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Statement I: The smaller the orbit of a ...

Statement I: The smaller the orbit of a planet around the Sun, the shorter is the time it takes to complete.
Statement II: According to Kepler's third law of planetary motion, square of time period is proportional to cube of mean distance from Sun.

A

Statement I is True, Statement II is True: Statement II is a correct explanation for Statement I.

B

Statement I is True, Statement II is True: Statement II is Not a correct explanation for Statement I.

C

Statement I is True, Statement II is False.

D

Statement I is False, Statement II is True.

Text Solution

AI Generated Solution

The correct Answer is:
To solve the question, we need to analyze both statements and determine their validity based on Kepler's laws of planetary motion. ### Step 1: Analyze Statement I **Statement I:** "The smaller the orbit of a planet around the Sun, the shorter is the time it takes to complete." - According to Kepler's laws, particularly the third law, the time period (T) of a planet's orbit is related to its distance (a) from the Sun. The law states that the square of the time period is proportional to the cube of the semi-major axis of the orbit: \[ T^2 \propto a^3 \] - This implies that if the semi-major axis (mean distance from the Sun) is smaller, the time period (T) will also be smaller. Therefore, Statement I is **true**. ### Step 2: Analyze Statement II **Statement II:** "According to Kepler's third law of planetary motion, square of time period is proportional to cube of mean distance from Sun." - This statement directly reflects Kepler's third law, which can be mathematically expressed as: \[ T^2 = k \cdot a^3 \] where \( k \) is a constant for all planets orbiting the same star (in this case, the Sun). - Since this statement accurately describes Kepler's third law, Statement II is also **true**. ### Step 3: Determine the Relationship Between Both Statements - Since both statements are true, we need to evaluate if Statement II provides a correct explanation for Statement I. - Statement II explains that a smaller mean distance (smaller orbit) leads to a shorter time period (Statement I). Thus, Statement II does indeed provide the correct explanation for Statement I. ### Conclusion Both statements are true, and Statement II correctly explains Statement I. Therefore, the correct answer is: **Both Statement I and Statement II are true, and Statement II is the correct explanation for Statement I.** ### Final Answer **Option 1:** Statement I is true. Statement II is true. Statement II is the correct explanation for Statement I. ---
To solve the question, we need to analyze both statements and determine their validity based on Kepler's laws of planetary motion. ### Step 1: Analyze Statement I **Statement I:** "The smaller the orbit of a planet around the Sun, the shorter is the time it takes to complete." - According to Kepler's laws, particularly the third law, the time period (T) of a planet's orbit is related to its distance (a) from the Sun. The law states that the square of the time period is proportional to the cube of the semi-major axis of the orbit: \[ ...
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