A planet moves around the sun. It is closest to sun to sun at a distance `d_(1)` and have velocity `v_(1)` At farthest distance `d_(2)` its speed will be

Options

A. `(d_(1)^(2)v_(1))/(d_(2)^(2))`
B. `(d_(2)v_(1))/(d_(1))`
C. `(d_(1)v_(1))/(d_(2))`
D. `(d_(2)^(2)v_(1))/(d_(1)^(2))`

Explanation

To solve the problem of finding the velocity of a planet at its farthest distance from the sun, we can use the principle of conservation of angular momentum. Here's the step-by-step solution:

### Step 1: Understand the Concept of Angular Momentum
Angular momentum (L) of a planet moving around the sun is given by the formula:
\[ L = m \cdot v \cdot r \]
where:
- \( m \) is the mass of the planet,
- \( v \) is the velocity of the planet,
- \( r \) is the distance from the sun.

### Step 2: Set Up the Problem
At the closest distance to the sun (\( d_1 \)), the planet has a velocity (\( v_1 \)). At the farthest distance (\( d_2 \)), we need to find the velocity (\( v_2 \)).

### Step 3: Apply Conservation of Angular Momentum
According to the conservation of angular momentum:
\[ L_1 = L_2 \]
This means:
\[ m \cdot v_1 \cdot d_1 = m \cdot v_2 \cdot d_2 \]

### Step 4: Simplify the Equation
Since the mass \( m \) of the planet is constant and appears on both sides of the equation, we can cancel it out:
\[ v_1 \cdot d_1 = v_2 \cdot d_2 \]

### Step 5: Solve for \( v_2 \)
Rearranging the equation to solve for \( v_2 \):
\[ v_2 = \frac{v_1 \cdot d_1}{d_2} \]

### Conclusion
Thus, the velocity of the planet at its farthest distance \( d_2 \) is given by:
\[ v_2 = \frac{v_1 \cdot d_1}{d_2} \]

### Final Answer
The correct option is \( \frac{d_1 \cdot v_1}{d_2} \).

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