If an artificial satellite is moving in a circular orbit around earth with speed equal to one fourth of `V_(e)` from earth, then height of the satellite above the surface of the earth is

Options

A. 3R
B. 5R
C. 7R
D. 8R

Answer

To solve the problem of finding the height of an artificial satellite above the surface of the Earth when it is moving with a speed equal to one-fourth of the escape velocity, we can follow these steps:

### Step 1: Understand the given information
- The speed of the satellite \( V_o \) is \( \frac{1}{4} V_e \), where \( V_e \) is the escape velocity.
- The radius of the Earth is denoted as \( R \).
- The height of the satellite above the Earth's surface is denoted as \( h \).

### Step 2: Write the formula for orbital velocity
The orbital velocity \( V_o \) of a satellite in a circular orbit is given by the formula:
\[ V_o = \sqrt{\frac{GM}{R + h}} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

### Step 3: Write the formula for escape velocity
The escape velocity \( V_e \) is given by:
\[ V_e = \sqrt{\frac{2GM}{R}} \]
Squaring both sides gives:
\[ V_e^2 = \frac{2GM}{R} \]

### Step 4: Substitute \( V_o \) into the orbital velocity formula
Since \( V_o = \frac{1}{4} V_e \), we can substitute this into the orbital velocity equation:
\[ \left(\frac{1}{4} V_e\right)^2 = \frac{GM}{R + h} \]
This simplifies to:
\[ \frac{V_e^2}{16} = \frac{GM}{R + h} \]

### Step 5: Substitute \( V_e^2 \) from the escape velocity formula
From the escape velocity formula, we know:
\[ V_e^2 = \frac{2GM}{R} \]
Substituting this into the equation gives:
\[ \frac{1}{16} \cdot \frac{2GM}{R} = \frac{GM}{R + h} \]

### Step 6: Simplify the equation
Cancelling \( GM \) from both sides, we get:
\[ \frac{2}{16R} = \frac{1}{R + h} \]
This simplifies to:
\[ \frac{1}{8R} = \frac{1}{R + h} \]

### Step 7: Cross-multiply to solve for \( h \)
Cross-multiplying gives:
\[ R + h = 8R \]
Thus:
\[ h = 8R - R = 7R \]

### Conclusion
The height of the satellite above the surface of the Earth is \( h = 7R \).

### Final Answer
The height of the satellite above the surface of the Earth is \( 7R \).

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