For example, making the frame more rigid improved the outcome here, but if you put the whole thing on wheels you would likely still get more increase because the weight could take a more direct path downward as the frame moves.
That should change how long it takes the arm to drop, but it shouldn't change the amount of potential energy released (and thus, projectile speed).
Or at very high altitude.
Or have ablative heat shielding.
But atmospheric drag would probably prevent you from getting anything like those speeds with this kind of mechanism.
>"In Project HARP, a 1960s joint United States and Canada defence project, a U.S. Navy 16 in (410 mm) 100 caliber gun was used to fire a 180 kg (400 lb) projectile at 3600 m/s or 12,960 km/h (8,050 mph), reaching an apogee of 180 km (110 mi), hence performing a suborbital spaceflight. However, a space gun has never been successfully used to launch an object into orbit or out of Earth's gravitational pull."
That's still rather fast ...
As you can probably guess, I'm not a physicist. Maybe someone with expertise can chime in here.
I don't think relativistic effect would enter the equation for escape velocity (there's a 5 orders of magnitude difference), your issue will be basic material science, namely:
* That you'll need to add enough headroom that the payload is still at escape velocity after going through the atmosphere, which means the projectile needs to be launched way beyond escape velocity in a very dense atmosphere. Which means it will instantly burn up / blow up as it leaves the launch system, probably alongside the sling and part of the arm…
* except for the counterweight and material rigidity you need for the arm to not be feasible either, you'd need a long arm multiple miles long and a counterweight in the megaton range… before even accounting for the length of the long arm itself, or how you'll actually bring the bloody arm down