Two satellites `S_(1)` and `S_(2)` resolve round a planet in coplaner circular orbit in the same sense. Their period of revolution are `1` hour and `8` hour respectively. The radius of the orbit of `S_(1)` is `10^(4) km`. When `S_(2)` is closest to `S_(1)`, find
(a) The speed of `S_(2)` relative to `S_(1)`,
(b) The angular speed of `S_(2)` actually observed by an astronaut is `S_(1)`

Explanation

`T^(2) prop r^(3)`
`(T_(1)^(2))/(T_(2)^(2))=(r_(1)^(3))/(r_(2)^(3))`
`(1/8)^(2)=((10)/(r_(2)))^(3)`
`(1/8)^(2//3)=(10^(4))/(r_(2))`
`r_(2)=4xx10^(4) km`
`T=(2pir)/v implies v=(2pir)/T`
(a) Speed of `S_(2)` relative to `S_(1)`
`|v_(2)-v_(1)|=(2pir_(2))/(T_(2))-(2pir_(1))/(r_(1))=2pi((r_(2))/(T_(2))-(r_(1))/(T_(1)))`
`=2((4xx10^(4))/8-(10^(4))/1)=pixx10^(4)km//hr`
(b) Angular speed of `S_(2)` relative to `S_(1)`
`(v_(2)-v_(1))/(r_(2)-v_(1))=(-pixx10^(4))/(3xx10^(4))=-(pi)/3 rad//s`