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i think you have used the wrong formula... basically what you should think is... that the work done to the catapult... will convert... totally into kinetic energy of the stone...force in the formula (F e ) is F ..and you are explicitly writing it 1/2 F... is already given to us as the avg...force.. we don't need to take the avg..of the avg force...so... F e = 1/2mv^2and now.. you will get it right...