By how much percent does the speed of a satellite orbiting in circular orbit be increased so that it will escape from the gravitational field of the earth ?

Options

A. `11.2 ` %
B. `41.4` %
C. `27.5` %
D. `34.4` %

Answer

To solve the problem of how much percent the speed of a satellite orbiting in a circular orbit must be increased to escape from the gravitational field of the Earth, we can follow these steps:

### Step 1: Understand the Kinetic and Potential Energy
The kinetic energy (KE) of a satellite in a circular orbit is given by:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the satellite and \( v \) is its orbital speed.

The gravitational potential energy (PE) of the satellite is given by:
\[ PE = -\frac{G M m}{r} \]
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.

### Step 2: Total Energy in Orbit
The total mechanical energy \( E \) of the satellite in orbit is the sum of its kinetic and potential energy:
\[ E = KE + PE = \frac{1}{2} m v^2 - \frac{G M m}{r} \]

### Step 3: Escape Velocity
The escape velocity \( v_e \) from the gravitational field is given by:
\[ v_e = \sqrt{\frac{2GM}{r}} \]

### Step 4: Orbital Velocity
The orbital velocity \( v_i \) of the satellite in a circular orbit is given by:
\[ v_i = \sqrt{\frac{GM}{r}} \]

### Step 5: Relationship Between Escape Velocity and Orbital Velocity
From the equations for escape velocity and orbital velocity, we can see that:
\[ v_e = \sqrt{2} v_i \]

### Step 6: Calculate the Required Increase in Speed
To find the increase in speed required, we calculate:
\[ \Delta v = v_e - v_i = \sqrt{2} v_i - v_i = (\sqrt{2} - 1) v_i \]

### Step 7: Calculate the Percentage Increase
The percentage increase in speed is given by:
\[ \text{Percentage Increase} = \frac{\Delta v}{v_i} \times 100 = \frac{(\sqrt{2} - 1) v_i}{v_i} \times 100 = (\sqrt{2} - 1) \times 100 \]

### Step 8: Numerical Calculation
Calculating \( \sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \]
Thus,
\[ \text{Percentage Increase} = (1.414 - 1) \times 100 \approx 0.414 \times 100 \approx 41.4\% \]

### Final Answer
The speed of the satellite must be increased by approximately **41.4%** to escape from the gravitational field of the Earth.

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