Question

If the distance between the Sun and Earth is doubled, then the duration of the year on Earth will be:
[Given actual duration of the year = \(T\)]

• A. \(2\sqrt{2}\,T\)
• B. \(\frac{T}{2}\)
• C. \(\sqrt{2}\,T\)
• D. \(\frac{T}{\sqrt{2}}\)

Hint:

Kepler’s Third Law: \(T^2 \propto r^3\). If distance changes by factor \(k\), time changes by \(k^{3/2}\).

Answer 1

The Correct Option is A

Step 1: Use Kepler’s Third Law.
According to Kepler’s Third Law of planetary motion:
\[ T^2 \propto r^3 \]
where \(T\) is time period and \(r\) is distance from the Sun.

Step 2: Write proportional relation.
\[ \frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3} \]

Step 3: Substitute given condition.
If distance is doubled:
\[ r_2 = 2r_1 \]
So,
\[ \frac{T_2^2}{T^2} = \frac{(2r)^3}{r^3} \]
\[ \frac{T_2^2}{T^2} = 8 \]

Step 4: Take square root.
\[ \frac{T_2}{T} = \sqrt{8} \]
\[ \frac{T_2}{T} = 2\sqrt{2} \]

Step 5: Final result.
\[ T_2 = 2\sqrt{2}\,T \]
\[ \boxed{2\sqrt{2}\,T} \]