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S(1) and S(2) are two satellites revolvi...

`S_(1)` and `S_(2)` are two satellites revolving around a planet P in coplanar circular orbits in anticlockwise direction. Their period of revolution are 50 minutes and 400 minutes respectively. The radius of orbit of `S_(2)` is `5 xx 10^(4) km`.
(a) Find the radius of orbit of `S_(1)`
(b) When `S_(2)` is closest to `S_(1)`, then find (i) speed of `S_(1)` relative to `S_(2)` and (ii) angular speed of `S_(1)` as observed by astronaut in `S_(2)`.

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To solve the problem step by step, we will follow the instructions given in the video transcript. ### Part (a): Finding the radius of orbit of \( S_1 \) 1. **Understanding Kepler's Third Law**: According to Kepler's Third Law, the square of the period of revolution of a satellite is directly proportional to the cube of the radius of its orbit. This can be expressed as: \[ T^2 \propto R^3 \] ...
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