If I jump and stay in the air for a long enough time, will my position be changed since the Earth is rotating?
You will land ever so slightly behind where you started from, as the earth rotates under you. Yes, in general, at all times you are moving at the same tangential velocity (with respect to the center of the earth) as the ground and the atmosphere. However, when you jump, you attain a higher elevation and therefore a lower angular velocity compared to the ground you jumped off.
Say you are at the equator where everything is moving the fastest with respect to the center of the planet. If you jumped 1m, your angular velocity drops to 99.999984% of the rest of the earth. So while the surface of the earth zooms past at 463m/s, you lag behind by 73nm for each second you spend at 1m altitude. That's a thousandth of the width of a human hair. Now to land far back to even a measurable degree, we're talking either crazy altitude jumps, or long air time. That's some crazy air time! Or should I say..hair time...
When you jump, you're already travelling in roughly the same direction as the Earth so obviously the Earth is not going to move along a few metres. This is the same as if you jump in a train - you don't land a metre behind where you took off.
Having said that, there is a slight effect due to the Coriolis Force. This is an apparent force that arises due to motion in a rotating reference frame. The maths is quite complex (you'd usually study this two or three years into a university degree). The strength of the effect depends on the latitude, the direction of the jump and how fast you go.
While on ground, you are moving with the Earth and hence have some horizontal momentum. If you jump upwards, you don't lose this horizontal momentum just because you have lost contact with the ground. So, you will continue to move horizontally approximately in step with the Earth. Also, the atmosphere is moving (approximately) in step with the Earth and it will help you along.
Consider that you are on a train. The train is moving fast at a constant speed on a smooth, straight track. There is a table in front of you and a coffee cup on it. If you push the cup off the table, will it fall to the fall near to you or shoot to the back of the carriage? You are facing forwards and there is a passenger opposite you. If he drops his knife, does it fall on the table below or shoot backwards and into you?
With sufficiently accurate calculation or measurement, you might find a slight variation in your position but much less than you would expect from the rotational speed of the Earth.
No, there won't be any significant change in your position. To make a significant shift in your position, you might have to jump as high as some orbit of a satellite revolving around earth.
Because the speed of rotation of earth and your horizontal velocity do not have comparable magnitude, the displacement produced will be negligible.
Simple, the highest you jump (related to the time counted), the more mm(s), cm(s) or meter(s) you've got. In other words, the final position is based on how high you jump. Try to jump up to the sky :)
Perhaps, that could be described as follows:
Df = Differential Position in mm(s);
Hi = Height (jumping) in meter(s);
Ti = Constant of Time = 24 hours
Note: (24 hours x 60 minutes = 1440 minutes)
If you jump 1 meter (1000 mms), then;
Df = Hi * Ti/1440
Df = 0.0166666666666667 mms
Note: you also must think the interference of wind blows :D more complex. Or, for instance, the interference of atmosphere is ignored (or null). but, That's only illustration! :)