Two satellite `S_1 and S_2` are orbiting around the earth in circular orbits in the same direction. Time period for the two satellite are 1h and 8h respectively. The radius of the orbit of satellite `S_1 "is" 10^4km`. If satellite `S_1 and S_2` are on the same side of the earth, find the linear and angular speed of `S_2` with respect to `S_1`.

Explanation

If the radii of satellites `S_1 and S_2 "are" r_1 and r_2` and their respective time periods are `T_1 and T_2`, then
`T_1^2 propr_1^3 and T_2^2prop r_2^3`
`therefore (T_1^2)/(T_2^2)=(r_1^3)/(r_2^3) or (1/8)^2=((10^4)/(r_2))^3`
`or, 1/(64) =((10^4)/(r_2))^3 or,(1/4)^3=((10^4)/(r_2))^3`
`therefore r_2=4xx10^4km`
The speed of satellite `S_1`,
`v_1=(2pir_2)/(T_2)=(2pixx4xx10^4)/(8)`
`=pixx10^4km*h^(-1)`
`therefore` Linear speed of `S_2` with respect to `S_1`
`=|v_2-v_1|=pixx10^4 km *h^(-1)`
and angular speed of `S_2` with respect to `S_1`
`omega=(|v_2-v_1|)/(r_2-r_1)=(pixx10^4)/(4xx10^4-10^4) =(pi)/3rad*s^(-1)`.