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I'm struggling with a problem, in which I want to determine initial condition (initial velocity, and angle) given that I know my ball has to reach (final velocity, location), but I want to make sure that my ball does not go through a wall. Image below is of soccer freekick, which I want to model.

enter image description here

so far let's say (2D model => x,z only)

$\ x'' = -kx'$

$\ z'' = -kz' - g $

Note : Assume K to be constant.

and my wall constraint are

when $\ 10 < x < 11 $ then $\ z > 10 $

And known values are

$\ x_{t_{2}} = X_{f} $

$\ z_{t_{2}} = Z_{f} $

$\ t_{0} = 0 $

How do you go from here? In actual model, there would be Magnus effect making impossible to solve analytically, is there any way to solve such systems, with constraint ? Any solution that satisfies given known should work. Only way I can think of is to brute force the solution, but I don't even think that is even possible.

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  • $\begingroup$ This is a tough problem. You can reduce the problem by writing out the kinematic equations with constraints at the initial soccer wall (some z and x values) and then constraints with the final goalie wall. I attempted to write this out but it got very complicated quickly. Ignore things like air resistance and the Magnus effect. They won't meaningfully change your answer $v_0, \theta, \phi$ $\endgroup$
    – John M
    Mar 14, 2015 at 20:25
  • $\begingroup$ Your simplification is leading you astray, I think. The first term doesn't make much sense. When viewed from above, the angle arcsin (x'/y') remains constant. x' only goes to zero when the "forward" motion of the ball goes to zero, so the final X value (10 < x 11) can be calculated from x',y', and Yf. Furthermore, from y'' = -ky' you can calculate the elapsed time until the ball hits the wall/goal, and work back to Zf. Without knowing y0 and Yf, you can't solve the problem. In effect you don't know the launch angle or the angular target size. $\endgroup$
    – WhatRoughBeast
    Mar 14, 2015 at 22:34

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