Newton's old formula is more useful than it looks
This white paper comes from Dr. Ken Oehler, founder of Oehler Research. The original is written in a conversational tone and its title borrows from an American country song, so the change of register is no surprise when you open the source PDF. The technical argument underneath it is very clear: modern ballistic solving chases ever finer correction terms, and in doing so it lets people lose sight of the most basic thing of all, gravity.
In meters and seconds the free-fall approximation is distance ≈ 5 × time² (that is, ½gt²). Fall for 2 seconds and you fall 20 meters. Turn it around: throw a stone straight up to 20 meters and it rises for 2 seconds, falls for 2 seconds, 4 seconds in all. It looks unrelated to trajectory work, and it is in fact the pivot of the whole paper.
The author is not dismissing the three centuries of progress since Newton. He quotes ballistician Bob McCoy's observation that for flat-fire, small-yaw ground small-arms trajectories the older traditional methods are adequate for all practical purposes. Get the basics right first, then discuss the details.
4 seconds of flight: eight projectiles, the same 20 meters
Oehler opens the US Army firing tables for the 105 mm tank gun and the 25 mm cannon, but he does not start from the range column. He starts from the time column, finds the row where time of flight is close to 4 seconds (2 seconds up plus 2 seconds down), and then reads off the corresponding range and maximum ordinate, the highest point above the line of sight. He puts the large-caliber rounds, a 7.62 mm match load, a .22 rimfire and a football with 4 seconds of hang time into one table:
105 mm M735 APDS: about 5500 meters, maximum ordinate 19.2 meters. 105 mm HEAT: about 2950 meters, 20.7 meters. 105 mm M450 TP training round: about 2950 meters, 20.6 meters. 25 mm M791 APDS-T: about 3900 meters, 19.3 meters. 25 mm M793 TP-T: about 2150 meters, 19.9 meters.
7.62 mm Federal 175 grain (2600 fps muzzle velocity): about 1600 meters, 20.3 meters. .22 rimfire Federal Match: about 800 meters, 21.4 meters. A football with 4 seconds of hang time: about 50 meters, 19.6 meters.
Range runs from 50 meters to 5500 meters, a factor of 110, yet every maximum ordinate falls between 19.2 and 21.4 meters, which is the 20 meters Newton's formula predicts. How far the projectile travels horizontally has almost no bearing on the drop; the drop is governed by time of flight. (The figures above are all taken from the table in the source paper. The projectile designation on the 105 mm HEAT row is hard to read after the PDF was reflowed, so only the round type is kept here.)
The shooter sees distance, not time
The difficulty is that the ballistician thinks in time while the shooter can only see distance. The two have to be connected, and the connecting tool is the drag function or drag table. These tables have never matched the particular bullet in your hand, yet they remain in general use: out to the distance at which a rangefinder becomes mandatory, the prediction is usually good enough to get on target. Further out, it often fails.
With no air drag the relationship is almost boringly simple: a bullet at 3000 fps covers 1000 yards in exactly 1 second, and distance against time is a straight line. Add drag and the velocity no longer holds; the curve bends with time, and how much speed is lost at each instant depends on the current velocity and on the assumed drag function. Only a very high ballistic coefficient in thin air comes close to that straight line.
In recent years the practice promoted by instructor Todd Hodnett is known as truing: the student actually shoots a far target, then goes back and adjusts the assumed muzzle velocity or ballistic coefficient in the solver until the prediction hits that far target, and ties the resulting numbers to this rifle with this lot of ammunition. The method is simple and it genuinely works, but there is room to improve it.
G1 or G7? The two curves nearly overlap
Take a bullet leaving at 3000 fps that actually covers 1000 yards in 1.500 seconds. Fit it in the G1 family and the curve with BC1 = 0.471 passes through that measured point. Switch to the G7 family and the curve with BC7 = 0.238 passes through it as well. One problem, two answers: which do you use?
Plot both curves on the same chart and the G1 line is completely hidden behind the G7 line, indistinguishable all the way past 1000 yards (about 1.5 seconds). Compare the drop and wind deflection the two predict and the difference inside 1000 yards stays within 0.05 mil. In other words, provided you have actually trued at 1000 yards, choosing G1 or G7 makes no practical difference.
