Editing Projectile motion (section)

==== Numerical solution ====
A projectile motion with drag can be computed generically by [[numerical methods for ordinary differential equations|numerical integration]] of the [[ordinary differential equation]], for instance by applying a [[Ordinary differential equation#Reduction to a first-order system|reduction to a first-order system]]. The equation to be solved is

:<math>\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}x \\ y \\ v_x \\ v_y\end{pmatrix} = \begin{pmatrix}v_x \\ v_y \\ -\mu\,v_x\sqrt{v_x^2+v_y^2} \\ -g-\mu\,v_y\sqrt{v_x^2+v_y^2}\end{pmatrix}</math>.

This approach also allows to add the effects of speed-dependent drag coefficient, altitude-dependent air density (in product <math>c(v)\rho(y)</math>) and position-dependent gravity field <math display="inline">g(y)=g_0/(1+y/R)^2 </math> (when <math display="inline">y \ll R: g \lesssim g_0/(1+2y/R) \approx g_0(1-2y/R)</math>, is linear decrease).