nomenclature: vi = initial speed, z = height x,y are the 'ground', φ = elevation angle of the barrel, α. = α 'dot' = dα/dt, α: = double dot show that an artillery cannon can hit a target anywhere in the surface given by: g²r²=(vi²)²-2gz(vi²) ignore air resistance. no wind. note: this is not intended to be done as a surface integral. so what i did was decided that it can shoot something directly above it: zmax = vi²/2g, zD = 0 then considering the 2-d case (y=0) and then as φ decreases, so rotates the point of max height (up to φ = ╥/2, but then the max height moves back inside this trajectory). This is going to make a paraboloid from (0,zmax) to (?,0). The equation of tha paraboloid is z = Ax + Bx² + C at zmax x=0 so zmax = C = vi²/2g z. = Ax. + Bx.² z: = Ax: + Bx:² = -g, but x: = 0 so the equation for the paraboloid is wrong??? help please.