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 Forces Acting on the Satellite
The satellite is in circular motion around the Earth. The gravitational force acting on the satellite provides the necessary centripetal force for its circular motion.

### Step 2: Write the Gravitational Force Equation
The gravitational force \(F\) acting on the satellite is given by:
\[ F = \frac{GMm}{r^2} \]
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 3: Set the Radius
In this case, the radius \(r\) is \(5R\) (where \(R\) is the radius of the Earth):
\[ r = 5R \]

### Step 4: Apply the Centripetal Force Condition
For circular motion, the centripetal force required is given by:
\[ F = \frac{mv_0^2}{r} \]
where \(v_0\) is the orbital velocity of the satellite.

### Step 5: Equate the Forces
Setting the gravitational force equal to the centripetal force, we have:
\[ \frac{GMm}{(5R)^2} = \frac{mv_0^2}{5R} \]

### Step 6: Simplify the Equation
Cancelling \(m\) from both sides and simplifying gives:
\[ \frac{GM}{25R^2} = \frac{v_0^2}{5R} \]
Multiplying both sides by \(5R\):
\[ \frac{GM}{5R} = v_0^2 \]

### Step 7: Calculate the Kinetic Energy
The kinetic energy \(K\) of the satellite is given by:
\[ K = \frac{1}{2}mv_0^2 \]
Substituting \(v_0^2\) from the previous step:
\[ K = \frac{1}{2}m \left(\frac{GM}{5R}\right) \]
\[ K = \frac{GMm}{10R} \]

### Step 8: Determine the Total Energy
The total mechanical energy \(E\) of the satellite in orbit is the sum of its kinetic energy and potential energy \(U\):
\[ U = -\frac{GMm}{r} = -\frac{GMm}{5R} \]
Thus, the total energy \(E\) is:
\[ E = K + U = \frac{GMm}{10R} - \frac{GMm}{5R} \]
\[ E = \frac{GMm}{10R} - \frac{2GMm}{10R} = -\frac{GMm}{10R} \]

### Step 9: Minimum Kinetic Energy to Escape
To escape the gravitational field, the total energy must be zero. Therefore, we need to provide additional kinetic energy \(E\) such that:
\[ E + K = 0 \]
\[ K = \frac{GMm}{10R} \]

Thus, the minimum kinetic energy required to be provided to the satellite is:
\[ \boxed{\frac{GMm}{10R}} \]