An earth satellite of mass `m` revolves in a circular orbit at a height `h` from the surface of the earth. R is the radius of the earth and `g` is acceleration due to gravity at the surface of the earth. The velocity of the satellite in the orbit is given by

Options

A. `(gR^(2))/(R+h)`
B. `gR`
C. `(gR)/(R+h)`
D. `sqrt((gR^(2))/(R+h))`

Explanation

To find the velocity of a satellite in a circular orbit at a height \( h \) from the surface of the Earth, we can follow these steps:

### Step 1: Understand the gravitational force acting on the satellite
The gravitational force acting on the satellite of mass \( m \) at a height \( h \) from the Earth's surface is given by Newton's law of gravitation:
\[ F = \frac{G M m}{(R + h)^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Set up the centripetal force equation
For the satellite to maintain its circular orbit, the gravitational force must provide the necessary centripetal force:
\[ F = \frac{m v^2}{R + h} \]
where \( v \) is the orbital velocity of the satellite.

### Step 3: Equate gravitational force to centripetal force
Setting the gravitational force equal to the centripetal force gives:
\[ \frac{G M m}{(R + h)^2} = \frac{m v^2}{R + h} \]

### Step 4: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G M}{(R + h)^2} = \frac{v^2}{R + h} \]
Multiplying both sides by \( (R + h) \):
\[ \frac{G M}{R + h} = v^2 \]

### Step 5: Solve for \( v \)
Taking the square root of both sides gives us the expression for the velocity of the satellite:
\[ v = \sqrt{\frac{G M}{R + h}} \]

### Step 6: Relate \( G \) and \( g \)
We know that the acceleration due to gravity at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
From this, we can express \( G M \) as:
\[ G M = g R^2 \]

### Step 7: Substitute \( G M \) into the velocity equation
Substituting \( G M \) into the velocity equation gives:
\[ v = \sqrt{\frac{g R^2}{R + h}} \]

### Final Answer
Thus, the velocity of the satellite in its orbit is:
\[ v = \sqrt{\frac{g R^2}{R + h}} \]