[Ether]

Unless I was careless with the algebra (it happens when I'm tired),
Steve's catapult can be modeled with an ODE of the form

θ'' = k₁ + k₂∙cos(θ) + k₃∙θ'

Attached is an Octave script that uses Octave's built-in ODE solver "lsode"
to numerically integrate arbitrary ODEs of the form x'' = f(t,x,x')