It takes a lot of force to get a rocket up to speed. And if there's not a human in it, why not do it a little easier. So here's the idea. Get a 2 mile tube. 1 mile below ground, 1 mile up. A tunnel is centered in the middle of this thing. A Smaller Rocket is sitting at the bottom, and with a series of hits of air pressure, can be shot really really fast into the sky. With a little rifling, it could even get a small spin going for stability. I'm hoping it will reach mach 1.5 before exiting the top. The idea is that this will shoot the rocket up high enough, 20,000 ft or so and then allow the rocket, already in motion, to ignite its thrusters, and continue on from there, saving some fuel, and not have to jettison parts on the way up. And maybe even having a single body rocket that can return to earth with a glide, and then be reloaded again for another shot.
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Impractical. There have been attempts to launch rockets from balloons but the size and weight of orbital rockets is too much for this type of thing.
True, but this wont be a large full scale rocket. This will be a small one. Figure 2/3 the size of the payload area. It doesn't need to completely make space, just about 100,000 ft. And if its already at say 30,000 ft from the cannon, with a climb ratio of 10,000 ft every 20 sec, then a quick powerful burn for 1:36min should get it up there, enough to wear it can push the payload out farther to be captured, or put into a descent orbit the satellite can recover from. Then a nice fall back down, and some wings or a chute pops out.
The engines shouldn't fire till the rockets speed decreases to just above mach 1.o, then burst to keep it up to speed so the climb ratio doesn't fall to far. It could be practical in this sense.
This is basically the same as using a gun to lauch a rocket. This has been looked at before using either gunpowder or an electromagnetic mass driver. This might work, but the cost of building the gun would be higher than the alternatives, such as air-launching from a large jet airplane. A large jet is expensive, but they're already available. An airliner could release the rocket at altitudes above 40,000 feet at least. It would be hard to match that with a gun , but it is theoretically possible
Rather than dig a mile deep hole and build a mile high tube (which would be the tallest man-made structure ever), it is usually suggested that the gun be built running up the side of a mountain.