A body is orbiting very close to the earth surface kinetic energy K.E. The energy required to completely escape from it is

Options

A. K
B. 2K
C. `K/2`
D. `(3k)/2`

Answer

To solve the problem step by step, we need to analyze the situation of a body orbiting close to the Earth's surface and determine the energy required for it to escape from that orbit.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- A body is orbiting very close to the Earth's surface. We denote the mass of the Earth as \( M \) and the radius of the Earth as \( R \).
- The gravitational force provides the centripetal force necessary for the orbit.

2. **Setting Up the Forces**:
- The gravitational force acting on the body is given by:
\[ F_g = \frac{GMm}{R^2} \]
where \( G \) is the gravitational constant and \( m \) is the mass of the orbiting body.

- The centripetal force required to keep the body in circular motion is:
\[ F_c = \frac{mv^2}{R} \]

3. **Equating Forces**:
- At the point of orbit, the gravitational force equals the centripetal force:
\[ \frac{GMm}{R^2} = \frac{mv^2}{R} \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{GM}{R^2} = \frac{v^2}{R} \]
- Rearranging gives:
\[ v^2 = \frac{GM}{R} \]

4. **Calculating Kinetic Energy**:
- The kinetic energy \( K.E. \) of the body is given by:
\[ K.E. = \frac{1}{2} mv^2 \]
- Substituting for \( v^2 \):
\[ K.E. = \frac{1}{2} m \left(\frac{GM}{R}\right) = \frac{GMm}{2R} \]

5. **Calculating Gravitational Potential Energy**:
- The gravitational potential energy \( U \) at a distance \( R \) from the center of the Earth is:
\[ U = -\frac{GMm}{R} \]

6. **Finding Total Energy**:
- The total mechanical energy \( E \) of the system is the sum of kinetic and potential energy:
\[ E = K.E. + U = \frac{GMm}{2R} - \frac{GMm}{R} \]
- Simplifying this gives:
\[ E = \frac{GMm}{2R} - \frac{2GMm}{2R} = -\frac{GMm}{2R} \]

7. **Energy Required to Escape**:
- The energy required to escape from the gravitational field (to reach zero potential energy) is equal to the negative of the total energy:
\[ \text{Energy required to escape} = -E = \frac{GMm}{2R} \]
- Since we found earlier that \( K.E. = \frac{GMm}{2R} \), we conclude:
\[ \text{Energy required to escape} = K.E. \]

### Final Answer:
The energy required to completely escape from the orbit is equal to the kinetic energy \( K.E. \).