Cartridge arithmetic

Energy, pressure, sectional density, calibre and velocity: the figures a cartridge record holds that must agree with one another.

The model

A cartridge record holds figures that describe one load and one case. They are restated measurements of each other, so arithmetic says whether they agree, and a record that disagrees with itself has at least one wrong figure.

E = \frac{1}{2} m_{b} v^{2}

p_{psi} = 145.038 p_{MPa}

SD = \frac{m_{b} / 453.59237}{{(d_{b} / 25.4)}^{2}}

\frac{| d_{name} - d_{b} |}{d_{b}} \leq 0.08

0.6 \leq \frac{v_{load}}{v_{typ}} \leq 1.6, unless v_{load} < 345 m / s

where

$E$: the typical muzzle energy, in J
$m_{b}, v$: the typical bullet's mass, in kg in equation 1 and g in equation 3, and its muzzle velocity, in m/s
$p_{MPa}, p_{psi}$: the maximum pressure in megapascals and in pounds per square inch
$SD$: sectional density, in pounds per square inch as ballistics tables state it
$d_{b}, d_{name}$: the bullet's diameter, in mm, and the calibre its name states (9x19mm is 9, .45 ACP is 11.43)
$v_{load}, v_{typ}$: a load's velocity and its cartridge's typical velocity

A name that follows the case rather than the bullet (.38 Special, a necked-down .450/.400) and a flechette that names the bore are known exceptions to equation 4. A subsonic load is slow on purpose: one that says so in its name, or one under the speed of sound with a bullet much heavier than the cartridge's typical, is left out of equation 5.

Table 4. How a cartridge record is judged against its own arithmetic. Rows are read in order; the first that applies decides.
#	When	Verdict	What it means
1	$\| \frac{E}{\frac{1}{2} m_{b} v^{2}} - 1 \| > 0.1$	Review	The energy does not follow from the bullet and the velocity: one of the three is from another load.
2	$\| \frac{p_{psi}}{145.038 p_{MPa}} - 1 \| > 0.03$	Review	The pressure in psi is not the pressure in MPa: one was copied from another cartridge, or a CUP figure was filed as psi.
3	$\| \frac{SD}{{SD}_{(3)}} - 1 \| > 0.08$	Review	The sectional density does not follow from the bullet's weight and diameter.
4	A name that follows the case, not the bullet (.38 Special, .450/.400), or a flechette	Set aside	Set aside from the calibre test: the name is not the bullet.
5	$\frac{\| d_{name} - d_{b} \|}{d_{b}} > 0.08$	Review	The bullet is far from the calibre its name states.
6	A subsonic load	Set aside	Set aside from the velocity test: slow on purpose.
7	$\frac{v_{load}}{v_{typ}} \notin [0.6, 1.6]$	Review	A load far from its cartridge's typical velocity.
8	Anything else	Holds	The figures agree with one another.

Worked example: the 7.62x39mm

From its record: a 7.97 g bullet at 715 m/s with 2010 J, 7.92 mm across, and a maximum pressure of 355 MPa and 51488 psi, confirmed on the C.I.P. data sheet.

Energy from bullet and velocity (1)	$E = \frac{1}{2} \times 0.00797 \times 715^{2} = 2037 J$
Pressure in psi (2)	$355 \times 145.038 = 51488 psi \approx 51488 psi$
Sectional density (3)	$SD = \frac{7.97 / 453.59237}{{(7.92 / 25.4)}^{2}} = 0.181$

The recorded energy should lie within

1834 J2241 Jrecorded: 2010 J

Answer: 2010 J lies inside 1834 to 2241 J, so the energy, bullet and velocity agree.

The model

Worked example: the 7.62x39mm

Checks that use this method