If earth is `1//4` of its present distance formt he sun, then what is the duration of the year?

Explanation

To solve the problem of finding the duration of the year if the Earth is at \( \frac{1}{4} \) of its present distance from the Sun, we can use Kepler's Third Law of planetary motion. This 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.

### Step-by-step Solution:

1. **Identify the Variables**:
- Let \( T_1 \) be the current period of the Earth (1 year).
- Let \( R_1 \) be the current distance from the Earth to the Sun (1 astronomical unit, AU).
- Let \( T_2 \) be the new period we want to find.
- Let \( R_2 \) be the new distance from the Earth to the Sun, which is \( \frac{1}{4} R_1 \).

2. **Apply Kepler's Third Law**:
According to Kepler's Third Law:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

3. **Substituting Known Values**:
We know \( T_1 = 1 \) year and \( R_2 = \frac{1}{4} R_1 \). Therefore:
\[ R_2 = \frac{1}{4} R_1 \]
Thus, we can express \( R_2^3 \):
\[ R_2^3 = \left(\frac{1}{4} R_1\right)^3 = \frac{1}{64} R_1^3 \]

4. **Setting Up the Equation**:
Substitute \( R_2^3 \) into the equation:
\[ \frac{1^2}{T_2^2} = \frac{R_1^3}{\frac{1}{64} R_1^3} \]
Simplifying gives:
\[ \frac{1}{T_2^2} = 64 \]

5. **Solving for \( T_2^2 \)**:
Taking the reciprocal:
\[ T_2^2 = \frac{1}{64} \]

6. **Finding \( T_2 \)**:
Taking the square root:
\[ T_2 = \frac{1}{8} \text{ years} \]

### Final Answer:
The duration of the year when the Earth is \( \frac{1}{4} \) of its present distance from the Sun is \( \frac{1}{8} \) years.

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