The same point appears in two more sets of plots in the source paper: at 3000 fps, three curves from the G1 family (BC1 = 0.750 / 0.500 / 0.375) and three from the G7 family (BC7 = 0.374 / 0.250 / 0.189). The two charts are almost impossible to tell apart inside 800 yards. What really decides accuracy is not which drag function you pick, but whether measured data has been used to pin the curve in the right place.
Below the transonic: stepped ballistic coefficients
The differences appear past 1000 yards. Sierra and others have provided velocity-dependent stepped ballistic coefficients for years to compensate for the offset between measured drag and the G1 prediction, but those stepped values usually cover only the supersonic segment, where the ballistic coefficient varies relatively little. Testing in the source paper indicates that the stepped approach can be extended to much longer distances, provided the step boundaries and the coefficients are set from long-range measured time of flight.
The example in the paper runs like this. If the bullet in question really behaves as G7 with BC7 = 0.238, its time of flight at 1400 yards is 2.57823 seconds. To describe the same bullet with G1, BC1 has to be changed from 0.471 to 0.331 below 1350 fps. With that segment added, the G1 curve is again completely hidden behind the G7 curve out to roughly 1500 yards.
The working procedure is to fire another test at a distance where the bullet has clearly dropped into the subsonic region, enter the first measured ballistic coefficient and the velocity boundary into a solver that handles stepped BC, then adjust the ballistic coefficient of the next segment until the predicted time of flight matches the measured time of flight. The stepped ballistic coefficients you end up with do not look elegant, but from the muzzle to the subsonic target the drop and wind deflection predictions are typically within 0.1 mil. However far you want the prediction to hold, that is how far you have to test.
The author also stresses that ballistic coefficient and drag function cannot be derived from published data, any more than muzzle velocity can be derived from factory specifications or a loading manual. They vary from rifle to rifle and from lot to lot, and can only be trued with your own rifle and your own ammunition. Truing on measured time of flight additionally removes the uncertainty of judging impacts by eye, wind-induced error, hold error and too few shots fired.
Putting the proving ground in a box: what you have to measure
The measured data this method needs is modest: weather data (barometric pressure and temperature above all, which a handheld weather meter will give you), an exact range to the target (a laser rangefinder first verified to 0.1% accuracy, used with a suitable reflective target), muzzle velocity, time of flight to the far target, and software able to analyze the raw data. Government proving grounds rely on expensive Doppler radar and a room full of ballisticians; what the author wants is a system one person can move to an unprepared range, operate alone and still get results from.
Time of flight is the hard part. The interval between two signals can be measured very precisely and the start signal is easy, since a skyscreen at the muzzle will trigger the timer. The difficulty is the stop signal downrange. A microphone can hear a supersonic bullet go by, but the delay between the bullet passing and the sound arriving introduces error, and even when passage or impact is detected at the far target, the signal still has to get back to the timer at the firing point. Running 1000 yards of cable usually works on the first day; moisture, vehicles and rodents see to it that cable left out overnight does not.
Oehler solved this timing and communication problem for proving grounds in the 1990s. System 86 acoustic targets linked the shooter and several remote targets by radio and supplied time of impact as a by-product. Because the bullet's passing position relative to the microphones was known, the delay could be corrected and time of impact converted into time of flight, and the ballistic predictions built on that turned out to be unexpectedly accurate. System 88 later inverted the relationship. Muzzle velocity and time of flight became the primary outputs and the target data the added value, with built-in radio and GPS timing replacing the cables downrange. Its software computes a ballistic coefficient for each shot directly from the measured values and the specified drag function, and a separate Extended Range Truing program automates the stepped-truing procedure.
The remote microphones can be arranged in a square array up to 10 feet on a side, which gives good target accuracy alongside time of flight, or in a line array, convenient as a fly-through terminal target or mounted on a vertical pole for an intermediate fly-over measurement. The source paper closes by trailing a simpler successor system: in practice more than one target is rarely used at once, and long-range time of flight measurement does not need microsecond precision, since the roughly 0.2 millisecond timing accuracy of a single-hop radio link is plenty for flight times over a few hundred meters. The whole system would cost about as much as a good riflescope.