What keeps a mid-air motionless object travelling with the Earth's rotation?

i.e. If there was no wind then it seems odd that this object wouldn't be still (motionless) in space/the universe, but disappear at 1000mph+ in the opposite direction the Earth was spinning.

I've been reading up about jet packs (limited technology though it is at present) and it got me questioning things like this (more on this at: http://www.jetpackinternational.com ). Any ideas? Hopefully you know what I mean; it's difficult to convey my thoughts here.
...And don't pick flaws in my example (it isn't perfect I know) but just imagine something...a flat disc motionless in the sky or anything. Imagine it disappearing over that horizon suddenly without any fuel whatsoever! What's stopping it doing that then?

Momentum.

Maybe gravity keeps it locked to the Earth's rotation. If the same object existed many thousands of miles above the earth, it would not move with the earth. Less gravity. Eg The moon.

the earth's gravitational pull

Inertia and gravity. Earth's gravity keeps it from going up. Inertia and wind resistance keep it in place horizontally. It would take a lot of energy to make it accelerate from motionless (WRT the earth's surface) to 1000 mph.

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Try a mental exercise. If the earth had its current atmosphere but no spin the air and earth would be stationary. If a giant motor then brought the earth quickly up to speed (one rotation per day) the earth's surface would pass through the air (creating a great wind at the equator but not much of anything at the poles). However, the mountains and even desert surfaces would cause the air at the surface to spin with the earth. This would slowly but surely cause all the atmosphere above the surface to spin with the earth due to friction. After a while the earth's surface and all of its atmosphere would spin at the same speed. The earth would be without wind except that the uneven heating of the sun (day and night, etc.) causes changes in the density of the air which results in winds relative to the surface. Any helium filled balloon lost by a kid at the fair travels almost straight up because the earth's surface and the atmosphere are spinning at the same speed and the balloon only goes up at an angle if there is a windy day caused by sunshine (or lack of it).

This has been explained by math for a long time. It is a topic called relative motion. It has to do with the fact that when you left the earth you inheritted a speed from its rotation and movement through space. You know how if you shoot a bullet at 100 mph it goes 100mph. But if you are on a car going 100 mph then the bullet fired at 100mph really travels 200mph relative to someone not on your car but only 100 relative to you. So unless you slow yourself down in the oposit direction the earth made you spin then you will stay in geosyncronis orbit with the earth. Fact!

This happens because the object maintains its eastward momentum even after you release it. This is an argument that Galileo made; and later Newton wrote it into his laws of motion.

When you're holding a baseball in your hand, the baseball and you are both travelling eastward at several hundred miles per hour due to the earth's spin. You and the baseball both have eastward momentum.

When you toss the ball upward, the laws of motion say that the ball's horizontal momentum will be the same as before; in other words, it still has eastward momentum, even when nothing's touching it, carried over from when you were holding it. Because it maintains its momentum, the ball continues eastward at the same rate as yourself. So from your point of view, the ball went straight up and then straight down.