To help me design a more efficient pneumatic gun I have been doing a little bit of ballistic modeling. The goal is to get a certain range with as little gas consumption as possible.

Let me note immediately that I'm not an aerodynamicist. I am a mechanical engineering student, but I'm only a sophomore. I haven't found basic external ballistics to be exceedingly difficult but it is new to me.

Models of the internal ballistics of a pneumatic gun are rather well developed and I intend to study them later. However, I'm having a difficult time finding information about the drag coefficients of a cylinder and a cylinder with a more aerodynamic nose. I can find drag coefficients for the shapes I'm interested in at different aspect ratios, but I know nothing about the Reynolds number or velocity (which essentially would have the same meaning here) that goes along with those drag coefficients. I've read many times that the drag coefficients vary greatly at low speeds/Reynolds numbers, so I'm unsure how reasonable it would be to use a constant drag coefficient.

For those who might wonder, my current exterior ballistics model makes many assumptions but seems reasonable for what I intend to do. For that reason I doubt this would be useful for other spud gunners. The gun I intend to use this analysis on will be a Nerf gun, so I can make many assumptions that others couldn't make about other projectiles because the center of gravity is so far ahead of the center of pressure. That essentially means the dart will be very stable over the entire trajectory because any deviation in the angle of attack will create a relatively large torque about the center of gravity, pushing the dart back to zero angle of attack.

My model essentially is a system of equations of the force of gravity and drag on a point mass. I reduced the two second order non-linear ODEs to a system of 4 first order non-linear ODEs and used the basic 4th order Runge-Kutta method to approximate the solution. The results seem reasonable for a constant Cd, but I'm sure once I try to fit some actual data to the model I'll notice some problems.

Any specifics or perhaps a description of the trend would be very helpful. At worst I suppose I'll have to figure out how the Cd varies with velocity after the testing I plan to do. Having a decent idea of what to expect can't hurt.

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