# Projectile Motion with Drag without Numerical Analysis

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.

• 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. Dec 24, 2016 at 0:51

## 1 Answer

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.

• 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? Dec 24, 2016 at 1:19
• Oops, sorry, you are correct. I'll edit my answer.
– Pirx
Dec 24, 2016 at 1:22
• Note on edit; I have changed the coefficients to $k$ Dec 24, 2016 at 2:27
• 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.
– Pirx
Dec 24, 2016 at 18:19
• 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.
– Pirx
Dec 26, 2016 at 15:17