A comet revolves around the sun in an eliptical orbit. When it is closest to the sun at a distance d, its corresponding kinetic energy is `k_(0)`. If it is farthest from the sun at distance 3d then the corresponding kinetic energy will be

Options

A. `(k_(0))/(9)`
B. `(8k_(0))/(9)`
C. `(k_(0))/(4)`
D. `(4k_(0))/(9)`

Answer

To solve the problem, we need to analyze the kinetic energy of the comet at two different points in its elliptical orbit around the sun.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The comet is closest to the sun at a distance \( d \) and has a kinetic energy \( K_0 \).
- At its farthest point from the sun, the distance is \( 3d \).
- We need to find the kinetic energy at this farthest point.

2. **Using Kepler's Second Law**:
- Kepler's second law states that the line joining a planet (or comet) to the sun sweeps out equal areas in equal times. This implies that the product of the radius and the velocity of the comet at different points in its orbit is constant.
- Mathematically, this can be expressed as:
\[ r_1 v_1 = r_2 v_2 \]
- Where \( r_1 \) and \( v_1 \) are the radius and velocity at the closest point (distance \( d \)), and \( r_2 \) and \( v_2 \) are the radius and velocity at the farthest point (distance \( 3d \)).

3. **Setting Up the Equation**:
- At the closest point:
\[ r_1 = d, \quad v_1 = v \]
- At the farthest point:
\[ r_2 = 3d, \quad v_2 = ? \]
- According to Kepler's second law:
\[ d \cdot v = 3d \cdot v_2 \]
- Dividing both sides by \( d \) (assuming \( d \neq 0 \)):
\[ v = 3v_2 \implies v_2 = \frac{v}{3} \]

4. **Calculating Kinetic Energy**:
- The kinetic energy \( K \) is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
- At the closest point, the kinetic energy is:
\[ K_0 = \frac{1}{2} m v^2 \]
- At the farthest point, substituting \( v_2 \):
\[ K_2 = \frac{1}{2} m v_2^2 = \frac{1}{2} m \left(\frac{v}{3}\right)^2 = \frac{1}{2} m \frac{v^2}{9} = \frac{1}{9} \left(\frac{1}{2} mv^2\right) = \frac{K_0}{9} \]

5. **Conclusion**:
- Therefore, the kinetic energy of the comet when it is farthest from the sun at a distance \( 3d \) is:
\[ K_2 = \frac{K_0}{9} \]

### Final Answer:
The corresponding kinetic energy when the comet is farthest from the sun at a distance \( 3d \) is \( \frac{K_0}{9} \).

---