Ballistics is an ancient art—one that was already old when longbows, catapults, and bronze smoothbore cannons were the high-tech missile projectors of their respective day. "External" ballistics undertakes the prediction of the trajectory of the flight of projectiles (arrows, bullets, cannonballs, air-gun pellets, etc.) from their launcher (bow, gun, etc.) to the target. Recently made more quantitative and accessible by the advent of computers, external ballistics depends on accurate knowledge of many parameters. Two of the most important parameters are the initial speed of launch (e.g., MV = gun muzzle velocity) and the "ballistic coefficient" (BC) of the projectile (i.e., the dimensionless ratio of the weight/drag of the projectile to the weight/drag of a standard reference projectile defined to represent BC = 1.0).
Inexpensive optical chronographs are available that will accurately measure MV (see "Optically Measure Projectile Velocity With An Ohmmeter," Electronic Design, Oct. 29, 2001, p. 57). But determination of BC is more complicated, requiring projectile-velocity measurements not only at the muzzle but also at a minimum of at least one point downrange. Downrange velocity measurements typically in-volve costlier and more specialized instrumentation that may be beyond the means of the amateur archer or marksman.
This IFD presents an alternative BC-measurement scheme that combines a PC with inexpensive acoustic sensors (electret microphone cartridges) to measure projectile time-of-flight (i.e., time between the sounds of launch and target impact) over two different target ranges. These data points are sufficient to compute both MV and BC. Software access to the PC's internal 8254 realtime clock is used as a ±10-µs precision timebase.
Figure 1 shows the signal-conditioning and interface circuitry between the electret mikes and the PC parallel port. The output of each mike is input to a comparator (LM339) with an adjustable trigger threshold. When the microphone element detects a sound transient loud enough to exceed the threshold bias, the comparator triggers and generates a greater-than 100-µs pulse on the connected parallel-port status bit. "Time of Flight" (TOF) software (the listing can be viewed or downloaded at www.elecdesign.com under Ideas for Design) running on the PC captures the interval between the muzzle blast (mike #1) and target impact (mike #2 for the near target or mike #3 for the far target) and converts the TOF pairs to back-extrapolated muzzle velocity (MV) and ballistic coefficient (BC).
Conversion of TOFs to BC and MV can be done in a number of ways. The method used in this example assumes subsonic (i.e., < 900 ft/s and non-compressible aerodynamics) projectile flight and, therefore, a square-law relationship of projectile deceleration to airspeed: DP = V2/C for some C proportional to the mass/drag ratio of the projectile. This square-law drag functionality implies the existence of a "Natural" or "Napierian" range (NR). This range is defined as the incremental flight distance required for air drag to reduce velocity by a factor of "e," where "e" is the familiar Napierian logarithm base = 2.71828....
Interestingly, DP = V2/NR. Computation of NR is easiest if the TOF is measured over two ranges in a ratio of 2:1. Let "near" = the near target range (ft), "tN" = the TOF to the near target, and "tF" = the TOF to the far target. Note that tF > 2tN, since air drag continuously slows the projectile during its flight, so flying twice as far takes longer than twice the time.
NR = near / log((tF/tN) − 1).
With the Napierian range NR (ft) in hand, MV (ft/s) and BC become available as:
MV = NR / (tN(e(near / NR) − 1))
BC = NR / 24,000.
The concepts presented here are particularly useful for the calibration of BC and MV in air rifles. By shooting alternately over the two target ranges, accurate values for BC and MV are quickly acquired. This enables realistic trajectory prediction using any of several ballistic software packages. This is helpful in preparation for long-range marksmanship competitions.
Subsequent to finishing this project, it was brought to the author's attention that the seminal idea of acoustic measurement of airgun projectile TOF was previously described in The Air Gun from Trigger to Muzzle, G.V. Cardew, G.M. Cardew, and E.R. Elsom, Martin Brothers (Printers) Ltd., Birmingham, May 1976.
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