Which of the diagrams shown in figure. Most closely shows the variation inkinetic energy of the earth as it moves once around the sun in its elliptical orbit ?

Options

A.
B.
C.
D.

Explanation

To determine the variation in kinetic energy of the Earth as it moves once around the Sun in its elliptical orbit, we can follow these steps:

### Step 1: Understanding the Orbit
The Earth travels in an elliptical orbit around the Sun. In this orbit, there are two key points to consider:
- **Perigee (P)**: The point where the Earth is closest to the Sun.
- **Apogee (A)**: The point where the Earth is farthest from the Sun.

### Step 2: Angular Momentum Conservation
The angular momentum of the Earth as it orbits the Sun is conserved. This means that the product of the radius and the velocity at these two points must be equal:
\[ m \cdot r_P \cdot v_P = m \cdot r_A \cdot v_A \]
Where:
- \( m \) is the mass of the Earth (which cancels out),
- \( r_P \) and \( r_A \) are the distances from the Sun to the Earth at perigee and apogee, respectively,
- \( v_P \) and \( v_A \) are the velocities at perigee and apogee, respectively.

### Step 3: Relating Distances and Velocities
From the conservation of angular momentum, we can derive:
\[ \frac{r_A}{r_P} = \frac{v_P}{v_A} \]
Since \( r_A > r_P \), it follows that:
\[ v_P < v_A \]
This means that the velocity of the Earth is greater at perigee than at apogee.

### Step 4: Kinetic Energy Calculation
Kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
From our previous step, since \( v_P > v_A \), we can conclude:
- Kinetic energy at perigee \( (KE_P) \) is greater than kinetic energy at apogee \( (KE_A) \):
\[ KE_P > KE_A \]

### Step 5: Total Mechanical Energy Conservation
The total mechanical energy (TME) of the Earth-Sun system is conserved:
\[ TME = KE + PE \]
Where PE is the potential energy. As the Earth moves closer to the Sun, potential energy decreases, and kinetic energy increases, and vice versa when moving away.

### Step 6: Graphical Representation
From the above analysis, we can summarize:
- Kinetic energy is maximum at perigee and minimum at apogee.
- The kinetic energy will never be zero as the Earth is always in motion.

### Conclusion
The correct graphical representation of the variation in kinetic energy as the Earth orbits the Sun is one that shows kinetic energy increasing as the Earth approaches the Sun (perigee) and decreasing as it moves away (apogee). Among the given options, the diagram that best represents this behavior is option D.

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