I have a 2D controllable rocket for which I want to plot the parabolic motion. Additionally, the parabola should be able to change due to the thrust of the rocket accelerating it in either direction. My understanding is that an initial velocity would not make sense if the rocket can change its velocity against gravity. Additionally, I'm not sure what difference knowing the time value would make. I have tried using a series of equations to find the vertex of the trajectory arc.
$$time_{max} = \frac{v_{current}}{g}\space \space(1)$$ $$x_{max} = x_{current} + v_{x} * time_{max}\space \space(2)$$ $$y_{max} = y_{current} + \frac{{v_{y}}^2}{2a}\space \space(3)$$ $$parabola: a(x-x_{max}) + y_{max}\space \space(4)$$
This method resulted in a parabola vertex that increased/decreased based on whether the rocket's vertical velocity was changing. This makes sense as the y-max is tied to the vertical velocity in the equations above, but in reality, the maximum height doesn't change as the object slows down due to gravity.
