Hi Varun. I took a look at the notebook. I don't follow the physical logic. I only see NDSolve in use to solve the trebuchet equations. To elaborate on my response of 1 day ago, the solution for ballistic flight of the projectile would go like this:
1) Solve the trebuchet just as done by Erik Mahieu in the demonstration. This solution terminates at a final time tMax, which is when the projectile leaves the pouch to begin ballistic flight. At that time tMax we have numerical values for the 3 generalized coordinates and their derivatives. These are the arm angles defining the configuration of the trebuchet as it goes through its motion.
2) Erik also gives the equations for the x, y location x[t], y[t] and the velocity components of the projectile in the pouch as derived from the general coordinate angles. So at the point where the projectile leaves the pouch it is at x[tMax], y[tMax] and has velocity components x'[tMax] and y'[tMax].
3) The ballistic flight of the projectile after leaving the pouch is a completely separate problem from the trebuchet problem. The trebuchet equations have nothing to do with it. The equations for ballistic flight without drag are the usual: xx''[t]==0, yy''[t]==-g. These are solved using NDSolve, and the connection to the trebuchet problem is in the initial conditions xx[tMax]==x[tMax], xx'[tMax]==x'[tmax], yy[tmax]==y[tmax], yy'[tmax]==y'[tMax]. (Notice that x and y refer to the values derived from the trebuchet solution while xx and yy are the coordinates for the ballistic flight.) The ballistic flight is solved with NDSolve with a "StopIntegration" when yy[t]==0, which is when the projectile hits the ground. So the ballistic flight is solved for {t, tMax, Infinity} with Event Location or WhenEvent used to stop the simulation on impact.
4) So you see there are two separate solves for two different differential equation systems. First for the trebuchet, and then for the ballistic flight with initial conditions provided by the end point of the trebuchet solution. These will provide separate interpolating functions for the solutions: the first set for coordinates of the trebuchet, and the derived values for the location of the projectile in the pouch; and the second set for the coordinates of the projectile in ballistic flight after leaving the pouch. Both are required for plotting the full path of the projectile.
If you don't mind, I would like to make a few suggestions about using Mathematica. First, your notebook puts all the calculations in a single cell. It is easier to find your way through a problem if you put each calculation in its own cell, and you evaluate and look at the result as you go along. Second, if you are going to put something in Manipulate, first solve it without. For example, in the method I outline above, do each of these things in separate cells, one at a time, and look at the result by plotting the NDSolve solutions using Plot and ParametricPlot, so you can see that you are getting what you expect. Do the same for the following graphics. Once you have everything going as you expect, then you can delete the output cells, merge the separate cells into one with semicolons as separators, and then encapsulate in animations or Manipulations.
Best regards, David