A planet is revolving in an elliptical orbit around the sun. Its closest distance from the sun is r and the farthest distance is R. If the velocity of the planet nearest to the sun be v and that farthest away from the sun be V. then v/V is

Options

A. `R^2//r^2`
B. `r^2//R^2`
C. `R//r`
D. `r//R`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum of a planet revolving around the Sun is given by the product of its mass, its velocity, and its distance from the Sun.

### Step-by-Step Solution:

1. **Identify the positions and velocities:**
- The closest distance from the Sun (perihelion) is \( r \) with velocity \( v \).
- The farthest distance from the Sun (aphelion) is \( R \) with velocity \( V \).

2. **Write the expression for angular momentum:**
- Angular momentum at the closest point (perihelion) is given by:
\[ L_{perihelion} = M \cdot v \cdot r \]
- Angular momentum at the farthest point (aphelion) is given by:
\[ L_{aphelion} = M \cdot V \cdot R \]
Here, \( M \) is the mass of the planet.

3. **Apply the conservation of angular momentum:**
- Since there are no external torques acting on the planet-sun system, the angular momentum at perihelion must equal the angular momentum at aphelion:
\[ M \cdot v \cdot r = M \cdot V \cdot R \]

4. **Cancel the mass \( M \):**
- Since \( M \) is present on both sides of the equation, we can cancel it out:
\[ v \cdot r = V \cdot R \]

5. **Rearrange the equation to find the ratio \( \frac{v}{V} \):**
- We can rearrange the equation to express the ratio of the velocities:
\[ \frac{v}{V} = \frac{R}{r} \]

6. **Conclusion:**
- Therefore, the ratio of the velocity of the planet at its closest distance to the velocity at its farthest distance is:
\[ \frac{v}{V} = \frac{R}{r} \]

### Final Answer:
The ratio \( \frac{v}{V} \) is \( \frac{R}{r} \).