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I was wondering whether relative speed of geostationary satellite with respect to earth is zero or not. Since angular velocity of satellite about earth is same as angular speed of earth about its own axis but distance of surface of earth from axis of rotation is different from distance of satellite from axis, it implies that speed of both (observer on the surface of earth watching satellite and satellite itself) is different (Since $v=rw $). It means that relative speed of geostationary satellite with respect to earth should not equal to zero. But shouldn't it should be zero since to an observer on earth geostationary satellite would appear to be at rest.

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No the tangential speed of an observer on the surface of the earth and the satellite relative to each other is not the same. The tangential velocity of the satellite is much more greater than that of an observer on earth's surface, just as you pointed out in the question. It's the relative angular velocity that is same in both the cases that's zero, as both cover 2pi radians(One revolutin) in 24hrs.

Also note that both of them are accelerating, so the observer seeing the satellite at rest does not imply 0 relative tangential velocity.

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If the geostationary satellite is stationary in the observer's frame on earth, (which it actually is), then it's relative velocity will be zero.

The problem lies in writing the expression for relative velocity correctly. The observer's frame is a rotating frame, which is to say that the imaginary coordinate axes fixed to him are rotating. So relative velocity is not simply the difference of tangential velocities. I leave the vector calculation to you.

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