Here's a good thought experiment I thought of that makes you think about gravity and the nature of objects in motion:
When you jump up in the air, the Earth is beneath you when it lands. No matter how high you jump, the location you jumped from will be beneath you considering you jumped at rest relative to the Earth.. but what if you hypothetically jumped straight up hundreds of kilometers? Is the location on Earth that you jumped from still going to be directly beneath you? If it's always directly beneath you, and the Earth is rotating, wouldn't that from the frame of reference of everything else in the solar system, you'd appear to be flying around the Earth in a circle at impossible speeds, completing a distance of one circle around the Earth along with the Earth as it makes a full rotation?
That can't happen.. can it? Well, it actually does happen - it's happening to objects in orbit around the Earth right now. The reason that the same spot on the Earth below doesn't stay the same for, say, a spacecraft in orbit above Earth, is because the velocity required to stay in orbit around the Earth without falling back down to it changes depending on altitude, and your relative velocity with the rotation of the Earth that you had initially after jumping accounts for less and less of the velocity required to stay in orbit the higher up you get.
For spacecraft taking off from Earth, their trajectory upwards from the Earth into space will appear as a curved path going with the Earth's rotation due to Earth's gravity, as if what we proposed earlier is actually true! =) But, due to the increase in velocity required to keep the spacecraft in orbit, the location that is directly below it on Earth ends up changing.