What is the equation family of the projectile-motion-with-air-resistance eqn?

Asked Oct 16 '14 at 15:37

Active Jun 09 '15 at 22:45

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The general form of the equations of projectile motion with air resistance are (from here)

$s_y(t) = -\frac{mg}{k}t + \frac{m}{k}\left(v_{yo} + \frac{mg}{k}\right)\left(1 - e^{-\frac{k}{m}t}\right)$

and

$s_x(t) = \frac{m}{k}v_{xo}\left(1-e^{-\frac{k}{m}t}\right)$

The question I have is: to which general family of equations do these solution to ODE belong? I need to fit this equation to data, and would like to use the generic form.

Thanks!

UPDATE

@Tony Piccolo suggested this presentation that offers the following derivation (thanks!):

$y=\left( \frac{mg \sec\theta}{bV_0}+\tan \theta \right)x+\frac{m^2g}{b^2}\ln \left(1-\frac{b \sec\theta}{mV_0}x\right)$

which was actually what I was looking for. Thanks Tony!

edited Jun 09 '15 at 22:45

Vlad

6,938

asked Oct 16 '14 at 15:37

Oliver Amundsen

Why did you use the tags conic-sections and hyperbolic-functions ? – Tony Piccolo Oct 16 '14 at 20:50
updated. thanks. Any ideas on an answer? – Oliver Amundsen Oct 17 '14 at 16:38
The trajectory is not a standard curve and has not a special name. It has a vertical asymptote (terminal velocity): however, for $k \to 0$, it becomes a parabola. – Tony Piccolo Oct 17 '14 at 17:11
I'm interested in describing the assymetry of the trajectory, so in my case $k \neq 0$ – Oliver Amundsen Oct 17 '14 at 20:05
1

This can be useful. – Tony Piccolo Oct 17 '14 at 23:08
It actually is very useful since it expresses $y=f(x)$! Thanks Tony! – Oliver Amundsen Oct 20 '14 at 10:36

What is the equation family of the projectile-motion-with-air-resistance eqn?

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