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If a planet is spinning east to west and there is a satellite spinning from west to east...

Can the satellite travel at a speed sufficient to make the planet appear, from the vantage point of the satellite, to be rotating from west to east?

What kind of calculations could be made to determine that speed?

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Suppose we take the sign convention to be that the velocity of the planet's surface (i.e. East to West) is positive, then because the satellite is moving in the opposite direction (West to East) it's velocity will be negative. We need to consider angular velocities, $\omega$, given by $v = r\omega$, where $r$ is the distance from the centre of the planet, but this doesn't change the argument because $v$ and $\omega$ always have the same sign.

The relative angular velocity of the planet's surface as seen from the satellite is:

$$\omega_{rel} = \omega_{planet} - \omega_{satellite}$$

and because we've already decided that $\omega_{satellite}$ must have a negative sign, $\omega_{rel}$ is always positive regardless of how fast or slow the satellite is orbiting i.e. the surface is always moving East to West..

A more interesting question is if the planet and satellite are moving in the same direction (let's take this to be East to West). In this case there is a geostationary orbit at which the satellite is moving at exactly the same angular velocity as the planet, so the planet's surface appears to be stationary. If the satellite moves further away it's angular velocity decreases and it will see the planet rotate East to West, but if it moves nearer it's angular velocity increases it will see the planet rotate West to East.

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You are right. That interesting, and really the question I was looking for. Thanks for your answer. My study on angular velocity begins today. – Hinkler Apr 28 '12 at 14:49

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