If you are standing on earth's surface, in what direction (and at what speed) should you throw a 1Kg uniform sphere of radius 0.1 meters in order to put it into lower earth orbit? Assume that there is air, but it is not moving relative to the earth
If we ignore air resistance for a moment, then all orbits in an inverse square force like gravity are closed. this means that if you throw something hard enough it will complete one orbit then return to its starting point i.e. your hand.
So if you throw the object downwards it obviously hits the Earth, and if you throw it straight upwards it goes up then down and hits the Earth. For all the angles in between the object will go into orbit, though it will be an exceedingly low orbit since once an orbit it will pass through the point where you released it.
If you want to put the object into a roughly circular orbit you can't do this just by throwing it. Satellites are launched using a rocket trajectory like this:
And this requires a continuous boost from the rocket. You can't replicate this with a throw.
Adding air now means that if you throw the object hard enough to get it out of the atmosphere it will simply burn up, just as satllites burn up when the reenter the atmosphere.