If the Earth is spinning or rotating at a really fast speed, why haven't we seen any videos from space of it spinning when we get a lot of photos of it?
[Physics] Video of Earth spinning
earthrotation
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Your argument is actually more or less right, but some of the details are wrong.
First you have to realize that Newtonian mechanics and general relativity have different definitions of an inertial frame. According to Newtonian mechanics, the coffee cup sitting on my desk right now defines a (very nearly) inertial frame, but a falling rock is extremely noninertial, because the rock has an acceleration of 9.8 m/s2. According to GR, free-fall is the preferred inertial state, so the rock is considered to define an inertial frame, but the coffee cup has a proper acceleration of 9.8 m/s2.
The Newtonian definition is actually impossible to define 100% rigorously, but traditionally the "fixed stars" have been taken as a pretty good standard for Newtonian frames. Any frame in which the stars have a very small acceleration is considered a very good inertial frame.
So if you have the Newtonian definition in mind, then your argument only goes wrong at the end, where you refer to a "very fast speed." What's relevant is the stars' acceleration, not their speed. If a rocket ship is gliding through our solar system at 1,000,000 m/s, then it's an inertial frame. It doesn't matter that the stars have a velocity of -1,000,000 m/s in its frame; what matters is that they have a=0.
According to the Newtonian definition, a frame of reference fixed to a point on the earth's surface is not an inertial frame. You can tell this because in that frame, the stars have large centripetal accelerations. However, the earth-fixed frame comes very close to being inertial, because you can find other frames that are inertial and that differ from it only by a very small acceleration. Therefore experiments on the earth's surface need to be pretty sensitive in order to detect any noninertial effects. The classic example of such an experiment is the Foucault pendulum.
In GR, a frame of reference fixed to a point on the earth's surface is not an inertial frame, and it doesn't even come close to being one. It differs from a valid (free-falling) inertial frame by a huge amount -- an acceleration of 9.8 m/s2. Even an extremely crude experiment can determine this. For instance, I can tell because I feel pressure from my chair on the seat of my pants. A secondary issue is that the earth's frame is rotating, and GR does consider rotating frames to be noninertial as well. (There was a lot of historical confusion on this point, including some early mistakes by Einstein, who thought GR would embody Mach's principle better than it actually did.)
Because you were also in orbit around the sun with the Earth and still have that velocity.
You may be imagining this in terms of stepping off of a slow moving vehicle on the Earth: you jump off, you come to a stop relative the ground and watch the trolley car go it's merry way. But that is a feature of friction between you and the ground. There is no such thing as a absolute reference frame in the universe and when you "leave the Earth" you don't come to stop relative anything so that you can watch the Earth fly away.
Newton's laws apply here: "a body in motion (that's the you or the planet) will continue in motion unless acted on by an external force". You just keep going except for changed induced by your drive.
Best Answer
First of all, let's calculate the rotational velocity of the Earth, at the equator. The diameter of the earth is 12,756 km. Therefor, the speed is ${12,756km*\pi}/{24 hrs}=1669 km/hr$.
Given that we know the rate of spin of the Earth, what else would be required to see this change?
Just to compare this, I pulled the speed of the moon orbiting the Earth from wikipedia. That speed is 1.022 km/s, or about 3,680 km/hr. That is a much higher speed than the Earth rotational speed. Given that the distance requirements are similar to see the rotational speed of the Earth, it seems that one would have to notice the moon orbiting if one could notice the rotation of the Earth, in real time. (Note, it's a bit easier to see the Earth rotating, due to the easy comparison, but...)