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Good day!

I am interested in the solution of the following problem, could you please help me.

Suppose that we have some solid airplane. We throw this airplane at some angle $\theta$ to the horizon with some initial velocity $v_{0}$. At the beginning, the airplane is oriented parallel to the velocity $v_{0}$. What is the height of the maximal lift of the airplane, and at what distance from the throwing point this airplane will land?

We assume that all the necessary parameters are known. The initial speed $|v_0|$ is less than 50 $\frac{m}{s}.$

I would like to remark that an approximate solution is also accepted, but the error shouldn't be very large :) Good estimates will be also very useful)

If you don't know the exact answer to my question but you know some article or book that may help me to figure this problem out, I would be also very glad to receive your advice :)

Thank you!

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  • $\begingroup$ Is this school problem with writing $x$ and $y$ speeds and distances? $\endgroup$
    – yW0K5o
    Commented Sep 16, 2023 at 11:03
  • $\begingroup$ Well, if, instead of an airplane, we had a simple projectile, then yes, it would become an easy school problem. $\endgroup$
    – Liubimov Oleksandr
    Commented Sep 16, 2023 at 11:08
  • $\begingroup$ If so, the answer is the maximum lift $y_{max}=\frac{v_0^2sin^2\theta}{2g}$ and the distance $x_{max}=\frac{v_0^2sin2\theta}{g}$. $\endgroup$
    – yW0K5o
    Commented Sep 16, 2023 at 13:41
  • $\begingroup$ Thank you for the answer, but, unfortunately, it is not what I am interested in. These formulas work only for the simple projectiles. The issue with an airplane is that it is also affected by the aerodynamical forces like drag and the lift force which change the result very much $\endgroup$
    – Liubimov Oleksandr
    Commented Sep 16, 2023 at 14:35
  • $\begingroup$ I understand now. $\endgroup$
    – yW0K5o
    Commented Sep 16, 2023 at 14:46

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In this case "Assume that the necessary parameters are known and express the space of answers" means the same thing as "Write a multi-volume treatise on airplane design."

Try tossing a balled up piece of paper, a paper airplane with the backs of the wings folded down slightly, and a paper airplane with the backs of the wings folded up slightly.

The only thing we can know from the given information is that energy is conserved, so maximum height is some value less than the height which corresponds to its initial kinetic energy in the frame of the air: $mg(h-h_0) \lt .5mv^2$ where $v$ is the initial airspeed, all variables positive.

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  • $\begingroup$ Thank you for the answer. Is it possible to construct an approximate solution with some simplifications and without using very complicated theory from aeromechanics? Estimates with not so large errors will be useful too :) $\endgroup$
    – Liubimov Oleksandr
    Commented Sep 15, 2023 at 21:40
  • $\begingroup$ you need some details about center of gravity and aerodynamic center of the airplane. Then, maybe you need an approximation of no oscillatory motion around the trajectory. Tomorrow or the day after I could try to provide an answer, if I can recall enough from my studies of flight dynamics in aeronautical engineering $\endgroup$
    – basics
    Commented Sep 15, 2023 at 21:47

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