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) {

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

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

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

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

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

  • $\begingroup$ Let us see your formulae for solving this backwards. Depending on how you did it, the answer likely involves a bit of integration or deriviation. $\endgroup$
    – dotancohen
    Sep 13, 2017 at 13:11
  • $\begingroup$ I edited my question and added the requested formulae for the "backwards" problem. $\endgroup$
    – al3xst
    Sep 13, 2017 at 14:50

1 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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.