Assertion: The time period of revolution of a satellite close to surface of earth is smaller then that revolving away from surface of earth.
Reason: The square of time period of revolution of a satellite is directely proportioanl to cube of its orbital radius.

Options

A. If both assertion and reason are true and reason is the correct explanation of assertion
B. If both assertion and reason are truebut reason is not the correct explanation of assertion
C. If assertion is true but reason is false
D. If both assertion and reason are false.

Explanation

To solve the question, we need to analyze the assertion and the reason provided.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that "the time period of revolution of a satellite close to the surface of the Earth is smaller than that revolving away from the surface of the Earth."
- This means that satellites that are closer to Earth complete their orbits faster than those that are further away.

2. **Understanding the Reason**:
- The reason states that "the square of the time period of revolution of a satellite is directly proportional to the cube of its orbital radius."
- This can be mathematically expressed as \( T^2 \propto r^3 \), where \( T \) is the time period and \( r \) is the orbital radius.

3. **Deriving the Relationship**:
- We know that for a satellite in orbit, the gravitational force provides the necessary centripetal force. This can be expressed as:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the orbital radius.
- From this equation, we can derive the velocity \( v \) of the satellite:
\[ v = \sqrt{\frac{GM}{r}} \]

4. **Finding the Time Period**:
- The time period \( T \) of the satellite can be calculated using the formula:
\[ T = \frac{2\pi r}{v} \]
- Substituting the expression for \( v \):
\[ T = \frac{2\pi r}{\sqrt{\frac{GM}{r}}} = 2\pi \sqrt{\frac{r^3}{GM}} \]

5. **Analyzing the Relationship**:
- Squaring both sides gives:
\[ T^2 = \frac{4\pi^2 r^3}{GM} \]
- This shows that \( T^2 \) is directly proportional to \( r^3 \), confirming the reason provided.

6. **Conclusion**:
- Since the assertion is true (satellites closer to Earth have a smaller time period) and the reason is also true (the relationship \( T^2 \propto r^3 \) is valid), we conclude that both the assertion and reason are correct, and the reason correctly explains the assertion.

### Final Answer:
Both the assertion and reason are true, and the reason correctly explains the assertion.

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