I have the following problem: I want to determine the initial velocity of an object which is for example dropped from a moving plane. I know the position, from where the object is dropped and I know where the object landed. Lets say the initial position is $x=0$ and $y=300$. The object hits the ground at $x=700$ and $y=0$. What is the initial velocity $v_{x0}$?
Since I have to consider air resistance, I also know the mass of the object: $2kg$ and it's round, so I can use $c_w=0.45$ and the diameter of my sphere is $A=0.30m^2$ and the temperatur of the air is everywhere 20 °C so the air density is: $\rho=1.2041\frac{kg}{m³}$ resulting in $F_m= 0.0813 * v²$
I find lots of solutions which ignore the air resistance. This is easy, but I really would like to solve this problem with air resistance.
With eulers method I can calculate where this object will land, when I know the initial velocity, now I want to solve this "backwards". How can I formulate the problem mathematically, so I can solve this for $v_{x0}$?
I appreciate any help!
Edit: As requested, my program which solves the problem for unknown impact, but with known initial velocity and initial position:
$kmass = -\frac{\frac{1}{2}c_wA\rho}{m}$
$\Delta t$ is small step size
while(y > 0) {
$v_{xnew} = kmass \sqrt{v_x^2 + v_y^2} * v_x * \Delta t$
$v_{ynew} = (kmass \sqrt{v_x^2 + v_y^2} * v_y - g) * \Delta t$
$x_{new} = x + \frac{v_x + v_{xnew}}{2} * \Delta t$
$y_{new} = y + \frac{v_y + v_{vynew}}{2} * \Delta t$
$v_x = v_{xnew}; v_y = v_{ynew}; x = x_{new}; y = y_{new};$
}