[David Fort]

Thanks everyone for the suggestions and questions. Here is some additional info.

First, as I said, the test was unscientific, and that is a profound understatement. We are pulling a catapult back with a rope. We can pull on that rope while standing on a scale, and see that we pull the rope with about 50 pounds of force.

The direction of pull is not perfectly normal to the scale, maybe 10 degrees off.

There is one pulley in that system to turn the rope up away from the floor so it can be pulled up, pushing our feet down into the scale.

In our catapult system, we have a 2" bore cylinder, and thus think it should have an area of pi.

We are pulling, rather than pushing. The rod has diameter 0.625.
The effective area is the annular area, the cylinder diameter minus the rod cross section area.

Thus, the effective area for retracting is 0.307 square inches.

(all the dimensions in this discussion come from the FIRST Pneumatics Manual. We haven't actually measured them).

The conclusion that we are "getting less than 100 pounds" of force is based on the highly rigorous observation that we can pull the catapult back by hand with 50 pounds of rope tension, as measured by the bathroom scale, and that when we use the 2" cylinder, we can't pull it back. We did not measure the force exerted by the cylinder. Not even with the bathroom scale. (we couldn't figure out how to do that. An idea comes to mind now, but at the time, not).

(the additional 2x is from a 1:2 pulley system which divides the cylinders force in half to get twice the rope travel. Assuming massless frictionless pulleys and all that).

here is a chart of our force/area calculations (which match those listed in the FIRST pneumatics manual):
CylinderDiameter RodDiameter CylinderArea RodArea AnnularArea PushForceLbs RetractForceLbs
0.750 0.250 0.442 0.049 0.393 26.507 23.562
1.500 0.437 1.767 0.150 1.617 106.029 97.030
2.000 0.625 3.142 0.307 2.835 188.496 170.088

There are 3 (fixed position) pulleys in the rope between the cylinder and the catapult to route the rope over the river and through the woods as required to get the whole thing to fit.

Final disclosure: It wasn't a bathroom scale. It was pinched from the wrestling team.

Thanks for your thoughts.