Can I launch an object into orbit using only a cannon? My friends and I were reminiscing about our intro physics classes and decided to revisit an old problem were we were required to calculate the orbital velocity of a cannonball fired from 1 meter above an airless earth.
We extended this problem to a debate about whether this same hypervelocity cannon on Earth can launch an object into a stable orbit above the atmosphere without any additional propellant in the object itself. So far, we haven't found a solution, but we are looking at the possibility of atmospheric skipping or using the gravitational influence of the moon to adjust our object's trajectory.
Is it possible for our hypothetical cannon to launch something into a stable orbit around the Earth from Earth's surface without any additional thrust?
 A: Simple geometry says no. Consider a projectile which is launched from the earth's surface at some angle other than vertical, ignoring the effects of air friction. Once the launch force is terminated, if the object is travelling less than escape velocity, the object will assume a closed elliptical orbit around the center of the earth (well, OK, around the mutual center of mass - we're not talking about launching a planetary-scale object here, are we?). Since the orbit intersected the launch point (by definition), all subsequent orbits will make the same angle of intersection with the earth's surface. At this scale, the earth is essentially a smooth sphere.
Slightly more complicated geometry says maybe, but still probably not. A sufficiently high path will interact with the moon, which would conceivably produce a slingshot effect sufficient to produce a "stable" orbit, but by the same argument as before, this is likely to be perturbed, sooner or later, by the moon.
As for the question of speed, well, that's a pretty unequivocal yes, but with a catch - you need a nuke. The Pascal-B shot of Operation Plumbob http://nuclearweaponarchive.org/Usa/Tests/Plumbob.html launched a 1-ton steel plate out of the bore hole at an estimated (lower limit) 66 km/sec, about 6 times escape velocity. We simply (as far as I know) have no experience with making a projectile which will handle this in atmosphere, so payload design is purely hand-waving. But in principle...
