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

• A. 4:1
• B. 1:4
• C. 2:1
• D. 1:2

Hint:

Remember: Orbital velocity \( v \propto T^{-1/3} \). If one planet takes more time to complete an orbit, its speed is lower.

Answer 1

The Correct Option is D

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 \( \boxed{1:2} \).