Lets say we fire a bullet from a very powerful gun at a height of 1 metre parallel to the surface of the Earth. Assume there is no air resistance and the only force acting on the bullet is from Earth's gravity. Also assume that the Earth is perfectly spherical and that nothing will get in the bullets way when it travels around the globe.
How fast must the bullet be fired such that is circumnavigates the Earth once and lands at our feet?
I know that its speed must be close to the speed needed for it to stay at an orbit height of $1$ metre ($7925$ m/s), but I am not sure how I can work the exact speed out.
I first thought the curvature of the earth wouldn't matter in the calculation and that I could just work out the answer for a flat distance of $40 000$ km (circumference of Earth), however I quickly realised that this would not work.
Is this situation I have described even possible? Would the orbit be circular or elliptic?
Any help with this question would be greatly appreciated.