paper: Ballistic trajectory with air friction drag and Magnus

*Better late than never.

For students (and teachers) interested in the physics, math, and computer simulation aspects of the ballistic trajectory problem.

Ballistic trajectory with air friction drag and Magnus
by: Ether

Rebound Rumble ballistic trajectory with air friction drag and Magnus Effect (backspin/topspin lift/dive). For students interested in the physics, math, and computer simulation aspects of this problem. Free-body force diagram, derivation of differential equations of motion, and C pseudo-code for trapezoidal numerical integration.

Rebound Rumble ballistic trajectory with air friction drag and Magnus Effect (backspin/topspin lift/dive).

For students interested in the physics, math, and computer simulation aspects of this problem.

Free-body force diagram, derivation of differential equations of motion, and C pseudo-code for trapezoidal numerical integration.

Note: In this paper, I have used the sign convention that “g” (acceleration due to gravity) is negative. So if you are trying the integration algorithm, make sure to set g equal to -9.8 m/s/s (or -32 ft/s/s).

ballistic trajectory with drag & magnus.pdf (105 KB)
Appendix A.pdf (33.1 KB)
Appendix B.pdf (1.4 MB)

*Appendix A (2nd order integration) added.

*Appendix B added. Graphs of simulation runs.

http://www.chiefdelphi.com/media/papers/2725

Pretty interesting. Unfortunately it looks like just a 2D simulation. The 3rd dimension can really play into the aerodynamics, for example, the surface velocity of a spinning ball spins relatively slower the closer you get to the “poles” of the sphere. This would be valid for an infinitely long, smooth cylinder. Not to say this doesn’t provide valuable insight into how the Magnus effect affects rotating bodies in air.

It’s still valid for a sphere, provided you use the appropriate coefficients.

Not to say this doesn’t provide valuable insight into how the Magnus effect affects rotating bodies in air.

Yes, insight. This was the purpose of the paper, to use Rebound Rumble as an “excuse” to get students interested in the physics, math, and computer simulation aspects of ballistic trajectories using a hopefully-not-too-difficult simplified model.

For Rebound Rumble, a simulation was not necessary and perhaps not even all that helpful for the purpose of building a winning robot. This is because, as many folks have already pointed out here on CD, the major design challenge of the shooter for Rebound Rumble was to get a shooter that was consistent from shot to shot. This was a big challenge due many factors, a key one being the variation in the ball due to manufacturing tolerances and changes in the ball’s physical characteristics as it got worn. Once you’ve built the most consistent shooter you can, it’s a simple matter to test-fire it to determine its performance.

Hmm, perhaps you’re right (capital D instead of lowercase D in CD). However, finding the correct constants aren’t so trivial.

If I had a shooter that could launch a ball with the same (known) launch speed and launch angle with and without spin1, I would try the following:

Set um=0 and adjust ud to match the trajectory1 with no spin.

Leaving ud at that setting, adjust um to match the trajectory with the expected spin. (repeat to get um’s for various spins if needed).

Then I’d launch the ball at various angles (and speeds) and see how well the simulations matched the actual trajectories1.

It would make an interesting science project.

1Probably need a high-speed frame-timestamped video camera to determine launch speed, angle, spin, and trajectory
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