The time period of an artificial satellite in a circular orbit of radius `R` is `2` days and its orbital velocity is `v_(0)`. If time period of another satellite in a circular orbit is `16` days then

Options

A. its radius of orbit is 4R and orbital velocity is `v_(0)`
B. its radius of orbit is 4R and orbital velocity is `(v_(0))/(2)`
C. its radius of orbit is 2R and orbital velocity is `v_(0)`
D. its radius of orbit is 2R and orbital velocity is `(v_(0))/(2)`

Explanation

To solve the problem step by step, we will use Kepler's third law of planetary motion, which states that the square of the time period of orbiting bodies is directly proportional to the cube of the semi-major axis (or radius in the case of circular orbits).

### Step 1: Understand the relationship between time period and radius
From Kepler's third law, we know that:
\[ T^2 \propto R^3 \]
This can be expressed as:
\[ \frac{T_1^2}{R_1^3} = \frac{T_2^2}{R_2^3} \]

### Step 2: Set the known values
Let:
- \( T_1 = 2 \) days
- \( R_1 = R \) (the radius of the first satellite)
- \( T_2 = 16 \) days
- \( R_2 \) (the radius of the second satellite, which we need to find)

### Step 3: Substitute the values into the equation
Substituting the known values into the equation:
\[ \frac{(2)^2}{R^3} = \frac{(16)^2}{R_2^3} \]

### Step 4: Simplify the equation
Calculating the squares:
\[ \frac{4}{R^3} = \frac{256}{R_2^3} \]

### Step 5: Cross-multiply to solve for \( R_2^3 \)
Cross-multiplying gives:
\[ 4R_2^3 = 256R^3 \]

### Step 6: Solve for \( R_2^3 \)
Dividing both sides by 4:
\[ R_2^3 = 64R^3 \]

### Step 7: Take the cube root to find \( R_2 \)
Taking the cube root of both sides:
\[ R_2 = 4R \]

### Step 8: Find the orbital velocity
The orbital velocity \( v \) of a satellite in circular orbit is given by:
\[ v = \sqrt{\frac{GM}{R}} \]
For the first satellite:
\[ v_0 = \sqrt{\frac{GM}{R}} \]
For the second satellite with radius \( R_2 = 4R \):
\[ v_2 = \sqrt{\frac{GM}{4R}} = \frac{1}{2} \sqrt{\frac{GM}{R}} = \frac{1}{2} v_0 \]

### Final Results
- The radius of the second satellite's orbit \( R_2 = 4R \)
- The orbital velocity of the second satellite \( v_2 = \frac{1}{2} v_0 \)