If a satellite is moving around thee earth in an orbit of 5 R radius , here R = radius of the earth . The minimum kinetic energy required to be provided to the satellite such that it escapes the gravitational field of the earth is ( M and m are masses of earth and satellite respectively )

Options

A. `(GMm)/(5R)`
B. `(GMm)/(15R)`
C. `(GMm)/(10R)`
D. `(GMm)/(sqrt(2)R)`

Answer

To solve the problem of finding the minimum kinetic energy required for a satellite in an orbit of radius \(5R\) (where \(R\) is the radius of the Earth) to escape the gravitational field of the Earth, we can follow these steps:

### Step 1: Understand the Energy Conservation Principle
The total mechanical energy of the satellite must be zero for it to escape the gravitational field. This means that the sum of its initial potential energy and kinetic energy must equal zero when it escapes.

### Step 2: Write the Expression for Potential Energy
The gravitational potential energy (\(U\)) of the satellite at a distance \(r\) from the center of the Earth is given by:
\[ U = -\frac{GMm}{r} \]
where:
- \(G\) is the gravitational constant,
- \(M\) is the mass of the Earth,
- \(m\) is the mass of the satellite,
- \(r\) is the distance from the center of the Earth.

For our case, since the satellite is at a distance of \(5R\), we have:
\[ U = -\frac{GMm}{5R} \]

### Step 3: Write the Expression for Kinetic Energy
The kinetic energy (\(K\)) of the satellite in orbit is given by:
\[ K = \frac{GMm}{2r} \]
For the satellite in an orbit of radius \(5R\), we have:
\[ K = \frac{GMm}{2(5R)} = \frac{GMm}{10R} \]

### Step 4: Apply the Conservation of Energy
According to the conservation of energy, the initial potential energy plus the initial kinetic energy must equal zero at the point of escape:
\[ U + K + K_{given} = 0 \]
Substituting the expressions for \(U\) and \(K\):
\[ -\frac{GMm}{5R} + \frac{GMm}{10R} + K_{given} = 0 \]

### Step 5: Solve for the Given Kinetic Energy
Rearranging the equation gives:
\[ K_{given} = \frac{GMm}{5R} - \frac{GMm}{10R} \]
To combine the fractions:
\[ K_{given} = \frac{2GMm}{10R} - \frac{GMm}{10R} = \frac{GMm}{10R} \]

### Conclusion
Thus, the minimum kinetic energy required to be provided to the satellite so that it escapes the gravitational field of the Earth is:
\[ K_{given} = \frac{GMm}{10R} \]