Say that America/NASA has developed a small city with around 100 or so inhabitants on the Moon.
Would a cannon or slingshot or trebuchet type machine be able to launch cargo out of Earth's orbit and have it reach the Moon for them to grapple it and reel it in?
My main question is if the prospect of delivering interplanetary cargo without a rocket is feasible, just by building some sort of aerodynamic shell and giving it enough force to exit Earth.
There was a man called Gerald Bull who, with others, developed the HARP system for intended to be a stepping stone to a super-cannon that could launch a (small) payload to space where it would (possibly) inject itself into a proper orbit. The main attraction is the relatively small cost to get small payloads to orbit.
I think if you were able to ask the late Mr. Bull if your proposed system were possible he'd stop talking and start designing.
Whether it's possible to make a gun capable of launching a sufficiently large payload to space which could then inject itself into an orbit approaching the moon is debatable. I'd say not now, but maybe with advances in propulsion and further super-gun development it's not inconceivable, although far fetched.
Note that by "cargo" I'm thinking in terms of very small items. Less than a few kilos (and I'm that's pushing it). That might sound small, but if you're on the Moon and that circuit board that keep life support going fails, a few kilos would be fine.
You're using the wrong tool for the job. You can get far higher velocities far more economically from a linear motor. (Think of a maglev train that just keeps speeding up.) Boosting something from LEO to LTO is no big deal if you have enough power available.
However, there's nothing to be gained by putting such a system in orbit. A rocket doesn't require burning fuel, anything thrown out the back counts--and your cargo heading for the moon is something thrown out. Your station goes the opposite direction with just as much energy, all it saves you is that your rocket stays on the station rather than being launched into space.
If you put it on the Earth you have some pretty major engineering problems due to the atmosphere. If you build a long track it has to be horizontal--and that means an awful lot of atmosphere. You'll need a very, very big payload with a heat shield better than anything NASA has built in order to do it. Also, nobody has managed to design a system that won't destroy itself with the shockwaves.
Alternately, you dig straight down and use a very high boost. The drag problem is lowered to simply incredible. Still no answers for the shockwaves, though.
Note that there has been a version of Jules Verne's gun built. There is no doubt that the "payload" (merely of scientific curiosity, it was completely inert) was launched at far above escape velocity. While the instruments recorded very little there is no doubt it was vaporized before leaving the atmosphere. If it hadn't been vaporized it would have beaten Sputnik to space. While in theory it could be done again with a capsule engineered to survive the passage I doubt it ever will be--the charge in the "gun" was a fairly large hydrogen bomb, the "payload" a plug in the hole drilled for it.
Rockets are required because the energy that would need to be fed into this super advanced "trebuchet" would essentially be a massive explosion. Destroying not only machine itself, but also the cargo.
In other words: Rockets slowly use up their fuel by bringing it along with them in order to not have to use a single ground-based detonation that would blow everything up.
There is a solution to all this however: Lasers. By focusing a high intensity laser emitted from the surface onto sails attached to whatever you want to send you can push it along without needing to use rockets. NASA is already developing this technology today.
"...to launch cargo out of Earth's orbit and have it reach the Moon"?
Oh heavens no, definitely not. Let's do some quick back of the napkin math:
It takes about $7.8 km/s$ to reach Low Earth Orbit, and around $11 km/s$ for a Trans Lunar Injection.
The Shuttle takes about 8 minutes to reach LEO velocity.
Meaning, it spreads out the $7.8 km/s$ velocity change over 480 seconds.
$1g = 9.8m/s^2$
$G-force = \frac{\text{Acceleration in }m/s^2}{9.8}$
So, the Shuttle's acceleration $= \frac{\text{7800 }m/s}{480\text{ seconds}} = 16.25$
and $\frac{16.25}{9.8} = 1.65g$
So, very roughly, on average, the Shuttle's acceleration over time is 1.65g.
Now an artillery projectile does not contain its own propellant, so it is not accelerating after it leaves the barrel. It gains all its velocity at once in the beginning, meaning this cannon/catapult/trebuchet must impart $11 km/s$ $\Delta v$ instantaneously.
$\Delta v$ = literally, change in velocity
Lets round up and say this cannon imparts this velocity change over 1 second exactly, instead of milliseconds.
Projectile's acceleration $= \frac{\text{11000 }m/s}{1\text{ second}} = 11000$
$\frac{11000}{9.8} = 1122.44g$ = LOL
What about just getting to orbit?
$\frac{7800}{9.8} = 795g$
Pretty safe to say just by looking at the numbers, doing some quick rough estimates, it's never going to happen.
The problem you see with g-force is the time over which you accelerate. Acceleration says low as long as you take longer time to reach your desired speed. (In this case, orbital or TLI velocity.)
You can experience this directly in your car: accelerate from 0 mph to 50 mph pushing the pedal as hard as you can. You will reach your desired speed rather quickly, but will feel more acceleration forces on your body as you go.
Press the pedal lightly from 0 to 50 mph, and you will eventually reach the same target speed over a greater period of time, and thus feel much less acceleration.
So the problem with the orbital cannon is two-fold, and likely why one was attempted but never built: 1) it can only impart change in velocity once, at the beginning 2) orbital velocity is so great (never mind TLI right now) that nothing hollow could survive the g-forces. (Seriously, you might even flatten a solid piece of steel.)
I suggest a two-stage system to get your cargo to the moon. Some brief research suggests every lunar mission has involved a second "launch" from Earth's orbit.
Stage One can be your space gun or maglev rail designed to put standard payloads into orbit: https://en.wikipedia.org/wiki/Non-rocket_spacelaunch#Projectile_launchers
Stage Two offers choices at different price ranges and delivery times, from a rocket that would take 2-3 days, to an ion thruster that would take over a year. http://www.universetoday.com/13562/how-long-does-it-take-to-get-to-the-moon/
Non-parishable supplies could be scheduled and shipped cheaply a year in advance, but the colony would never be more than a few days away from an emergency delivery.
In general, a single “gun” to launch will not produce an orbit. A rail or long tube on the Moon can no doubt get a payload to its escape velocity (the opposite direction from what you asked), it will not be in orbit; that requires another burn in-flight, at the very least.
Now with multiple bodies, the situation is more complex. Imagine shooting the payload in the direction opposite the Moon’s orbital motion, cancelling it out. Then it drops straight down and hits Earth at 25000 miles per hour, which is a bit much to “catch”!
Brian Wang of the NextBigFuture blog has brought up the ultimate "gun"; the so called Jules Verne Cannon which uses a nuclear charge to fire a massive payload into orbit or directly impacting on the moon.
This is essentially an ORION nuclear pulse drive turned upside down, the nuclear charge is inside a salt dome or similar formation and the energy of the explosion is transferred via pusher plate to a cargo pod, which punches through the atmosphere and into orbit.
Reading the article demonstrates that like the ORION spacecraft, bigger really is "better", and nuclear pulse drives of all sorts don't scale "down" very well. Of course given the bone crushing thousands of "g" acceleration, the sorts of cargoes that you could deliver are essentially solids (you could send a kiloton of coal, for example, if you need carbon for industrial scale processing), so sending thousands of tons of computer parts or machinery is essentially a no go. Of course, a thousand ton ingot of high quality steel or titanium could be sent to make machine parts, but eventually the neighbours would complain about the noise the cannon makes and it will be much cheaper to mine ores from convenient Near Earth Objects instead.
