The time period of a satellite of earth is 7hours if the seperation between the earth and the satellite increase two times the previous value . what will be its new time period

To determine the new time period of a satellite when its distance from Earth is doubled, we can use Kepler's Third Law of planetary motion. This law states that the square of the orbital period (T) of a planet (or satellite) is directly proportional to the cube of the semi-major axis (r) of its orbit. In simpler terms, if you know the time period of a satellite at a certain distance, you can find the new time period when the distance changes.

Understanding Kepler's Third Law

Kepler's Third Law can be expressed mathematically as:

• T² ∝ r³

This means that if you double the distance (r), the relationship between the time period (T) and the distance will change accordingly. Let's break this down step by step.

Step 1: Initial Conditions

We know the initial time period (T₁) of the satellite is 7 hours. The initial distance (r₁) is some value that we don't need to know explicitly to solve the problem.

Step 2: Doubling the Distance

If the separation between the Earth and the satellite is increased to two times its previous value, we can express the new distance as:

• r₂ = 2 * r₁

Step 3: Applying Kepler's Law

Now, we can relate the new time period (T₂) to the new distance using Kepler's Law:

• T₁² / T₂² = r₁³ / r₂³

Substituting r₂:

• T₁² / T₂² = r₁³ / (2 * r₁)³

This simplifies to:

• T₁² / T₂² = r₁³ / (8 * r₁³)

Thus, we have:

• T₁² / T₂² = 1 / 8

Step 4: Solving for the New Time Period

From the equation above, we can rearrange it to find T₂:

• T₂² = 8 * T₁²

Taking the square root of both sides gives us:

• T₂ = √(8) * T₁

Since T₁ is 7 hours, we can calculate:

• T₂ = √(8) * 7 hours ≈ 2.828 * 7 hours ≈ 19.8 hours

Final Result

The new time period of the satellite, when its distance from Earth is doubled, will be approximately 19.8 hours. This demonstrates how significantly the orbital period increases with distance, illustrating the principles of orbital mechanics effectively.