Determine initial velocity of an object which was thrown (WITH air resistance)

Question

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) {

$v_{xnew} = kmass \sqrt{v_x^2 + v_y^2} * v_x * \Delta t$

$v_{ynew} = (kmass \sqrt{v_x^2 + v_y^2} * v_y - g) * \Delta t$

$x_{new} = x + \frac{v_x + v_{xnew}}{2} * \Delta t$

$y_{new} = y + \frac{v_y + v_{vynew}}{2} * \Delta t$

$v_x = v_{xnew}; v_y = v_{ynew}; x = x_{new}; y = y_{new};$

}

Answer 1

You have the "forward" function x=f(v), and you want to know for what value of v you get the desired value of x which I will call $x_t$ (target).

You could use a simple Newton-Raphson consecutive approximation. Start with the velocity $v_1$ you calculate for the no-friction case and obtain a value for $x_1$ (it will be too small). Then double the velocity to $v_2 = 2v_1$, and compute $x_2$. You can then compute your next estimate of velocity, $v_3 = v_2 + \frac{v_2-v_1}{x_2-x_1}(x_2-x_t)$

Now evaluate $x_3$, pick the two closest values and repeat. For a well behaved function this approach will quickly converge - and at any rate you always have the bounds (upper/lower) of your answer as soon as the result ($x_i-x_t$) changes sign between the two nearest estimates. Keep going until they are "as close as you need".