In 1875, the following experiment was made at the National Armory for the purpose of determining the length of bore which gives the maximum muzzle velocity with the Springfield rifle, using the service ammunition. A rifle was made so that the length of bore could be increased from five inches to one hundred and twenty-two inches by additional barrels. The muzzle velocity was measured with each length of bore, each muzzle velocity being the mean of ten shots. The ammunition was, of course, the same throughout the experiment. The following table, taken from the report of the experiment. Army Ordnance Notes No. XXXVIII, shows the lengths of bore and corresponding muzzle velocities:
Length of Bore (inches) Muzzle Velocity (feet)
5 704
12 1100
22 (Carbine) 1210
26 1254
32.6 (Service rifle) 1320
42 1345
62 1397
92 1416
102 1410
112 1420
122 1411
In order to find the length of bore corresponding to the maximum muzzle velocity, I tried to find a function which would satisfy the values given in the table. Calling the muzzle velocity v and the
length of bore x, the following form was first tried:
v = a + bx + cx2 + fx3
The values of the constants a, b, c and f were determined from the values of x and v given in the table by the Method of Least Squares. The values of v were then calculated for the values of x given in the table, and the calculated and measured values were compared, but the agreement was not sufficiently close to warrant the use of this form. The form v = ea + bx + cx2 was then tried in the same way. It more nearly satisfied the conditions and showed that this form was better adapted to the purpose than the preceding one, but the differences between the calculated and measured velocities were still too great to make it of any use as an empirical formula. The values of the constants are a = 6.5867, b = .02175 and c =— .0001421.
The calculated and measured velocities are compared in the following table:
Lengths of Bore Inches. | Muzzle Velocity (Calculated) Feet. | Muzzle Velocity (Measured) Feet. | Diff’s. + | Diff’s. — |
5 | 806 | 704 | 102 | … |
12 | 923 | 1100 | … | 177 |
22 | 1093 | 1210 | … | 117 |
26 | 1160 | 1254 | … | 94 |
32.6 | 1268 | 1320 | … | 52 |
42 | 1408 | 1345 | 63 | … |
62 | 1618 | 1397 | 221 | … |
92 | 1612 | 1416 | 196 | … |
102 | 1521 | 1410 | 111 | … |
112 | 1395 | 1420 | … | 25 |
122 | 1243 | 1411 | … | 168 |
|
|
| +693 | -633 |
In order to make the agreement closer another term was added to the exponent so that the form was
v = ea + bx + cx2 + fx3
The calculated curve has now two maximum ordinates. The minimum between them was found in the neighborhood of 102 inches, due apparently to the fact that the measured muzzle velocity for 102 inches is less than that on either side of it. This, of course, is an accidental error, and it would seem to preclude the use of any equation of as high a degree as this unless we reject the result for the length of bore of 102 inches. This minimum is magnified in the calculated curve and the maximum ordinate between it and zero is found between 42 and 62 inches; so that the agreement is not as good as with the previous equation of a lower degree.
The next form tried was one deduced from Sarrau's formula for the muzzle velocity, which, considering all the quantities constant except the muzzle velocity and length of bore.
The constants a and d were determined, as before, by the Method of Least Squares, and were found to be a = .0021869 and b = .050552. The following table gives the calculated and measured velocities:
Lengths of Bore Inches. | Muzzle Velocity (Calculated) Feet. | Muzzle Velocity (Measured) Feet. | Diff’s. + | Diff’s. — |
5 | 772 | 704 | 68 | … |
12 | 1019 | 1100 | … | 81 |
22 | 1201 | 1210 | … | 9 |
26 | 1249 | 1254 | … | 5 |
32.6 | 1311 | 1320 | … | 9 |
42 | 1371 | 1345 | 26 | … |
62 | 1434 | 1397 | 37 | … |
92 | 1437 | 1416 | 21 | … |
102 | 1423 | 1410 | 13 | … |
112 | 1404 | 1420 | … | 16 |
122 | 1380 | 1411 | … | 31 |
|
|
| +165 | -151 |
This agreement is much closer than with any of the preceding forms, although this one contains but two constants. The fair agreement throughout such a great variation in length of bore confirms, as far as this experiment goes, the employment of this function to express the relation between length of bore and muzzle velocity. From the manner in which it was determined we should expect that it would agree better than a purely general form, but how much better can only be determined by applying it to the results of experiments such as this.
From the formula
.0021869v = x4/10 [1 - .050552x1/2]
the maximum muzzle velocity is found to be 1446 feet, corresponding to a length of bore of 77.3 inches. The form v = ea + bx + cx2 gives the position of the maximum ordinate nearly the same, namely, at 76.5 inches, but the corresponding muzzle velocity is too great. Since the acceleration is zero where the muzzle velocity is a maximum, this value of x=77.3 inches marks the point at which the remaining pressure of the powder gas on the bullet is just equal to the friction in the bore.
Many other deductions could be made from this empirical formula, but they involve either the supposition that the curve represents the relation between v and x far beyond 'the values by which it is determined, or the taking of successive derivatives which may depart widely from the truth.