A planet revolving around the Sun reaches the covered region A in 2 days, A in 3 days and A in 6 days. Then the relation between A. A, and A, will be

To understand the relationship between the distances covered by a planet revolving around the Sun in different time intervals, we can apply Kepler's laws of planetary motion, particularly the law of areas. This law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Let's break down the problem step by step.

Understanding the Problem

We have a planet that covers a region A in 2 days, region A in 3 days, and region A in 6 days. Let's denote the areas covered in these time intervals as A1, A2, and A3 respectively. According to Kepler's second law, the areas covered in equal time intervals are proportional to the time taken to cover them.

Setting Up the Relationship

From the information given, we can express the areas covered in terms of the time taken:

• For the first interval (2 days): A1 = Area covered in 2 days
• For the second interval (3 days): A2 = Area covered in 3 days
• For the third interval (6 days): A3 = Area covered in 6 days

Applying Kepler's Law

According to Kepler's second law, the areas covered in these time intervals can be expressed as:

• A1 / 2 = A2 / 3 = A3 / 6

This means that the ratio of the areas to the time taken is constant. We can set a constant k such that:

• A1 = 2k
• A2 = 3k
• A3 = 6k

Finding the Relationship

Now, we can express the relationships between A1, A2, and A3:

• A1 : A2 : A3 = 2k : 3k : 6k

To simplify this ratio, we can divide each term by k:

• A1 : A2 : A3 = 2 : 3 : 6

Final Ratio

Thus, the relationship between the areas A1, A2, and A3 can be summarized as:

• A1 : A2 : A3 = 2 : 3 : 6

This means that for every 2 units of area covered in the first interval, 3 units are covered in the second, and 6 units in the third. This proportionality reflects the nature of the planet's motion around the Sun, adhering to Kepler's laws.

Conclusion

In summary, the relationship between the areas covered by the planet in the specified time intervals is 2:3:6. This illustrates the fundamental principles of planetary motion and how time and area are interconnected in celestial mechanics.