Are you having difficulty deriving mathematical expressions for the coefficients to the parabola? These two videos may help.
First, we’ll start with the case where the balloon provides buoyancy which exactly cancels the weight of the projectile. In doing so, I hope you see how to solve it with and without buoyancy.
After determining the “parabola” for the case when B=1 as described in the video above, I hope that you are now ready for the general case of a parabola with or without buoyancy.
If you still have some lingering confusion, you can consult the following video (coming soon).