A plenet moving along an elliptical orbit is closest to the sun at a distance `r_(1)` and farthest away at a distance of `r_(2)`. If `v_(1)` and `v_(2)` are the linear velocities at these points respectively, then the ratio `(v_(1))/(v_(2))` is

Options

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

Explanation

To find the ratio of the linear velocities \( \frac{v_1}{v_2} \) of a planet moving in an elliptical orbit around 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 (\( L \)) for a planet in orbit is given by the formula:
\[ L = m \cdot v \cdot r \]
where:
- \( m \) is the mass of the planet,
- \( v \) is the linear velocity of the planet,
- \( r \) is the distance from the Sun (the radius of the orbit).

### Step 2: Set Up the Angular Momentum Equations
At the closest point (perihelion), the angular momentum is:
\[ L_1 = m \cdot v_1 \cdot r_1 \]
At the farthest point (aphelion), the angular momentum is:
\[ L_2 = m \cdot v_2 \cdot r_2 \]

### Step 3: Apply Conservation of Angular Momentum
Since angular momentum is conserved, we can equate \( L_1 \) and \( L_2 \):
\[ m \cdot v_1 \cdot r_1 = m \cdot v_2 \cdot r_2 \]

### Step 4: Simplify the Equation
We can cancel the mass \( m \) from both sides of the equation (assuming it is constant):
\[ v_1 \cdot r_1 = v_2 \cdot r_2 \]

### Step 5: Solve for the Ratio of Velocities
To find the ratio \( \frac{v_1}{v_2} \), we can rearrange the equation:
\[ \frac{v_1}{v_2} = \frac{r_2}{r_1} \]

### Final Result
Thus, the ratio of the linear velocities at the closest and farthest points in the orbit is:
\[ \frac{v_1}{v_2} = \frac{r_2}{r_1} \]

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