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The ratio of ...

Question

The ratio of orbital radii of two satellites of a planet is 1:2. What is the ratio of their time periods?

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Solution

$F r o m K e p l e r s l a w T^{2} i s p r o p o t i o n a l t o a^{3} w h e r e a i s l e n g t h o f s e m i m a j o r a x i s . \frac{T_{1}^{2}}{T_{2}^{2}} = \frac{a_{1}^{3}}{a_{2}^{3}} \frac{T_{1}}{T_{2}} = (\frac{a_{1}}{a_{2}})^{3 / 2} \frac{T_{1}}{T_{2}} = (\frac{1}{2})^{3 / 2} = \frac{1}{2 \sqrt{2}}$

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