Calculate time to go from one point to another point [closed]

Question

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.

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

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