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in a 2D football game I want to calculate cycle(time) until ball go from one point to another point.

in this link exist all fomulas that we need for my problem but anyone does not have any parameter that shows the effect of decay on ball speed.

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closed as too localized by David Z Sep 18 '12 at 17:33

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1 Answer 1

If you know the acceleration $a(v)$ as a function of speed (as in aerodynamic drag) then

$$ \Delta t = \int_{v_1}^{v_2} \frac{1}{a(v)}\,{\rm d}v $$

For example, a free falling body with air resistance has acceleration (positive is up)

$$ a(v) = \kappa v^2 - g $$

where $\kappa$ is some constant. The time to reach speed $v$ from standstill is

$$ \Delta t = \int_v^0 \frac{1}{\kappa v^2-g}\,{\rm d} v = -\frac{ \ln\left( \frac{\sqrt{g}-\sqrt{\kappa} v}{\sqrt{g}+\sqrt{\kappa} v}\right)}{2 \sqrt{\kappa g}} $$

or the inverse

$$ v(t) = \sqrt{\frac{g}{\kappa}} \, \frac{1-{\rm e}^{-\sqrt{\kappa g}\, 2 t}}{1+{\rm e}^{-\sqrt{\kappa g}\, 2 t}} $$

with terminal velocity

$$ v(t=\infty) = \sqrt{\frac{g}{\kappa}} $$

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thanks,can you please say about what is g and k? –  Arash Sep 18 '12 at 18:10
$g$ is acceleration due to gravity, and $\kappa$ is an aerodynamic constant, typically calculated by $\kappa = \frac{1}{2} \rho A C_d$ with $A$ the swept area, $\rho$ the air density, and $C_d$ the coefficient of drag. –  ja72 Sep 18 '12 at 18:16
in soccer simulation 2d that is a field of robotic, we do not have gravity.only agent that effect on speed of ball is ball decay and it has a constant value in each cycle –  Arash Sep 18 '12 at 18:22
You will need a simulation to solve this problem then. That's what computers were invented for, literally the first computers were developed by the army math department to create projectile tables. –  ja72 Sep 19 '12 at 13:29
You could use the principles in this paper, faculty.kfupm.edu.sa/AE/aymanma/images/… –  ja72 Sep 19 '12 at 13:33
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