Say, a person with a jetpack is in the moon. If the person is balanced over his weight by an upward force generated by the jetpack above the surface of the moon (kind of hovering for some time and at considerable height), can this lead to a relative motion between the person and the moon? ( I referred to it as moon, which does not have atmospherical effect)
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$\begingroup$ From a practical standpoint, it is extremely unlikely that you manage to direct the jetpack exactly downwards. Instead, you will do more or less permanent adjustments. Any misalignment will give you some sideway acceleration ... $\endgroup$– Hagen von EitzenCommented Nov 4, 2019 at 10:18
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$\begingroup$ Thanks for your consideration of this matter from a practical point of view! $\endgroup$– Jason LeninCommented Nov 4, 2019 at 10:35
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1 Answer
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To start with, he is rotating with the Moon at the same tangential speed as the Moon-surface.
If he rises up in elevation, he is at a larger orbit than the surface. If he wants to finish this larger orbit in the same time as the surface, then he must speed up sideways (larger distance over the same time requires larger speed). The relation is:
$$v_\parallel=r\omega$$
where $v_\parallel$ is the tangential speed (parallel to the surface), $r$ is orbit radius and $\omega$ is rotational speed. It is the rotational speed $\omega$ (the number of radians per second) that must stay constant and equal to that of the Moon. So, for a larger $r$, $v_\parallel$ must increase.
Conclusion: With no sideways acceleration to speed him up, he will get a relative motion compared to the surface, when he flies upwards. The surface will simply move away from him, not because it faster, but because it finishes its circular motion faster.
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$\begingroup$ Thanks a lot for the descriptive answer. Depending on your clarification, it can also let the person land on somewhere else after some time I guess! $\endgroup$ Commented Nov 4, 2019 at 10:00
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$\begingroup$ @JasonLenin Sure. If you lift off, and the surface then moves away underneath your feat ad described, then when you land, you are somewhere else. If you time it, you can land where you want to land, yes. $\endgroup$– SteevenCommented Nov 4, 2019 at 10:07
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$\begingroup$ What if we do the same process on earth? How much it is practical to maintain a horizontal motion relative to earth in that way? $\endgroup$ Commented Nov 4, 2019 at 10:29