How do you make bullets go faster? You
lengthen the barrel, giving more time for
the expanding gasses to push the bullet.

How do you give a catapult more throwing
distance? You lengthen the swing of the
arm. I believe that trebuchets had about
almost 180 degrees of swing to gain
speed
before they let their projectiles go,
but they depended on gravity, so anything
beyond that would have started to slow
the projectile down. Some catapults (if not
all?) used energy in tortion. Bunches of
ropes spun and spun, would spring back
when released, giving catapults their
energy, but these only have no more than
90 degrees to reach their top momentum.
It is time for the next evolution of
catapult.

Picture this: A ferris wheel. Take all the
carriages off the ferris wheel except for
two of them, which are at opposite sides
of eachother. Start up he ferris wheel, and
make it go faster, faster and faster. Get
that thing up to... 200 RPM?... and at the
right moment to take advantage of a 45
trajectory, release both carriages at the
same time. The projectile carriage will fly
for an incredible distance, way farther than
any catapult or trebuchet. The other
carriage is jolted to the ground and
destroyed, if it were allowed to stay
attached, it would create an imbalance
that would probably destroy the wheel.

Take this same concept, and design a
catapult that allows for a continuous
rotation until high RPM is attained.
Release, and watch fly. I don't know how
to calculate how many RPM's would be
needed to fly a pumpkin farther than a
mile, if the arm or rope that the pumpkin
was attached to was 25 fee from center.
Anybody know how to calculate this?

potential problem: G forces might break
up the pumpkin before reaching desired
RPM's and launching.

I did some number crunching for you:
If your wheel is 7.62 meters (about 25 feet) in radius (that is, your pumpkin is 25 feet from the center of the wheel) and it is half a meter off the ground at its lowest point, then the optimum launch point is when the pumpkin is moving upwards and it 13.51 meters off the ground. When released here, it will travel with an angle of 45 degrees to the horizontal.

If you spin the wheel up to 200 RPM, the pumpkin will be hurled at a velocity of about 159.6 meters per second. If we ignore factors like wind resistance and spin on the pumpkin, you can expect it to land a good 2612.58 meters (about 1.62 miles) away. As a side note, it will achieve a maximum height of about 663.31 meters (about 2176.21 feet) and have a total flight time of around 23 seconds.

The only downside I can think of for this is that in order to achieve a rotation of 200 RPM on a wheel that is 50 feet in diameter, you'd have to accelerate it to around 574.52 Km/h (about 357 Mph).

Might like to try experimenting with different angles of release...

You get my croissant, not only for the idea but also for getting me to do some maths, which I haven't in ages :)

I think you may be on your way to (re-)inventing the flywheel catapult. It's used on aircraft carriers and to accelerate roller coasters, now, but I have trouble finding a good link.

yeah.. the punkin chunkin contests baked that a few years ago... i like the cannons.. kinda crazy that some random people pretty much have field artillery chillin in their garage..

//It's used on aircraft carriers //
All modern aircraft carriers use Steam catapults. I suppose a flywheel type design would work but it would be quite different than this idea, I would guess a clutch of some sort connected to a cable system.

One could adapt this to be done with an upside-down car wheel. What you would lose in diameter you might make up for in speed. [Emjay], perhaps you would be good enough to crunch the numbers?

The counterweight could be water, and it would be released by opening a vessel. Much safer.

[muses] I wonder if one of these could be made human powered, using a bike? And gears. Lots of gears. It would take a while to get it up to speed.

I wonder if there is any off the shelf wheel the size specified by [twitch], capable of moving at 200 RPM?

If it weren't so darned exciting to see a
pumpkin fly a mile from a cannon, i'd like
them set a rule that it has to be man
powered, no huge amounts of energy
stored in an air compressor.

oh my bad, I just checked out the first link.
Sooo baked in deed. But I wanted to get it
here, I was curious about the math.

Just calculate linear velocity of a point at the end of a circle of Diameter A. Then using the Weight of the Pumpkin, the Linear Velocity and the Angle, Height and Vector of departure from the wheel will allow you to calculate the ballistic trajectory of the projectile. Pretty much you will break down the Verticle and Horizontal Vector components of the Projectiles vector of departure and then apply the acceleration due to gravity against the vertical vector and the that will allow you to calculate the total time of the flight(accounting for wind resistance and the curvature of the earth of course) then you will use this time to calculate the distance traveled along a horizontal vector based on you initial velocity. Once again the reverse vector of Wind resistance will need to be applied along with any gravitational, Magnetic, and airflow effects, of Course. Its really just that simple.
;-)

If you want the short answer, for a wheel with a 25 foot radius you would need to achieve a launch velocity of 415 ft/sec at 45 degrees launch angle. The means the wheel must be spinning at 156 rpms. The projectile will leave the sling at about 280mph.