the center of the planet to the satellite.

3. **Kinetic Energy at Escape Velocity**:
The kinetic energy when the satellite escapes is:
\[ KE_{\text{escape}} = \frac{1}{2} m V^2 \]

4. **Relate Escape Velocity and Orbital Velocity**:
The orbital velocity \( V_0 \) can be expressed as:
\[ V_0 = \sqrt{\frac{GM}{R}} \]
Therefore, the square of the escape velocity can be expressed as:
\[ V^2 = \frac{2GM}{R} \]

5. **Calculate the Kinetic Energies**:
- For the orbital kinetic energy:
\[ KE_{\text{orbital}} = \frac{1}{2} m V_0^2 = \frac{1}{2} m \left(\frac{GM}{R}\right) \]
- For the escape kinetic energy:
\[ KE_{\text{escape}} = \frac{1}{2} m V^2 = \frac{1}{2} m \left(\frac{2GM}{R}\right) \]

6. **Find the Change in Kinetic Energy**:
The change in kinetic energy when transitioning from orbit to escape is:
\[ \Delta KE = KE_{\text{escape}} - KE_{\text{orbital}} = \frac{1}{2} m \left(\frac{2GM}{R}\right) - \frac{1}{2} m \left(\frac{GM}{R}\right) \]
Simplifying this gives:
\[ \Delta KE = \frac{1}{2} m \left(\frac{2GM}{R} - \frac{GM}{R}\right) = \frac{1}{2} m \left(\frac{GM}{R}\right) \]

7. **Calculate the Percentage Increase**:
The percentage increase in kinetic energy is given by:
\[ \text{Percentage Increase} = \frac{\Delta KE}{KE_{\text{orbital}}} \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \frac{\frac{1}{2} m \left(\frac{GM}{R}\right)}{\frac{1}{2} m \left(\frac{GM}{R}\right)} \times 100 = 100\% \]

### Final Answer:
The percentage increase in the kinetic energy of a satellite to enable it to escape is **100%**.

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