2.3.1 Initial Velocity and Final Velocity Method

2.3.1.1 Measurement Procedure

This first method is illustrated in Figure 2.3-1. This method uses two chronographs for each bullet fired to measure an initial velocity and a final velocity at a measured range distance between the chronographs. The initial velocity chronograph is usually placed near the muzzle of the gun, as shown in Figure 2.3-1. A blast shield with a small hole for bullet passage usually is used to keep the muzzle flash or blast from disturbing the screens of the initial velocity chronograph. If the screens are photoelectric types, as is the usual case, the muzzle flash may trigger screen 1 before the bullet arrives. Also, because the powder gases exit the muzzle at about 1.5 times the bullet velocity, the gases can trigger screen 1. Or, the muzzle blast can cause screen 1 or 2 to bend or vibrate. Any of these effects will cause an erroneous measurement of the initial velocity, which will, in turn, cause an error in the measured BC value.

The final velocity chronograph is placed downrange at a carefully measured distance from the initial velocity chronograph. This range distance is measured from the center point between screens 1 and 2 to the center point between screens 3 and 4. This is because each chronograph really measures the bullet travel time between the two screens to which it is connected (i.e., between screens 1 and 2 or between screens 3 and 4). Then, the velocity is obtained by dividing the precisely measured distance between the pair of screens by the measured bullet travel time. This calculation is performed within the electronics of the chronograph. Because the bullet slows down a tiny bit as it travels between the two screens, the measurement of velocity is considered to be valid at the center point between the pair of screens. Of course, if the separation distance between screens 1 and 2 is the same as between screens 3 and 4, the range distance between the two chronographs may be measured from screen 1 to screen 3.

The measurement procedure for each bullet fired is to record the initial and final velocities as well as the range distance between the two chronographs. The altitude, temperature, barometric pressure, and relative humidity at the firing point must also be recorded. If the shooting range is not level, the elevation angle must be recorded, especially if it exceeds about 3 degrees either upward or downward. If there is any appreciable wind at the firing point, BC measurements should not be attempted.

With these data for each bullet fired, an exterior ballistics software program for a personal computer can be used to calculate the ballistic coefficient. Some exterior ballistics programs contain an optional routine for computation of the BC value, but that is not necessary. The normal trajectory computation routine can be used in an iterative fashion for each bullet fired. That is, first initialize the program by entering the altitude, temperature, barometric pressure, humidity and range elevation angle at the firing point. Then, for each round fired enter the measured initial velocity as the “muzzle velocity” for the trajectory calculation. Then, guess a BC value, calculate a trajectory over the measured range distance, and examine the calculated final velocity. If the calculated final velocity is higher than the measured number, the BC value is too high. (Conversely, if the calculated final velocity is lower than the measured value, the BC value is too low.) Then, reduce (or increase) the BC value a little, calculate another trajectory over the measured range distance, and examine the calculated final velocity. The calculated final velocity should be nearer the measured value. If the calculated value from this second iteration is higher (or lower) than the measured value, reduce (or increase) the BC value in the program and perform another iteration of the calculations. After a few iterations, this method will “home in” on a correct value for the measured BC for the first round fired. Of course, this is a BC value for which the calculated final velocity matches the measured final velocity as closely as possible. This resulting BC value is considered valid for a bullet velocity midway between the initial and final velocities for that round.

For the other rounds fired, the resulting BC for the first round can be used as the initial guess for the BC value, and fewer iterations will be required to reach a correct BC value for each of those rounds.

Figure 2.3-1 Test Range Setup for Initial and Final Velocity Method for BC Measurement

An example has been prepared to illustrate this method of determining the BC. Suppose we have developed a load for the 308 Winchester (7.62 x 51 mm NATO) cartridge in a bolt-action rifle that pushes a certain .308 diameter 160 grain bullet at a muzzle velocity of about 2750 fps. We do not know the BC of this bullet type and want to measure it on our local shooting range. (Actually, we do not need to know the weight of the bullet or even the caliber to determine the BC. We need only the firing test data as described below.) Suppose that our shooting range is located at an altitude of 790 feet above sea level, and we perform the shooting tests on a day when the temperature is 78° F, the barometric pressure is 30.15 inches of Mercury, and the relative humidity is 80%. Note that the barometric pressure is obtained from a barometer at the range or from a local weather report for the time of day when the firing tests take place. The test range is level, and the range distance between chronographs is 103 yards, which is measured precisely and accurately when we set up the range for the tests. We fire, say, ten rounds to obtain an average BC value for this bullet type at velocities in the vicinity of 2750 fps. We record the altitude, atmospheric conditions, range distance between the chronographs, and the initial and final velocities for each round fired. Then, we retire to our computer at home.

