A satellite of mass `m` is orbiting the earth in a cicular orbit of radius `r`. It starts losing its mechanical energy due to small air resistance at the rate of `k` joule/sec. The time taken by the satellite to hit the suface of the earth is (`M` and `R` are the mass and radius of the earth)

Options

A. `(GMm)/K(1/R-1/r)`
B. `(GMm)/(2K)(1/R-1/r)`
C. `(2GMm)/K(1/R-1/r)`
D. `(GMm)/K(1/R+1/r)`

Explanation

To solve the problem step by step, we will analyze the energy changes of the satellite as it loses mechanical energy due to air resistance and calculate the time taken for it to hit the surface of the Earth.

### Step 1: Understand the Initial and Final Energy of the Satellite

1. **Initial Energy (Ei)**: The gravitational potential energy of the satellite in a circular orbit of radius \( r \) is given by:
\[ E_i = -\frac{G M m}{2r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the satellite.

2. **Final Energy (Ef)**: When the satellite hits the surface of the Earth (radius \( R \)), its energy is:
\[ E_f = -\frac{G M m}{2R} \]

### Step 2: Calculate the Change in Energy (ΔE)

The change in energy (ΔE) as the satellite moves from its initial orbit to the surface of the Earth is:
\[ \Delta E = E_i - E_f = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{2r}\right) \]
This simplifies to:
\[ \Delta E = -\frac{G M m}{2R} + \frac{G M m}{2r} = \frac{G M m}{2} \left(\frac{1}{r} - \frac{1}{R}\right) \]

### Step 3: Relate Energy Loss to Time

The satellite is losing energy at a rate of \( k \) joules per second. Therefore, we can relate the change in energy to the time taken \( T \) to hit the surface of the Earth:
\[ k = -\frac{\Delta E}{T} \]
This implies:
\[ T = \frac{\Delta E}{k} \]

### Step 4: Substitute ΔE into the Time Equation

Substituting the expression for ΔE into the equation for T gives:
\[ T = \frac{\frac{G M m}{2} \left(\frac{1}{r} - \frac{1}{R}\right)}{k} \]
Thus, the time taken by the satellite to hit the surface of the Earth is:
\[ T = \frac{G M m}{2k} \left(\frac{1}{r} - \frac{1}{R}\right) \]

### Final Answer

The time taken by the satellite to hit the surface of the Earth is:
\[ T = \frac{G M m}{2k} \left(\frac{1}{r} - \frac{1}{R}\right) \]