Good morning mathematicians,

I would like to ask you for good resources for modelling projectile motion under air resistance. I have not been able to find any good sources.

Thank you in advance.

  • $\begingroup$ Do you have a specific type of object in mind? A bullet? An airplane? $\endgroup$
    – Matti P.
    Jan 28, 2019 at 8:06
  • $\begingroup$ @Matti P. I would stick with bullet or sphere to maintain it simple. Because plane would be too complicated object for me I guess, yet. $\endgroup$ Jan 28, 2019 at 8:15
  • $\begingroup$ Well, a good starting point is to consider the aerodynamic drag of the object. After that, the question comes to setting the value of the drag coefficient $C_D$. en.wikipedia.org/wiki/Drag_(physics) $\endgroup$
    – Matti P.
    Jan 28, 2019 at 8:23
  • $\begingroup$ @Matti P. I guess that a decant amount of physics background is required like understanding of magnus effect when deriving and modelling the equations? $\endgroup$ Jan 28, 2019 at 8:26
  • 2
    $\begingroup$ It all depends on what your requirements are for the model - or how sophisticated model you want. But I would say that if you have the lift and drag coefficients (if the object does not generate any lift, or is essentially not an aircraft, then $C_L=0$), you already have a pretty good model. Is the object spinning during its flight? $\endgroup$
    – Matti P.
    Jan 28, 2019 at 8:30

1 Answer 1


Perhaps the simplest model for modelling projectile motion that includes air resistance is what is known as the linearly resisted model or the linear drag model. Here the resistance to the motion of the projectile as it moves through air is considered to be proportional to its velocity.

For some accessible accounts of this model, see, for example:

  1. Projectile Motion with Resistance and the Lambert W Function

  2. Analysis of asymptotic projectile motion with air resistance using the Lambert W function

  3. An analytic approach to projectile motion in a linear resisting medium

  4. Linear resisted projectile motion and the Lambert W function

  5. Exact expressions for the range and the optimal angle of a projectile with linear drag


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