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I would think that you would want to spread the g forces out over time - g forces involved in the acceleration of the egg. A cannon would accelerate the egg over a very short period of time and you would have a g-force spike. I don't know how you could protect the egg from these forces.
I would create a spinning arm with the egg on one end. The longer the arm and the faster the spin, then the further the egg would go. I would have the other end of the arm, past the pivot/motor weighted so that the whole spinning mass is balanced. The tricky part is "how do you hold the egg?" and "how do you release it at the right moment and at the right angle?" The only thing I can think of is to have a bucket at the end of the arm with the end being a trap door. The trap door has a release mechanism that can be tripped via a string that goes back along the arm, near the axis of revolution. When the arm is spinning fast enough then a pin could be inserted into a bracket that would "catch" the string at the right position of the arm, tripping the trap door. Have fun. Do a google search for
"Bad to the Bone" pumpkin
and you will see how well this concept can work.
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