Editing Projectile motion (section)

=== Trajectory of a projectile with Newton drag ===
[[File:Mplwp skydive trajectory.svg|thumb|250px|Trajectories of a [[skydiver]] in air with Newton drag: para-flight; bomber near(ly above) target. <math display="inline"> v.d=10^3.\nu=150 </math> cm<sup>2</sup>/s?]]
The most typical case of [[air resistance]], in case of [[Reynolds number]]s above about 1000, is Newton drag with a drag force proportional to the speed squared, <math>F_{\mathrm{air}} = -k v^2</math>. In air, which has a [[kinematic viscosity]] around  0.15&nbsp;cm<sup>2</sup>/s, this means that the product of object speed and diameter must be more than about 0.015 m<sup>2</sup>/s.

Unfortunately, the equations of motion can '''''not''''' be easily solved analytically for this case. Therefore, a numerical solution will be examined.

The following assumptions are made:
* Constant [[gravitational acceleration]] <math display="inline">g</math>
* Air resistance is given by the following [[drag equation|drag formula]],
::<math>\mathbf{F_D} = -\tfrac{1}{2} c \rho A\, v\,\mathbf{v}</math>

::Where:

:*''F<sub>D</sub>'' is the drag force,
:*''c'' is the [[drag coefficient]],
:*ρ is the [[air density]],
:*''A'' is the [[cross sectional area]] of the projectile. Again <math>\mu=k/m :=</math> <math>c\rho A/(2m)</math> <math>= c \rho/(2\rho_p l)</math>. Compare this with theory/practice of the [[ballistic coefficient]].

==== Special cases ====
Even though the general case of a projectile with Newton drag cannot be solved analytically, some special cases can. Here we denote the [[terminal velocity]] in free-fall as <math display="inline">v_\infty=\sqrt{g/\mu}</math> and the characteristic ''settling time'' constant <math display="inline">t_f\equiv1/(v_\infty \mu) =1/\sqrt{g\mu}</math>. (Dimension of <math>g</math> [m/s<sup>2</sup>], <math>\mu</math> [1/m])
*Near-'''horizontal''' motion: In case the motion is almost horizontal, <math>|v_x|\gg|v_y|</math>, such as a flying bullet. The vertical velocity component has very little influence on the horizontal motion. In this case:<ref name="Greiner2003">{{cite book|author=Walter Greiner|title=Classical Mechanics: Point Particles and Relativity|url=https://books.google.com/books?id=W7Rwdc6JMb8C|year=2004|publisher=Springer Science & Business Media|isbn=0-387-95586-0|page=181}}</ref>
::<math>\dot{v}_x(t) = -\mu\,v_x^2(t)</math>
::<math>v_x(t) = \frac{1}{1/v_{x,0}+\mu\,t}</math>
::<math>x(t) = \frac{1}{\mu}\ln(1+\mu\,v_{x,0}\cdot t)</math>
:The same pattern applies for motion with friction along a line in any direction, when gravity is negligible (relatively small <math>g</math>). It also applies when vertical motion is prevented, such as for a moving car with its engine off.

*Vertical motion '''upward''':<ref name="Greiner2003"/>
::<math>\dot{v}_y(t) = -g-\mu\,v_y^2(t)</math>
::<math>v_y(t) = v_\infty \tan\frac{t_{\mathrm{peak}}-t}{t_f}</math>
::<math>y(t) = y_{\mathrm{peak}} + \frac{1}{\mu} \ln \bigl(\cos\frac{t_\mathrm{peak}-t}{t_f} \bigr)</math>
:Here
::<math>v_\infty \equiv\sqrt{\frac{g}{\mu}}</math> and <math>t_f=\frac{1}{\sqrt{\mu g}},</math>
::<math>t_{\mathrm{peak}} \equiv t_f \arctan{\frac{v_{y,0}}{v_\infty}} = \frac{1}{\sqrt{\mu g}} \arctan{\left(\sqrt\frac{\mu}{g}v_{y,0}\right)},</math>
:and
::<math>y_{\mathrm{peak}} \equiv -\frac{1}{\mu}\ln{\cos{\frac{t_\mathrm{peak}}{t_f}}} = \frac{1}{2\mu} \ln{ \bigl(1+\frac{\mu}{g} v_{y,0}^2 \bigr)}</math>
:where <math>v_{y,0}</math> is the initial upward velocity at <math>t = 0</math> and the initial position is <math>y(0) = 0</math>.
:A projectile cannot rise longer than <math>t_\mathrm{rise}=\frac{\pi}{2}t_f</math> <!-- tan(t_peak/t_f) = v_y,0 sqrt(μ/g) = ? --> in the vertical direction, when it reaches the peak (0 m, y<sub>peak</sub>) at 0 m/s.

*Vertical motion '''downward''':<ref name="Greiner2003"/>
::<math>\dot{v}_y(t) = -g+\mu\,v_y^2(t)</math>
::<math>v_y(t) = -v_\infty \tanh\frac{t-t_{\mathrm{peak}}}{t_f}</math>
::<math>y(t) = y_{\mathrm{peak}} - \frac{1}{\mu} \ln(\cosh\biggl(\frac{t-t_\mathrm{peak}}{t_f}\biggr))</math> With ''hyperbolic'' functions
:After a time <math>t_f</math> at y=0, the projectile reaches almost terminal velocity <math>-v_\infty</math>.<!-- \tanh\frac{t-t_{\mathrm{peak}} }{t_f} is ca. 1 -->

==== 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).