The earth revolves around the sun in an elliptical orbit. It has a speed `v_(1)` when it is at the minimum distance `d_(1)` from the sun. When it is at the maximum distance `d_(2)` from the sun, its speed is

Options

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

Explanation

To solve the problem of finding the speed of the Earth when it is at the maximum distance \( d_2 \) from the Sun, we can use the principle of conservation of angular momentum. Here's a step-by-step solution:

### Step 1: Understand the system
The Earth revolves around the Sun in an elliptical orbit. At the minimum distance \( d_1 \) from the Sun, the speed of the Earth is \( v_1 \). We need to find the speed \( v_2 \) when the Earth is at the maximum distance \( d_2 \).

### Step 2: Apply the conservation of angular momentum
Since no external torque acts on the Earth-Sun system, the angular momentum is conserved. Therefore, we can write:

\[ L_1 = L_2 \]

Where \( L_1 \) is the angular momentum at distance \( d_1 \) and \( L_2 \) is the angular momentum at distance \( d_2 \).

### Step 3: Express angular momentum
The angular momentum \( L \) can be expressed in terms of linear velocity and distance from the Sun:

\[ L = m \cdot r \cdot v \]

Where:
- \( m \) is the mass of the Earth,
- \( r \) is the distance from the Sun,
- \( v \) is the linear velocity.

Thus, we can write:

\[ L_1 = m \cdot d_1 \cdot v_1 \]
\[ L_2 = m \cdot d_2 \cdot v_2 \]

### Step 4: Set the angular momentum equations equal
From conservation of angular momentum:

\[ m \cdot d_1 \cdot v_1 = m \cdot d_2 \cdot v_2 \]

### Step 5: Simplify the equation
Since the mass \( m \) of the Earth is constant and non-zero, we can cancel it from both sides:

\[ d_1 \cdot v_1 = d_2 \cdot v_2 \]

### Step 6: Solve for \( v_2 \)
Rearranging the equation to solve for \( v_2 \):

\[ v_2 = \frac{d_1}{d_2} \cdot v_1 \]

### Conclusion
Thus, the speed of the Earth at the maximum distance \( d_2 \) from the Sun is given by:

\[ v_2 = \frac{d_1}{d_2} \cdot v_1 \]