Two dimensional trajectories

In a 2d trajectory, the direction of the drag force is opposite to the speed vector ${\bf v}$. Newton's equations of motion for $x$ and $y$ components are written

$$m\frac{dv_x}{dt}=-F_dx;$$ (24)

$$m\frac{dv_y}{dt}=-mg-F_dy;$$ (25)

Using $F_d=kv^2$, $v_x=v\cos{\theta}$ and $v_y=v\sin{\theta}$, we find

$$\frac{dv_x}{dt}=-\frac{k}{m}vv_v,$$ (26)

$$\frac{dv_x}{dt}=-g-\frac{k}{m}vv_v,$$ (27)

where $v^2=v_x^2+v_y^2$. Hence, we cannot calculate the vertical motion of the object without reference to the horizontal component.