A satellite of mass `m` is just placed over the surface of earth. In this position mechanical energy of satellite is `E_(1)`. Now it starts orbiting round the earth in a circular path at height `h =` radius of earth. In this position, kinetic energy potential energy and total mechanical energy of satellite are `K_(2)`, `U_(2)` and `E_(2)` respectively. Then

Options

A. `U_(2) = (E_(1))/(2)`
B. `E_(2) = (E_(1))/(4)`
C. `K_(2) = - E_(2)`
D. `K_(2) = - (U_(2))/(2)`

Explanation

To solve the problem step by step, we will analyze the situation of the satellite at two different positions: when it is just placed on the surface of the Earth and when it is in orbit at a height equal to the radius of the Earth.

### Step 1: Calculate the Mechanical Energy when the Satellite is on the Surface of the Earth

When the satellite is just placed over the surface of the Earth, its total mechanical energy \( E_1 \) can be expressed as the sum of its potential energy \( U_1 \) and kinetic energy \( K_1 \).

1. **Potential Energy \( U_1 \)**:
\[ U_1 = -\frac{G M m}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( R \) is the radius of the Earth.

2. **Kinetic Energy \( K_1 \)**:
Since the satellite is at rest when placed on the surface, its kinetic energy is:
\[ K_1 = 0 \]

3. **Total Mechanical Energy \( E_1 \)**:
\[ E_1 = U_1 + K_1 = -\frac{G M m}{R} + 0 = -\frac{G M m}{R} \]

### Step 2: Calculate the Mechanical Energy when the Satellite is in Orbit

Now, we consider the satellite in a circular orbit at a height equal to the radius of the Earth.

1. **Height \( h \)**:
The height \( h \) is equal to the radius of the Earth, so the distance from the center of the Earth to the satellite is \( R + h = R + R = 2R \).

2. **Potential Energy \( U_2 \)**:
The potential energy when the satellite is in orbit is:
\[ U_2 = -\frac{G M m}{2R} \]

3. **Kinetic Energy \( K_2 \)**:
The orbital velocity \( v \) of the satellite can be calculated using the formula for circular motion:
\[ v = \sqrt{\frac{G M}{r}} = \sqrt{\frac{G M}{2R}} \]
The kinetic energy \( K_2 \) is then:
\[ K_2 = \frac{1}{2} m v^2 = \frac{1}{2} m \left(\frac{G M}{2R}\right) = \frac{G M m}{4R} \]

4. **Total Mechanical Energy \( E_2 \)**:
The total mechanical energy \( E_2 \) when the satellite is in orbit is:
\[ E_2 = U_2 + K_2 = -\frac{G M m}{2R} + \frac{G M m}{4R} \]
Simplifying this gives:
\[ E_2 = -\frac{G M m}{2R} + \frac{G M m}{4R} = -\frac{2G M m}{4R} + \frac{G M m}{4R} = -\frac{G M m}{4R} \]

### Step 3: Establish Relationships

Now we can establish the relationships between \( U_2 \), \( E_2 \), \( K_2 \), and \( E_1 \):

1. **Relationship between \( U_2 \) and \( E_1 \)**:
\[ U_2 = \frac{1}{2} E_1 \]

2. **Relationship between \( E_2 \) and \( E_1 \)**:
\[ E_2 = \frac{1}{2} E_1 \]

3. **Relationship between \( K_2 \) and \( E_2 \)**:
\[ K_2 = -\frac{1}{2} E_2 \]

4. **Relationship between \( K_2 \) and \( U_2 \)**:
\[ K_2 = -\frac{1}{2} U_2 \]