ARMET-TR-17002 Linear Velocity Decay Formula

(Created 15 February 2021)
(Revised April 2024)

This formula was initially found in the paper The 76mm Gun M1A2 and M1A2: An Analysis of US Anti-Tank Capabilities during World War II (2.7 MB PDF).

Important Note: That document has several typos at critically important sections which wasted a considerable amount of my time in trying to understand and validate the formulas contained within it. The biggest typo that stands out for me was that they give the wrong weight for the M62A1 APC shot – 11.09 pounds, as opposed to the actual figure they used in their calculations – 15.605 pounds.

This specific formula is a simple linear velocity decay formula that works best from Mach 0.8 (273.1 m/sec) to Mach 2.5 (853.44 m/sec) at sea level. Beyond that, it gets a little suspect.

It is good however, for simple dirty order of magnitude approximations of striking velocities at differing ranges, allowing you to calculate a tank gun range table quickly without messy calculus/integration.

The Linear Velocity Decay Equation is:

VR = V0 – (k * x)

where:

VR = Striking Velocity at Given Range (x). (fps or m/sec)
V0 = Muzzle Velocity. (fps or m/sec)
k = a constant derived via a calculation of the interaction of the projectile cross section with the surrounding air.
x = Given range (feet or meters)

The "k" constant equation is:

where:

p = Air Density at Firing Altitude (lbs/ft3 or kg/m3). At sea level it's 0.0751 lb/ft3 or 1.20298935 kg/m3.
S = Cross Sectional Area of Projectile (ft2 or m2)
m = mass of projectile (lbs or kg)
a = Speed of Sound at Firing Altitude (fps or m/sec). At Sea level, it's 1,120 fps or 341.376 m/sec
K = Derived Constant. Different for each type of shot.

Some values for "k" are:

5.5121678 – US 120mm M829A1 APFSDS (L/D of around 15-18)
0.841 – Generic value given in ARMET-TR-17002.
0.6399 – US 90mm M82 APC Shot (via tables in https://www.lonesentry.com/manuals/90-mm-ammunition/index.html)
0.82688 – US 90mm M304 (T30E16) HVAP Shot (via tables in https://www.lonesentry.com/manuals/90-mm-ammunition/index.html)

To generate a reasonably accurate K constant, you need to find an official range table with given striking velocities over a large period of values and play around with a calculator until you find a value that works over that period.

EXAMPLE:

A 100mm gun is firing a 15.6 kg (34.39211 lb) shell at 895 m/sec (2936.35 fps). What is the striking velocity at 2000 meters (6561.68 feet) if we use a K constant of 0.841?

First, we calculate the cross section of a 100mm gun:

100mm / 2 = 50mm radius
Pi * 50^2 = 7853.98163 mm2 = 0.00785398163 m2.

Now, we calculate K:

K = [ (1.20298935 * 0.00785398163) / (2 * 15.6] * 0.841 * 341.376
K = 0.0869412842

We then plug that into the LVD formula:

895 – (0.0869412842 * 2000)
VR = 721.1174316 m/sec at 2000 meters.

Pre-Built Calculator

: Shell Diameter (mm)
: Shell Mass (kg)
: Muzzle Velocity (m/sec)

: Striking Range (meters)
: Derived K Constant

Known Derived K Constants Are:
5.5121678 – US 120mm M829A1 APFSDS (L/D of around 15-18)
0.841 – Generic value given in ARMET-TR-17002.
0.6399 – US 90mm M82 APC Shot (via LONE SENTRY)
0.82688 – US 90mm M304 (T30E16) HVAP Shot (via LONE SENTRY)

Calculations:

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