Supergun Launching of Satellites

Other answers don't mention the fact that no single impulse (e.g, like being fired from a gun) can launch a projectile into orbit. A purely ballistic projectile fired from a gun must either crash back into the planet, or it must escape from the planet altogether.

In order to achieve orbit, at least two impulses must be applied to the projectile. The first one (from the gun) launches it into an elliptical trajetory that returns to the surface, and then the second impulse must be applied by a rocket motor to "circularize" the orbit at the moment when the projectile reaches the apogee of the initial ellipse.

Anything launched into orbit by such a gun needs to travel at orbital velocity (in fact above orbital velocity) in the lower atmosphere. That's generally undesirable, to put it mildly: there will be really serious heating.

Aside from the interior ballistic aspects of these various projects, it was quickly realized that any satellites launched by gun would have to withstand high g-loadings during firing of the gun and the size and mass of the satellite would be greatly constrained by the dimensions of the bore of the gun and the maximum impulse which could be provided by the propellant without damaging the gun.

Special designs for satellites were prepared so that sensitive electronics would not be damaged by being fired from a gun, and recognizing that the gun could not provide sufficient velocity to reach orbit, satellites with booster rockets were designed to fire after being flung aloft by the gun.

The project ended for various reasons, some budgetary, some political. The escalating war in Vietnam caused funds for a lot of research projects to be cut, and this project was originally a joint effort between the U.S. and Canada. When relations between the two countries hit a rough patch over differing policies regarding Vietnam, the project became ripe for being eliminated.

I think the heart of the question is whether one could arrange a continuous combustion of propellant along the length of the barrel. In that way the acceleration occurs along the length of the barrel in a more gentle way. Since the expanding gases from the propellant in a shell casing expand and the pressure of the expanding gases declines along the way it means the primary force or acceleration loading is not at the start of the projectile motion.

You still have a huge acceleration. Suppose the barrel is $100$m in length and assume the projectile has orbital velocity ($\simeq 10^4m/s^2$) at the end of the barrel. Then using the elementary equation $2ad = v_f^2 - v_i^2$ the acceleration is then $$ a = \frac{v^2}{2d} = \frac{10^8m^2/s^2}{200m} = 5.0\times 10^5m/s^2. $$ This is the average acceleration, which if you design the firing of propellant correctly it might be the actual acceleration that is nearly constant. This is considerable.

There is an additional problem. The projectile as it leaves the gun will be slowed by the large shock wave it produces in the atmosphere. So you would need to fire the projectile at a higher acceleration to account for this loss.

Let's say you have got such gun. Next logical step will be to install on the satellite a smaller gun that would shot-back several small shells and so accelerate the satellite, indeed? If this small on-board gun would use really many small shells (size of molecula) then your are getting just a traditional rocket. Apparently it is not much difference for energy required if you launch a rocket from traditional platform or gun it, but engineering difficulties for the latter technology are massive.