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Note: This might be a very dumb question

Considering Earth's rotation on its axis, I was thinking it made sense for the outer sides of the Earth to have a significantly higher tangential speed, considering how the radius is much larger, in order to keep the same rotational speed.

Having considered this, the question I was posing myself was: If you were to jump perfectly up, so no force is applied by you in the x-axis, since the atmosphere would have a slightly higher speed, would you fall slightly further away from where you jumped?

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    $\begingroup$ In fact you would fall back because you are slower than the atmosphere. But this would also happen without atmosphere. Check out Coriolis-Force. The derivation of that is simple $\endgroup$
    – Finn Eggers
    Commented Apr 24, 2019 at 13:43

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There is more to this when you consider inertial effects (major examples on Earth being Coriolis force and centrifugal force), but for the sake of your question I believe you want to ignore those effects.

If I understand the question correctly, you are wondering if jumping straight up would make you land further out, due to the change in wind speed because of the increased radius.

Ignoring the inertial effects, and assuming air carries you with it perfectly, the answer is approximately no, you would not expect to change your position on the surface at all. The reason that the wind must move faster when it is higher up is because it needs to cover (approximately) the exact same amount of surface distance in the same amount of time, but has to travel further because it has a greater radius. This means that the additional speed given at a larger radius is taking you further in the same amount of time as if you were standing on the surface; but you need to travel further in that same amount of time if you want to land in the same spot; because your path is now also longer.

This is all only approximate, because you actually won't get completely up to speed with the air, so in practice this is likely to make you move further back on the surface.

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