Magnus Effect Equation

So I’ve been looking into taking into account the Magnus effect for calculating range for our shooter. I stumbled across this document ece5bc8fb62d73248f77701cf40f070b114052da.pdf (104.6 KB) which takes the equation from an old Wikipedia page(if you go back in the edit history you can find it). I also found this which seems to contradict that equation. If anyone knows how to calculate the force applied from the Magnus effect, any help would be appreciated!

When I was looking into this earlier in the season I concluded this was one of the rare times that Ether’s math was wrong (or at least misleading).

The Magnus force equation should be proportional to (linear velocity x angular velocity), not linear velocity squared as in Ether’s paper. That said, if you assume that the ratio of linear velocity to angular velocity is fixed (as in a theoretically perfect hooded shooter), then the equation still gives you the right magnitude if you assume that what Ether calls “C_L” absorbs the scalar ratio between the two. But it’s highly non-obvious to me if that was Ether’s intention.


Ooof I’m torn, who to trust? @Ether or NASA? I’m thinking NASA, only because they showed their work more precisely (assuming the cylinder equation is correct, it seems to follow through).

Regardless, we did a bit of looking into this using the NASA page. The conclusion we came to: Calculating the exact constants was gonna be too hard to produce meaningful results.

In particular, the dependence on radius cubed scares me - slight compression or deformation in the ball would (per this model) change the magnus force by a proportionally large amount.

Additionally, since the ball is not quite uniform density, nor is the skin smooth, the rotation s seemed unpredictable over time.

Maybe we’re giving up too early - I suspect with a high speed camera, launching pinstripe balls against a grid background could gather enough data to inform the model reasonably…

Then again, as I was typing this, I saw Jared Russell typing a response, and figured I should probably let the pros do the talking.

Exactly. An equation like Kutta-Joukowski (or the drag equation) is useful for understanding relationships between various quantities - e.g. it’s useful to know that Magnus effect increases as linear and angular velocity increase, but not terribly useful to try to predict the actual magnitude from first principles.


Yeah, I was suspicious when I saw no mention of angular velocity in the equation. In the end I suspect we will just plot the distance from the target along with the required roller speed and use a regression that fits, that seems the easiest solution. I wanted to look into the actual math though just for fun, it seemed interesting.

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we’re doing something similar. I would recommend doing spline interpolation, as your results may not be quite linear

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Tough call. I’d be partial to @Ether, myself.

NASA is definitely correct on this one.

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