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. `sqrt(2) K.E`
B. `2K.E.`
C. `K.E. sqrt(2)`
D. None of these

Explanation

To find the energy required for a body orbiting very close to the Earth's surface to completely escape from it, we can follow these steps:

### Step 1: Understand the Kinetic Energy of the Orbiting Body
The kinetic energy (K.E.) of a body in orbit close to the Earth's surface can be expressed as:
\[ K.E. = \frac{1}{2} m v^2 \]
where \( v \) is the orbital velocity.

### Step 2: Determine the Orbital Velocity
The orbital velocity \( v \) for a body in a circular orbit close to the Earth's surface is given by:
\[ v = \sqrt{\frac{G M_e}{R}} \]
where \( G \) is the gravitational constant, \( M_e \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 3: Substitute the Orbital Velocity into the Kinetic Energy Formula
Substituting the expression for \( v \) into the kinetic energy formula:
\[ K.E. = \frac{1}{2} m \left(\sqrt{\frac{G M_e}{R}}\right)^2 = \frac{1}{2} m \frac{G M_e}{R} \]

### Step 4: Calculate the Gravitational Potential Energy
The gravitational potential energy (U) of the body at the Earth's surface is given by:
\[ U = -\frac{G M_e m}{R} \]

### Step 5: Determine the Total Energy of the Orbiting Body
The total mechanical energy (E) of the body in orbit is the sum of its kinetic energy and potential energy:
\[ E = K.E. + U \]
Substituting the expressions we have:
\[ E = \frac{1}{2} m \frac{G M_e}{R} - \frac{G M_e m}{R} \]
\[ E = \frac{1}{2} m \frac{G M_e}{R} - \frac{2}{2} m \frac{G M_e}{R} \]
\[ E = -\frac{1}{2} m \frac{G M_e}{R} \]

### Step 6: Energy Required to Escape
To escape the gravitational field, the total energy must be zero. Therefore, the energy required (ΔE) to escape from the orbit can be calculated as:
\[ \Delta E = 0 - E = \frac{1}{2} m \frac{G M_e}{R} \]

### Step 7: Relate the Required Energy to the Initial Kinetic Energy
Since we have already established that:
\[ K.E. = \frac{1}{2} m \frac{G M_e}{R} \]
Thus, the energy required to escape is equal to the initial kinetic energy:
\[ \Delta E = K.E. \]

### Conclusion
The energy required to completely escape from the gravitational field of the Earth for a body orbiting very close to the Earth's surface is equal to its initial kinetic energy.