The ratio of the distance of two planets from the sun is 1:2. The ratio of their periods of revolution is

Options

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

Answer

To solve the problem of finding the ratio of the periods of revolution of two planets based on their distances from the sun, we can use Kepler's Third Law of planetary motion. Here’s a step-by-step solution:

### Step 1: Understand Kepler's Third Law
Kepler's Third Law 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:
\[ T^2 \propto r^3 \]
This can also be written as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

### Step 2: Set Up the Ratios
From the problem, we know the ratio of the distances of the two planets from the sun:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]

### Step 3: Apply the Ratio to Kepler's Law
Using the ratio of the distances in Kepler's Third Law, we can write:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

### Step 4: Substitute the Values
Substituting the ratio of the distances into the equation:
\[ \frac{T_1^2}{T_2^2} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

### Step 5: Find the Ratio of the Periods
Taking the square root of both sides to find the ratio of the periods:
\[ \frac{T_1}{T_2} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} \]

### Step 6: Simplify the Ratio
To express the ratio in a more standard form, we can rationalize the denominator:
\[ \frac{T_1}{T_2} = \frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{4} \]

Thus, the ratio of the periods of revolution of the two planets is:
\[ T_1 : T_2 = \sqrt{2} : 4 \]

### Final Answer
The ratio of their periods of revolution is \( 1 : 2\sqrt{2} \).

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