## Theory

In a trebuchet, the potential energy of a counterweight is converted to the kinetic energy of the projectile.  The potential energy of the counterweight is given by the equation: (1)

where is the mass, in kilograms, of the counterweight, is the earth’s gravity, 9.8 m/s2 , and is the height, in meters, above the ground from which the mass falls.
The kinetic energy of the launched projectile is given by the equation: (2)

where is the mass of the projectile, in kilograms, and is the velocity, in meters per second, that the projectile acquires after launch.
In an ideal trebuchet, all of the potential energy is converted to kinetic energy as the projectile is thrown, therefore: (3)

And thus: (4)

Once the projectile leaves the sling it will undergo projectile motion, the only force acting on it will be gravity, as air resistance will be assumed to be negligible. Therefore, the equations for the x and y positions of the projectile will be: (5)

and (6)

respectively, where is the initial velocity of the projectile, in meters per second, is the time, in seconds, elapsed since launch, and is the angle at which the projectile was launched. The maximum range of the projectile will be on the x axis, or when (the projectile hits the ground). Plugging in into equation (6) and solving for gives: (7)

Plugging this value for into equation (5) gives the maximum value of or the maximum range of the projectile, to be:   (8)

Then, after solving equation (4) for , it is shown that: (9)

Inserting this expression for into equation (8) gives:   (10)

In order to achieve maximum range, the projectile will be released at a angle, at which both the sine and cosine functions equal so equation (10) becomes: (11)

To measure the efficiency of the trebuchet as a percentage, we will use: (12)

This equation returns the percentage of the potential energy that was converted into kinetic energy through the launching of the projectile.

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