Could we make a trebuchet that could launch objects to a stable orbit? Inspired by this xkcd, which calculated the energy requirements for accelerating individual humans to escape velocity (regardless of consideration for what that would do to your organs), I am interested to know if a trebuchet (or catapult) could be built that could launch things out off the Earth. Escape velocity isn't necessary, as I'd consider a stable orbit 'off earth'.

*

*What would be some design considerations / challenges? Do we have materials that could withstand this type of force?


*What would be maximum weight limits, ignoring air resistance (a tennis ball? a human? a satellite? a human in a metal survival pod?)


*What would be the speed at which these objects would need to reach at the earth's surface to be into a stable orbital velocity by the time they exit the atmosphere? (factoring gravitational slowdown and air resistance)


*Is it ever conceivable that we could 'launch' supplies to the ISS or orbit this way? What about launching satellites like this?
Edit
I misspoke when I said catapult, a string tension driven device. I meant a trebuchet, a gravity-powered heavy-object thrower. QuickLaunch, Inc. has plans to do just this launching a SSTO rocket at 6 km/s, which then fires after launch and provides the necessary correction for orbital insertion. Basically, I just need a trebuchet that will accelerate a mass (they're trying 1kg, 10kg, 50kg and 500kg masses) to 6 km/s at launch.

 A: *

*If the catapult accelerates the object from rest to $v_{e}$ over a length $L$ at a
constant acceleration $a_{e}$, then the acceleration will be
\begin{equation} a_{e}=\frac{v_{e}^{2}}{2L} \end{equation}
so the force during acceleration of an object of mass $m$ will be 
\begin{equation} F=m\frac{v_{e}^{2}}{2L} \end{equation}
For $v_{e}$ of Earth, when $L$ is 1000 meters (taller than the tallest building on Earth), and $m$ is 100kg, the force
would be
\begin{equation}
F=(100kg)\frac{(11.2*10^{3}m/s)^{2}}{2(1000m)}=6,544,000N
\end{equation}
during the launch.

*There is no weight limit for escape velocity. Velocity is all that matters. See answer to (1) to see why a large mass would be difficult to launch.

*If the object is above escape velocity at the surface, then it will also be above escape velocity at any altitude (ignoring air resistance). Escape velocity is a function of r, distance from the center of mass of earth.

*When you factor in air resistance, the previous answers will all be different. Even without air resistance, it would be very difficult.

A: Instead of using catapults, magnetic propulsion systems (especially superconducting) would be more useful for a Non-rocket space-launch. Particularly, the energy required for an average guy (70 kg) to get outta this world would be $43.904 × 10^8 J$
Your catapult is somewhat comparable to a Space gun. But, it cannot place the payload in a stable orbit, since gravity certainly wouldn't allow that...
Mass drivers which use superconducting coils would be a more efficient (more than 90%) of attaining such a large amount of kinetic energy (escape velocity) with a single lift. But, they're all in the future proposals list... They are so much powerful and probably don't have a weight limit. Yes, we could launch satellites directly into their geostationary orbits without the use of rockets. 
Even though it would be a lot of success, some million new bills should be invested to obtain it...
Also, I'll add that space elevators would be more useful when comparing budget. All those Wikipedia links are quite good in this subject...
Edit: Okay... A trebuchet consists of two arms - Projectile arm and Main weight arm. The ratio of the length of the main weight arm to the length of the projectile arm is typically between 1/2 and 1/5. The main weight arm has the counter-weight required to shoot the projectile and it should always be in multiples of the projectile weight (The height of main weight arms also matters here). The angle we use commonly for a projectile is 45°. Here to attain the escape velocity (at least close to), you must use at least a million ton counter weight, a 10 lbs projectile, and a trebuchet more than 2 mile long, and use an angle of 90° would be helpful..! Theoretically possible I would say...
Refer these Trebuchet range and Projectile range papers...
Edit: Your first two questions were reasonable. But now, It's quite impossible to attain. As I've already told, gravity wouldn't allow your projectile to be placed in a Geo-stationary orbit. Even though there's no air resistance (Friction in atmosphere) and you've broken the escape velocity, you'd escape into outer space and never return..! Hence in the absence of a rocket propulsion system, you require at least some kinda magnetic systems (not just wood and clockworks..!)
@Ehryk: If those papers were not useful, Here is an information from a simulation which gives an idea.
There are some software such as ATreb, WinTreb or Trebuchet Simulator... You could refer the List of Simulators provided...
A: For an object in low earth orbit (at 100+ miles above the earth's surface) the speed needed is about 17,000 miles per hour.  Even if a trebuchet could achieve that speed on the earth's surface, you would have at least three problems:


*

*The object would IMMEDIATELY burn up in our dense atmosphere. Think about the space shuttle which is going at orbital speed when it encounters the very tenuous atmosphere at very high altitudes. It needs special heat resistant ceramic tiles due to the heating caused by a very tenuous atmosphere. If the angle at which the first encounter the atmosphere were too steep it would completely incinerate.  So there is no material that you could use to build the satellite that would prevent it from immediately burning up.

*If you could magically make all the atmosphere disappear, you still could not launch a satellite with a trebuchet from the surface of the earth.  Well you could, but it would only complete less than one orbit. If you got the right speed, it would start out on a nice elliptical orbit, but the ellipse would bring you back to the launch point coming up through the crust of the earth. In other words the ellipse will pass through the earth such that in less than one orbit you will impact the earth's surface again.  To successfully launch, the satellite would need to have some kind of rocket motor onboard so that once it got to an appropriate altitude, it could change the velocity direction to be in an orbit that doesn't intersect the surface of the earth.

*The last problem that will make this Trebuchet impossible is the mass and required strength of the arm that will connect the heavy weight to the pivot point to the satellite.  I suspect that making this arm strong enough will make it too heavy to work.  So, for now let's assume the arm has zero mass and infinite strength. Then if we assume the heavy weight falls in say, about 1 second at about 1G, then to get the satellite to 17,000 miles per hour, the acceleration of the satellite would have to be 25,000 ft/sec^2 which means it would accelerate at 780Gs (so humans would be killed for sure).  That would mean that the length of the arm to the satellite would have to be 780 times longer than the short arm to the heavy weight.  So if the short arm were 10 feet, the long arm would have to be 7,800 feet which is 1.5 miles.  I think you can see that the arm requirements would make this totally impractical if not impossible.  For this to even work, the heavy weight would have to be greater than mass of the satellite times the long arm length divided by the short arm length by a very large factor (to insure the heavy weight falls at about 1G).  If we assume a 100kg satellite, then in this case that means the heavy weight would have to be something like 10 or 100 times (7800/10)*100 kg - thus something like 780,000kg to 7,800,000kg.  Imagine the strength of the arm that is required.  Then think about how heavy the arm would be and how that would make all of these requirements even more impossible since a heavy arm would greatly decrease the acceleration of the satellite.
So, no it CANNOT be done...
A: It actually would be possible to achieve an almost stable orbit IF you could make a trebuchet powerful enough to fire a projectile to the moon. You could get the projectiles to orbit several times at least. And as people pointed out, just making one to fire into space is practically impossible, but assuming you got past that step, you have another problem. The problem is raising the projectiles periapsis(lowest orbital position) once outside the atmosphere. After you launch, the periapsis would be as low as the trebuchetes altitude, meaning it will fall back to earth after 1 orbit. But if you could get a sling shot assist from the moon to gain the necessary orbital velocity to raise your periapsis, it should be able to get into a relatively stable orbit. I say relatively because it might be on a collision course with the moon in the future, or another sling shot, which would change it's orbit again.
