Editing Projectile motion (section)

==== 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 -->