Aquatic launcher,

A very long tube, starting deep underwater is evacuated of water and air, once a hard vacume is established a very large butterfly valve is opened allowing the ocean to rush in, propelling a ship to the top of the tube, a explosive system launches a rotating dish shaped cap, that uncaps the tube, then seperates into 3 or more pieces, clearing the exit, the heavier the ship, the less it will slow when hitting the atmosphere, a cylindrical rod gradually extends out of the nose cone, this rod is evaporated into plasma, this plasma is repelled, by a very strong electric or magnetic field, creating a supercavity. A rocket keeps the whole show from slowing down.

With current materials, how deep could that tube go? How thick does the wall need to be to withstand the pressure at a certain depth? How fast would the water rush in? What would the exit velocity be then? Needs more math and physics!

I am not sure about the scale, bigger ship needs more volume to reach near escape, once a ship can be designed a tube for that ship would need to he designed, you want the most Gs that the ship and crew can take, so the whole system would have to be optimized for launch weight, but I think it is possible, I am not sure how deep for what velocity, as the math there is new to me, I think you would use pressure and volume, over time, so the force would be greatest right at launch, but they have some very very deep oil pipes, that do not collapse when empty, but they have a very small cross section. The idea is there, but I am not sure about the math, help?

What about a very large funnel shape connected to the tube that is not as deep?

That gets the problem of a large surface area attempting to resist the high pressure.

One "cheap" way to build this would be to find a nice equatorial atoll somewhere, that is geologically stable and use a TBD (Tunnel Boring Device, like they used to dig the Chunnel) to dig a vertical shaft down thru the rock to act as your "barrel". You have a side shaft at the bottom as your water inlet. For extra oomph your atoll has a geothermal source that you can use to create a giant steam catapult.

Muhahahaha!