A satellite with kinetic energy `E_(k)` is revolving round the earth in a circular orbit. How much more kinetic energy should be given to it so that it may just escape into outer space

Options

A. `E_(k)`
B. `E_(k)//2`
C. `E_(k)//4`
D. `2E_(k)`

Explanation

To solve the problem of how much more kinetic energy should be given to a satellite in order for it to escape the gravitational pull of the Earth, we can follow these steps:

### Step 1: Understand the concept of escape velocity
The escape velocity is the minimum velocity an object must have to break free from the gravitational attraction of a celestial body without any further propulsion. For Earth, the escape velocity \( v_e \) can be expressed as:

\[ v_e = \sqrt{\frac{2GM}{R}} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Relate kinetic energy to escape velocity
The kinetic energy \( E_k \) of the satellite moving with velocity \( v \) is given by:

\[ E_k = \frac{1}{2} mv^2 \]

To escape, the satellite must reach the escape velocity \( v_e \). Thus, the kinetic energy required to escape is:

\[ E_{k_{escape}} = \frac{1}{2} m v_e^2 \]

### Step 3: Calculate the additional kinetic energy needed
The satellite currently has a kinetic energy \( E_k \). To find out how much more kinetic energy \( E \) needs to be added, we set up the equation:

\[ E_k + E = E_{k_{escape}} \]

Rearranging gives us:

\[ E = E_{k_{escape}} - E_k \]

### Step 4: Substitute the expression for escape kinetic energy
Substituting \( E_{k_{escape}} \):

\[ E = \frac{1}{2} m v_e^2 - E_k \]

### Step 5: Express the additional energy in terms of \( E_k \)
From the previous steps, we know that for the satellite to escape, its total mechanical energy must be zero. The total mechanical energy \( E_{total} \) is given by the sum of kinetic energy and potential energy. The potential energy \( U \) of the satellite in a gravitational field is given by:

\[ U = -\frac{GMm}{R} \]

For a satellite in a circular orbit, the total mechanical energy is:

\[ E_{total} = E_k + U \]

To escape, we need:

\[ E_k + U + E = 0 \]

From this, we can derive that:

\[ E = -E_k - U \]

### Step 6: Final expression for additional energy
Since we are interested in the additional energy required, we can conclude that the amount of kinetic energy that needs to be added to the satellite is equal to its current kinetic energy:

\[ E = E_k \]

### Conclusion
Thus, the additional kinetic energy required for the satellite to escape into outer space is equal to its current kinetic energy \( E_k \).