The acceleration depends on the power of the explosive charge and the mass of the missile. Think of the Orion with its nuclear pulse engine - I guarantee, if you used a nuclear bomb to propel too small a craft, it would experience g forces far beyond those experienced by a rifle bullet. But with something as large as an Orion, each bomb's energy is absorbed by enough mass that the change in velocity is such that the human body isn't atomized by it.
It's just a matter of sizing the parts correctly.
fasquardon
"Just". In this case I think the devil is in the details. What you say is correct as far as it goes, but that acceleration has to end up with the missile moving at a useful velocity. Otherwise, why are we bothering with the gun at all? And that velocity has to be achieved within a relatively short linear distance (the length of the gun barrel). I'm sure it is possible to build a gun that accelerates an ICBM gently enough for it to survive the process, I am NOT sure it would be possible to do so in any reasonable length (less than, say, a few kilometres).
Total delta-v requirements for LEO are about 10km/s. The gun has to provide a decent proportion of that or there's no point in using it. An ICBM might be able to provide 7km/s, which means the gun has to provide 3km/s. The Quicklaunch system Konrad provides a link to specifies a 1.1km barrel and a muzzle velocity of 6km/s, which gives their 1800g acceleration figure. Note that even at best they were only hoping for a 450kg projectile, some of which would have been the orbital circularisation motor. A Minuteman missile weighs a good deal more - 35 tonnes, in fact. My maths and physics knowledge is not up to the task, but I'm sure someone more knowledgeable could work out what sort of acceleration is necessary to get that up to around 3km/s.
Long story short, I don't think you're going to be using stock ICBMs for this.
*Edit*
I'm not sure what acceleration an ICBM is built to withstand - probably less than 10g, but for the sake of this exercise let's assume it's the same as a Sprint missile (100g). Having used
this site as a calculator, and assuming constant acceleration from 0m/s to 3000m/s, we're looking at about 450g of acceleration over a 1km barrel length. Our hypothetical ICBM second stage won't survive that. If we want to keep it to 100g then we need a barrel more than 4.5km long.
All of this assumes acceleration would be constant, which it wouldn't be. It would either come as a single shock at the start, or in several jolts as subsidiary charges fire while the projectile moves up the barrel (the V3 design). I doubt ICBMs could be used without modifications so extensive you might as well just build a complete new upper stage.