We start up the Sierra Infinity program, and it comes up automatically in the “Trajectory” mode of operation. Suppose that for the first round fired at the range, the initial velocity was 2742 fps and the final velocity was 2549 fps, as read from the initial and final velocity chronographs. In the Infinity program we select any 30 caliber bullet in the “Load Bullet” library, and transfer it to the “Active Bullets” list in the upper right corner of the blank part of the screen. Then, we initialize the trajectory computation as follows:

Trajectory Parameters

Units: Full English (since we are working in the English system of units) Muzzle Velocity: 2742 fps (for the first round fired) Maximum Range: 103 yds (this is as far as we need the trajectory to be computed) Range Increment: 1 yd (because the distance between chronographs is 103 yds, which is not divisible by any number other than 1) Zero Range: 103 yds (the distance between chronographs) Elevation Angle: 0 (because the test range is level) Sight Height: 1.5 inches (choice for telescope sight on the rifle)

Environment Parameters

Barometric Pressure: 30.15 in Mercury (from a barometer at the range or a local weather report) Temperature: 78° F (from a thermometer at the range or local weather report) Altitude: 790 ft (can be obtained from a topographical map or other source) Humidity: 80 % (relative humidity from a weather station at the range or from the local weather report) Wind Direction: Any number between 0 and 12 o’clock is OK

Horizontal Wind Velocity: 0 mph (no wind is very important) Vertical Wind Velocity: 0 mph (no wind is very important) At this point we have initialized a trajectory computation for some 30 caliber bullet (we don’t care which one) with the correct muzzle velocity, range distance between the chronographs, trajectory calculation parameters for our purposes, and environmental conditions at the firing point. But, we haven’t performed any trajectory computation yet, so there is nothing on the monitor screen yet. Now, although we will not use it explicitly, we must “Calculate” a trajectory so that the “Trajectory Variations” menu item will be available to us.

We then go to the Infinity toolbar at the top of the monitor screen and select “Trajectory Variations.” From the dropdown menu that appears, we select “Ballistic Coefficients.” In the sidebar at the right side of the monitor screen, we then see five values of ballistic coefficient listed. These values mean nothing to us since they are for the bullet that we chose to load into the “Active Bullet” list, not for the bullet that we are testing.

It is necessary now to make an initial guess for the BC value of our test bullet. If we guess well, we will not have to make many computation iterations to find the correct BC value. For a 30 caliber bullet that weighs 160 grains and has a Spitzer (sharp pointed) shape, the BC value at around 2600 fps should be somewhere near 0.5. So, let us choose this value as the initial guess. We then change the five numbers in the right-hand sidebar on the monitor screen to the value 0.5.

We change all the BC numbers for a particular reason. As we will explain later, Infinity allows the ballistic coefficient of each bullet type to change with bullet velocity as it flies downrange and slows down. This is because the measured BC of a bullet does change with velocity, and accounting for such changes can increase the accuracy of trajectory computations within Infinity. We use five velocity regions for this purpose. Within each velocity region there is a single value of BC valid for that region, and there is a value for each of the five regions. There are then four velocity boundaries separating these regions. When the velocity of a bullet falls through one of these boundaries, Infinity automatically changes the BC to the value for the new region. In our current case, we do not know whether our test bullet starting at 2742 fps and ending up at 2549 fps crosses a velocity boundary for the bullet we are using.Yes, we could look to see and make a more educated selection of the one or two BC values that we would need to change, but if we change all five of the values in the sidebar we will be sure to be safe.

After we change the BC numbers in the sidebar to 0.5, we are ready to begin the iterative search procedure for the correct BC value for the first round fired. Table 2.3-1 summarizes the computations in the search procedure. To begin the procedure, click the “Calculate” button on the bottom of the monitor screen. Infinity performs the first trajectory computation, the trajectory parameters appear on the screen, and we immediately scroll down to the final parameter values at 103 yds range. We find that the computed final velocity at 103 yds is 2564.8 fps for this first iteration (see Table 2.3-1). This is higher than the measured final velocity for this round 2549 fps, so the next guess for BC needs to be lower than 0.5.

At this point we have no idea how much to lower the next BC guess, but let’s try 0.4. We set the five BC numbers in the sidebar to 0.4 and click the “Calculate” button for the second trajectory computation. The computed final velocity at 103 yds for the BC equal to 0.4 is 2521.5 fps, which is too low compared to the measured 2549 fps. So for the third iteration, the guess for BC needs to be raised. Let’s try something halfway between 0.4 and 0.5; that is 0.45.

