The ratio of the distance of two planets from the sun is `1:2`. Then ratio of their priods of revolutions is

Options

A. `1:4`
B. `1:sqrt2`
C. `1:2`
D. `1:2sqrt2`

Explanation

To solve the problem, we will use Kepler's Third Law of planetary motion, which states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit around the sun. Mathematically, this can be expressed as:

\[ T^2 \propto R^3 \]

Given that the ratio of the distances of two planets from the sun is \( R_1 : R_2 = 1 : 2 \), we can denote the distances as:

\[ R_1 = 1 \text{ unit} \]
\[ R_2 = 2 \text{ units} \]

Now, we will find the ratio of their periods of revolution \( T_1 : T_2 \).

### Step 1: Write Kepler's Third Law for both planets
From Kepler's Third Law, we have:

\[ T_1^2 \propto R_1^3 \]
\[ T_2^2 \propto R_2^3 \]

### Step 2: Express the proportionality in terms of a constant
We can express these relationships as:

\[ T_1^2 = k R_1^3 \]
\[ T_2^2 = k R_2^3 \]

where \( k \) is a constant of proportionality.

### Step 3: Substitute the values of \( R_1 \) and \( R_2 \)
Substituting \( R_1 = 1 \) and \( R_2 = 2 \):

\[ T_1^2 = k (1)^3 = k \]
\[ T_2^2 = k (2)^3 = k \cdot 8 \]

### Step 4: Find the ratio \( T_1^2 : T_2^2 \)
Now, we can find the ratio of the squares of the periods:

\[ \frac{T_1^2}{T_2^2} = \frac{k}{8k} = \frac{1}{8} \]

### Step 5: Take the square root to find the ratio of the periods
To find the ratio of the periods \( T_1 : T_2 \), we take the square root of both sides:

\[ \frac{T_1}{T_2} = \sqrt{\frac{1}{8}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} \]

### Final Ratio
Thus, the ratio of their periods of revolution is:

\[ T_1 : T_2 = 1 : 2\sqrt{2} \]