How Can Doppler Radar Improve Drag Coefficient Measurements in Physics Labs?

[Dr. Courtney]

https://aapt.scitation.org/doi/10.1119/1.5064562

The availability of inexpensive Doppler radar has made it practical to accurately measure drag coefficients in undergrad physics labs.ABSTRACT
Undergraduate lab design balances several factors: 1) simple experiments connected with learning objectives, 2) experiments sufficiently accurate for comparisons between theory and measurements without gaps when students ascribe discrepancies to confounding factors (imperfect simplifying assumptions, measurement uncertainties, and “human error”), and 3) experiments capturing student attention to ensure due diligence in execution and analysis. Drag coefficient measurements are a particular challenge, though there has been some success using accurate measurements of terminal velocities. Video shows promise in several areas of kinematics, but the number of trials in a reasonable time is limited, and analysis techniques to determine drag coefficients often include numerical integration of differential equations. Here we demonstrate a technique with potential to measure drag coefficients to near 1% accuracy using an affordable 2.4 GHz Doppler radar system and round plastic pellets from an Airsoft launcher.

 

[jr-c1]

I’ve read your paper with great interest. I have a similar interest in calculating BCs from Doppler Radar, but I’m have difficulties getting the calculation correct.

I know I could use online calculators, I’d rather do the math myself. Could you share what formula you used?

 

[Dr. Courtney]

I’ve read your paper with great interest. I have a similar interest in calculating BCs from Doppler Radar, but I’m have difficulties getting the calculation correct.

I know I could use online calculators, I’d rather do the math myself. Could you share what formula you used?

I use the online JBM ballistic calculator to compute BCs (ballistic coefficients) from measured near and far velocities. There is not a closed form analytic formula, because of the drag models used for how drag changes with velocity require numerical integration of a differential equation. The JBM ballistics calculator (http://www.jbmballistics.com/cgi-bin/jbmbcv-5.1.cgi ) uses a guess-check iterative method to try a variety of BCs and iterate toward the correct one when integrating the differential equation gives the measured far velocity for the supplied near velocity.

BC is an industry standard metric for air drag of bullets. It is inversely proportional to cross sectional area and drag coefficient and directly proportional to mass. It allows the aerodynamics of different bullets to be more directly compared even if they have different masses or diameters. At sea level, it is roughly the fraction of 1000 yards at which a bullet loses half of its initial kinetic energy. In other words, a bullet with a BC of 0.500 loses close to half of its initial energy at a range of 500 yards.

Of course, it is also possible to use a guess-check iterative method to compute drag coefficients from near and far velocity measurements by numerically integrating the differential equation multiple times until the far velocity matches the measurement for a given near velocity and other conditions. We've done this for a number of trial cases and verified that the much simpler formula used in the above paper is accurate as long as the velocity does not change much over the flight path. If velocity changes too much over the flight path, error is introduced in two ways: 1) The average velocity used in the denominator is further from what is really needed (because V squared appears in the formula) and 2) The drag coefficient itself is changing with velocity.