We change all five BC numbers in the sidebar to 0.45 and click the “Calculate” button for the third trajectory computation. For this third iteration, we find that the computed final velocity at 103 yds is 2545.5 fps, which is closer but still lower than the measured 2549 fps. So, for the next iteration let’s raise the BC guess to 0.46.

Again, we change all five BC numbers in the sidebar to 0.46 and click the “Calculate” button for the fourth trajectory computation. For this fourth iteration, we find that the computed final velocity at 103 yds is 2549.7 fps (as shown in Table 2.3-1). This is just a little higher than the measured 2549 fps. So, for the next iteration we must lower the BC guess just a little.

Table 2.3-1 shows the final three iterations, each of which follows the same procedure. In each iteration we change the BC guess by a smaller amount so that the computed final velocity approaches the measured final velocity. The seventh iteration, which has a BC of 0.4583, produces a final computed velocity equal to the measured velocity, and this BC therefore is the correct value for this first fired round.

Table 2.3-1 Example of Ballistic Coefficient Iterative Search Procedure

Test Range Parameters:

Distance between chronographs: 103 yds Range altitude: 790 ft above sea level Temperature: 78º F

Barometric pressure: 30.15 in Mercury Relative humidity: 80% Exterior Ballistics Program: Sierra infinity

Test Round 1: Initial velocity 2742 fps; final velocity 2549 fps

Iteration   BC value   Computed final velocity  
1   0.5   2564.8  
2   0.4   2521.5  
3   0.45   2545.5  
4   0.46   2549.7  
5   0.458   2548.9  
6   0.459   2549.3  
7   0.4583   2549.0  

Test Round 2: Initial velocity 2751 fps; final velocity 2556 fps

1   0.4583   2557.6  
2   0.457   2557.1  
3   0.455   2556.2  
4   0.454   2555.8  
5   0.4545   2556.0  

 Table 2.3-1 also shows the iterations for the second fired round. In this example we suppose that the second round has a measured velocity of 2751 fps and a measured final velocity of 2556 fps. To initialize for the second round, we momentarily return to the “Operations” selection on the Infinity toolbar at the top of the monitor screen, select the “Trajectory” mode of operation, and change the “Muzzle Velocity” entry in the “Trajectory Parameters” sidebar to 2751 fps. Again, we must “Calculate” so that the “Trajectory Variations” menu item is available. Then, we return to the “Trajectory Variations” selection on the Infinity toolbar, again select “Ballistic Coefficients” on the dropdown menu. We verify that all five entries in the BC sidebar on the monitor have the value 0.4583 from the first round. Note that we have not changed any of the other Trajectory Parameters or Environment Parameters, since they all have the same values for our firing tests.

Table 2.3-1 shows the sequence of iterations for the second round. The first iteration with a BC guess of 0.4583 produces a computed final velocity of 2557.6 fps, higher than the measured final velocity of 2556 fps. The second iteration with a BC guess of 0.457 produces a computed final velocity of 2557.1, closer but still higher than the measured 2556 fps. The third iteration with a BC guess of 0.455 produces a computed final velocity of 2556.2 fps, even closer but still a little higher than the measured 2556 fps. The fourth iteration with a BC guess of 0.454 produces a computed final velocity of 2555.8 fps, which is lower than the measured 2556 fps. Since the results of the third and fourth iterations equally straddle the measured 2556 fps, the fifth BC guess is chosen as 0.4545, halfway between 0.455 and 0.454. This final iteration produces a computed final velocity equal to the measured 2556 fps, so this value is the correct value for the second bullet fired.

The same procedure should be followed for each of the remaining eight bullets in the test series that we fired at the test range. In this way, we will derive the measured BC values for all ten bullets and can apply statistical analysis to this limited sample of test bullets for this type of bullet at velocities between the initial and final values.

The computations in this example have been explained in some detail, so that the reader can repeat these calculations step by step if he or she uses Sierra’s Infinity program. If a different exterior ballistics program is used, the detailed steps of the procedure should be changed because any other program will function a little differently, but the basic method will not change. The idea is to find a BC value that makes the computed final velocity equal to the measured final velocity for each bullet tested. This will be an iterative, trial and error procedure. This example will still be useful as a guide even if a different exterior ballistics program is used, because BC values very close to the ones produced by using Infinity should result. This can serve as a check on the procedure developed for any other program.