The Physics of it

   

Gravitational Potential Energy is stored by raising the bucket, this energy is released creating a time varying torque, proportional to the vertical component of gravity acting on the bucket, which is transferred through the lever arm to the tip with mechanical advantage 

The tip on a trebuchet is also often attached with a non-rigid sling which actually contains the stone or whatever is being thrown and makes the system much more difficult to deal with as splitting the system up into lever arm and sling would require that the sling be attached to a non-inertial coordinate frame.

Newton's "three laws" are generally inapplicable to a trebuchet because the rotary acceleration is a function of time (i.e. non-constant which is a requirement for those three laws and their rotational counterparts) and a trebuchet also cannot be dealt with by the same small-angle approximation that allows a pendulum to be dealt with succinctly because the buckets often go through large arcs.

The Physics behind it

We have an object being thrown and want to know it's velocity (speed/distance in relation to time)
We can determine the velocity using  with a rate of g m/s per second. 
This is called the acceleration of gravity.   It uses absolute value for gravity (32.2 ft/s² ).  
Let's understand the abbreviations:  velocity = v, gravity =g and seconds(time) =t the falling body is dropped.
We can determine distance:
Given the velocity formula , with the integration results in: 
where x is the distance an object falls over a time t.
For example, an object falling for 3 seconds would travel 32 x 3 x 3 / 2 = 144 feet
Next, we can think about the velocity with respect to time- 
If all of this would be graphed, you would get a parabola! Opening downwards, of course!
If a body is projected upwards with an initial velocity v, then at some time t = v/g, it comes to rest and then begins to fall back. The motion is described by y = vt - gt²/2 at any time, so if this time is substituted, the height of the turning point is found to be y = v²/2g. By a proper choice of the three constants in the general quadratic y = at² + bt + c, motions under gravity (or any constant acceleration) in one dimension with arbitrary initial position, velocity and time can be described. 
So- Get your axes x horizontal and y vertical (going up!)