Using purely gravitational potential energy, what is the highest efficiency one can achieve with a trebuchet counter-weight type of machine? Efficiency defined here as transformation of potential energy in the counterweight to kinetic energy of the trajectory.
Edit: To be more specific, we can use the following idealization:
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no friction
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no air resistance
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no elastic/material losses
But any of the "standard" trebuchet designs are allowed: simple counterweight, hinged counterweight, vertical counterweight, etc.
Best Answer
The Wikipedia page on trebuchets links to a PDF paper which discusses exactly this question. It considers several models of varying complexity and finds a maximum range efficiency of 83% for a 100 pound counterweight, 1 pound projectile, a 5 foot long beam pivoted 1 foot from the point of attachment of the counterweight, and a 3.25 foot long sling. Here range efficiency is defined as the horizontal range of the realistic trebuchet model relative to the range of a "black box model" which is able to completely convert the gravitational potential energy of the counterweight into kinetic energy of the projectile.
In order to find the energy efficiency, defined as the fraction of the counterweight's gravitational potential energy that actually gets transferred to the projectile, you would need to use the relation
$$\frac{\epsilon_R}{\epsilon_E} = 2\sin\alpha\cos\alpha = \sin 2\alpha$$
where $\alpha$ is the angle of release of the projectile above the horizontal. Unfortunately, the paper doesn't give the value of $\alpha$ corresponding to the simulation that produced the maximum efficiency, so I can't give you a specific number without running the simulations myself. (Perhaps I'll do that when I have time; if anyone else gets to it first, feel free to edit the relevant numbers in.)