An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth. The height of the satellite above the surface of the earth is x R. Find the value of x.

Answer

To solve the problem, we need to find the height \( h \) of the satellite above the surface of the Earth in terms of the Earth's radius \( R \). The satellite is moving with a speed equal to half the escape velocity from the Earth.

### Step 1: Write down the formula for escape velocity
The escape velocity \( V_E \) from the surface of the Earth is given by the formula:
\[ V_E = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Determine the orbital velocity \( V_O \)
The problem states that the satellite's speed \( V_O \) is half the escape velocity:
\[ V_O = \frac{1}{2} V_E = \frac{1}{2} \sqrt{2gR} = \frac{\sqrt{gR}}{\sqrt{2}} \]

### Step 3: Write the formula for orbital velocity in terms of height
The orbital velocity \( V_O \) of a satellite in a circular orbit at height \( h \) above the surface of the Earth can be expressed as:
\[ V_O = \sqrt{\frac{GM_E}{R + h}} \]
where \( G \) is the gravitational constant and \( M_E \) is the mass of the Earth.

### Step 4: Relate \( GM_E \) to \( g \)
Using the relation \( g = \frac{GM_E}{R^2} \), we can express \( GM_E \) as:
\[ GM_E = gR^2 \]

### Step 5: Substitute \( GM_E \) into the orbital velocity equation
Substituting \( GM_E \) into the orbital velocity equation gives:
\[ V_O = \sqrt{\frac{gR^2}{R + h}} \]

### Step 6: Set the two expressions for \( V_O \) equal to each other
Now we have two expressions for \( V_O \):
\[ \frac{\sqrt{gR}}{\sqrt{2}} = \sqrt{\frac{gR^2}{R + h}} \]

### Step 7: Square both sides to eliminate the square roots
Squaring both sides results in:
\[ \frac{gR}{2} = \frac{gR^2}{R + h} \]

### Step 8: Simplify the equation
Cancelling \( g \) from both sides (assuming \( g \neq 0 \)):
\[ \frac{R}{2} = \frac{R^2}{R + h} \]

### Step 9: Cross-multiply to solve for \( h \)
Cross-multiplying gives:
\[ R(R + h) = 2R^2 \]
Expanding and rearranging:
\[ Rh + R^2 = 2R^2 \]
\[ Rh = 2R^2 - R^2 \]
\[ Rh = R^2 \]

### Step 10: Solve for \( h \)
Dividing both sides by \( R \):
\[ h = R \]

### Step 11: Find the height above the surface of the Earth
Since \( h \) is the height above the surface of the Earth and the radius of the Earth is \( R \), the height \( h \) above the surface is:
\[ h = R \]
Thus, the value of \( x \) (where \( h = xR \)) is:
\[ x = 1 \]

### Final Answer
The value of \( x \) is \( 1 \).

---