Recently, a question popped up on whether it would be possible to use a nuclear bomb-powered megacannon to accelerate a payload to void velocity from the surface of a planet. I felt it was a very interesting question that needed a bit of looking into.
Short answer: No. It is not even remotely possible using the level of technology available in the canon Frontier setting.
Long answer:
The fastest man-made object ever launched was a manhole cover.
Yup. That ~300-400 lb plate of 4-inch iron was launched to a velocity of over 66 km/sec by the 300-ton yield Pascal-B nuclear device on 27 August 1957.
The velocity of light is 299,792 km/sec.
1% of the velocity of light is 2,997.92 km/sec.
Any heavier object would not be as fast, any lighter object would likely vaporize. In any case, thermal deformation would mean that anything exiting the atmosphere will barely be recognizable as a technological object.
A larger yield nuclear device would just vaporize the object, and at some point, the stresses placed on the atmosphere would endanger the surrounding region.
To add to the problem, each “cannon” would be essentially a single-use mineshaft that must be carefully dug as an extremely precise angle in order to be able to be fired at a specific world in a system within range (no more than 21 light years away)... with a very narrow launch window of perhaps a few minutes; if this launch window is missed, the entire shaft becomes useless, since both worlds are spinning on their respective axis, orbiting around their respective stars, which are moving independently through space.
If that is not enough, even if you could somehow get nearly 3,000 kps from the launch, there is nothing to decelerate the payload (assuming it survives the launch) at the target system; it would be nearly impossible to coordinate an intercept at the void arrival point. Best solution would be to plot a decelerating orbit around the target star so that within a few months it would have a reasonable velocity as it crosses the orbit of the target world.
When all is said and done, it would be far simpler and far more economical to build a hull size 1 atomic drive starship, program it with the requisite astrogation coordinates, and send it with a robot pilot using its full ADF of 5 all the way, which would be detrimental to the health of any living creature therein.