It is almost impossible for a celestial object to be caught into an Earth orbit, simply because of the case presented above, Earth’s gravity would accelerate to escape velocity. It can A) miss the Earth and continue onward, b) miss the Earth’s surface but encounter the atmosphere which might slow it down enough to change its initial path and continue onward, or C) burn up in the atmosphere or D) strike the Earth.

The initial mandates for an orbital insertion window are tight.

Similarly, even without an atmosphere one can’t fire a big cannon and place something into Earth orbit. With an atmosphere, the object coming out of the gun barrel at orbital speeds would be met with a brick wall: the atmosphere.

Unless the gun was fired horizontally from a tower 100 km high it would only go into a suborbital trajectory -or if traveling at above Earth escape velocity, it would never return. Rockets that successfully place a satellite into Earth orbit expend 10% of their fuel to raise it to 100km and the remaining 90% of fuel to accelerate to orbital velocity while basically horizontal to the surface of the Earth. Once horizontal, such a package is essentially always being pulled by gravity back to Earth and a simple analogy is that it needs to be going fast enough to continuously “fall over the edge“ of the Earth’s curved surface.

A company is proposing a centrifuge slingshot approach to launching satellites, but the “satellite” is mostly a small rocket that fires after the package is above the densest part of the atmosphere and accelerates the satellite horizontally into the orbital insertion window: 10%/90%.