Two satellites `S_(1)` and `S_(2)` revolve around a planet in coplanar circular orbits in the same sense their periods of revolution are 1 hour and 8hours respectively the radius of the orbit of `S_(1)` is `10^(4)` km when `S_(1)` is closest to `S_(2)` the angular speed of `S_(2)` as observed by an astronaut in `S_(1)` is :

Explanation

At according to kepler's 3rd law `T^(2)propr^(3)`
`[(T_(1))/(T_(2))]^(2)=[(r_(1))/(r_(2))]^(3),i.e.,[(1)/(8)]^(2)=[(10^(4))/(r_(4))]^(3)`
or `r_(2)=4xx10^(4)`km
So, (a) As in cae of circualr motion
`T=(2pir)/(v), v=(2pir)/(T)`
, `|vecv_(2)-vecv_(1)|=2pi[r_(1)/(T_(1)~(r_(2))/(T_(2)))]`
i.e., `|vecv_(2)-vecv_(1)|=2pi[(4xx10^(4))/(8)~(10^(4))/(1)]=pixx10^(4)km//hr`
(b) When they are closest to each other and moving in same direction
`|v_("rel")|=|vecv_(2)-vecv_(1)|=pixx10^(4)km//hr`
and `r_("rel")=|vecr_(2)-vecr_(1)|=(4xx10^(4)-1xx10^(4))=3xx10^(4)km`
So, `omega_("rel")=(|v_(rel)|)/(|r_(rel)|)=(pixx10^(4))/(3xx10^(4))=(pi)/(3)` rad/hr