Question:
Two planets, A and B, orbit around a star such that the time period of A is 8 times the time period of B. The ratio of orbital velocities of the planets A and B is
Two planets, A and B, orbit around a star such that the time period of A is 8 times the time period of B. The ratio of orbital velocities of the planets A and B is
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Remember: Orbital velocity \( v \propto T^{-1/3} \). If one planet takes more time to complete an orbit, its speed is lower.
Updated On: Mar 18, 2026
- 4:1
- 1:4
- 2:1
- 1:2
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The Correct Option is D
Solution and Explanation
From Kepler’s third law, the square of the time period of a planet is proportional to the cube of the radius of its orbit:
\[
T^2 \propto R^3 \Rightarrow R \propto T^{2/3}
\]
Now, orbital velocity \( v \) is given by:
\[
\begin{align}
v = \frac{2\pi R}{T} \propto \frac{R}{T} \propto \frac{T^{2/3}}{T} = T^{-1/3}
\]
Let \( T_A = 8T_B \). Then the ratio of velocities is:
\[
\begin{align}
\frac{v_A}{v_B} = \left(\frac{T_A}{T_B}\right)^{-1/3} = (8)^{-1/3} = \frac{1}{2}
\]
So, the ratio of orbital velocities of A to B is \( \boxed{1:2} \).
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