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The distance of the two planets from the...

The distance of the two planets from the Sun are `10^(13)m` and `10^(12) m`, respectively. Find the ratio of time periods of the two planets.

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Here `r_(1)=10^(13)m` and `r_(2)=10^(12)m`
`(T_(1))/(T_(2))=?`
Now, `(T_(1)^(2))/(T_(2)^(2))=(r_(1)^(3))/(r_(2)^(3))`
`(T_(1))/(T_(2))=((r_(1))/(r_(2)))^(3//2)=((10^(13))/(10^(3//2)))^(3//2)=10^(3//2)`
Now, `v=(2pir)/T`
Hence `(v_(1))/(v_(2))=(r_(1)//r_(2))/(T_(1)//T_(2))=10/(10^(3//2))=1/(sqrt(10))`
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