A satellite of mass m is orbiting the earth of radius R at) a height h from its surface. The total energy of the satellite in terms of `g_(0)` the value of acceleration due to gravity at the earth 's surface, is

Options

A. `(2 mg_(0) R^(2))/(R + h)`
B. `(2 mg_(0) R^(2))/(R + h)`
C. `(mg_(0) R^(2))/ (2 R + h)`
D. `(mg_(0) R^(2))/(2 (R + h))`

Explanation

To find the total energy of a satellite of mass \( m \) orbiting the Earth at a height \( h \) from its surface, we can follow these steps:

### Step 1: Understand the variables
- Let \( R \) be the radius of the Earth.
- The height of the satellite above the Earth's surface is \( h \).
- The distance from the center of the Earth to the satellite is \( r = R + h \).

### Step 2: Write the formula for total energy
The total mechanical energy \( E \) of a satellite in orbit is given by the formula:
\[ E = -\frac{G M m}{2r} \]
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 to the satellite.

### Step 3: Substitute for \( r \)
Substituting \( r \) with \( R + h \):
\[ E = -\frac{G M m}{2(R + h)} \]

### Step 4: Relate \( G \) and \( g_0 \)
We know that the acceleration due to gravity at the Earth's surface \( g_0 \) is given by:
\[ g_0 = \frac{G M}{R^2} \]
From this, we can express \( G M \) as:
\[ G M = g_0 R^2 \]

### Step 5: Substitute \( G M \) into the energy equation
Now, substituting \( G M \) into the total energy equation:
\[ E = -\frac{g_0 R^2 m}{2(R + h)} \]

### Step 6: Final expression for total energy
Thus, the total energy of the satellite in terms of \( g_0 \) is:
\[ E = -\frac{m g_0 R^2}{2(R + h)} \]

### Summary
The total energy of the satellite in terms of \( g_0 \) is:
\[ E = -\frac{m g_0 R^2}{2(R + h)} \]