1
$\begingroup$

For a projectile on a horizontal plane, if given values for the angle of projection $\alpha$, the initial velocity $u$, some constant due to friction $k$ and an acceleration due to air resistance of: $$ a_x = k({v_x}^2 + {v_y}^2)\cos\beta $$ $$ a_y = -g - k({v_x}^2 + {v_y}^2)\sin\beta $$ where, at a given point $$ \beta = tan^{-1}\frac{v_y}{v_x} $$ Is it possible to calculate the usual parameters of time of flight, range, max height, landing velocity etc. by hand?

I am aware that one can solve for above by mapping a large series of positions over small increments of time using computer analysis, however I am intrigued to know if there is a more elegant solution than a brute-force algorithm.

EDIT: Are there any conditions with which this system would approximate a closed-form solution? E.g. making $\alpha$ small (and thus $v_y$) or changing the velocity in some way.

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ It turns out that very few realistic (i.e. complex) problems have anything close to "closed-form", analytic solutions. Just a side note (especially if you're searching for more information online): the general term people would use for solving this type of problem on a computer is "numerical methods", "numerical analysis" or "computational methods". People initially did this without computers, doing the calculations by hand --- but still using the same types of methods and approximations. $\endgroup$ – DilithiumMatrix Dec 24 '16 at 0:51
1
$\begingroup$

No, that is not possible, in general. I will note in passing that the equations you propose to describe your problem make no sense, and cannot possibly be correct. Apart from the mysterious origin of those $-7.6\times10^{-3}$ coefficients it's extremely unlikely that the coefficients for the $x$- and $y$- accelerations are the same.

Improve this answer
$\endgroup$
7
  • $\begingroup$ Thank you. Will the second term of $a_y$ not change sign with the sign of $v_y$ as sin and arctan are odd functions? $\endgroup$ – user120568 Dec 24 '16 at 1:19
  • $\begingroup$ Oops, sorry, you are correct. I'll edit my answer. $\endgroup$ – Pirx Dec 24 '16 at 1:22
  • $\begingroup$ Note on edit; I have changed the coefficients to $k$ $\endgroup$ – user120568 Dec 24 '16 at 2:27
  • 1
    $\begingroup$ It might be helpful to know what the purpose is you want this for. If it's ballistics you are interested in, that's an age-old problem of warfare technology, and there's lots of tables available for all sorts of projectiles. I hasten to add that that's just about all I know about the field of ballistics; you'd have to do some web research on this yourself. Other than that, while the differential equations can be integrated in closed-form for the one-dimensional case, those would be of no use at all to determine things like the time of flight, range, or maximum height of a projectile. $\endgroup$ – Pirx Dec 24 '16 at 18:19
  • 1
    $\begingroup$ If you are interested in the kind of parameter range treated in that NASA example (throwing of a baseball, I believe) then those solutions give a decent approximation, yes. Like I said, in general you will need to state the application and associated parameter range you have in mind for someone to be able to recommend a suitable approximation. $\endgroup$ – Pirx Dec 26 '16 at 15:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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