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. `2E_(k)`
C. `1/2E_(k)`
D. `3E_(k)`

Explanation

To solve the problem of how much more kinetic energy should be given to a satellite so that it may just escape into outer space, we can follow these steps:

### Step 1: Understand the total mechanical energy of the satellite in orbit
The total mechanical energy \( E \) of a satellite in a circular orbit is given by the formula:
\[ E = K + U \]
where \( K \) is the kinetic energy and \( U \) is the gravitational potential energy.

For a satellite in a circular orbit, the gravitational potential energy \( U \) is given by:
\[ U = -\frac{G M m}{r} \]
and the kinetic energy \( K \) is:
\[ K = \frac{1}{2} m v^2 \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite.

### Step 2: Relate kinetic energy and potential energy
In a stable circular orbit, the kinetic energy is related to the potential energy by:
\[ K = -\frac{1}{2} U \]
Thus, the total energy can be expressed as:
\[ E = K + U = K - 2K = -K \]
So, the total energy \( E \) of the satellite is:
\[ E = -\frac{G M m}{2r} \]

### Step 3: Determine the energy required to escape
To escape from the gravitational influence of the Earth, the satellite must have a total energy of zero. Therefore, the energy required to escape \( \Delta E \) can be calculated as:
\[ \Delta E = 0 - E = 0 - \left(-\frac{G M m}{2r}\right) = \frac{G M m}{2r} \]

### Step 4: Calculate the additional kinetic energy needed
Let the current kinetic energy of the satellite be \( E_k \). The additional kinetic energy \( \Delta K \) needed to reach the escape energy is:
\[ \Delta K = \Delta E - E_k \]
Substituting for \( \Delta E \):
\[ \Delta K = \frac{G M m}{2r} - E_k \]

### Conclusion
Thus, the additional kinetic energy that should be given to the satellite so that it may just escape into outer space is:
\[ \Delta K = \frac{G M m}{2r} - E_k \]