## To Calculate Actual Penetration Depth Is Impossible

Various variables exist between a bullet's impact point and cruising along its way to a Cape buffalo's heart and beyond. These varying influences of skin and bone and tendons and muscle limit the accuracy of pre-calculated, empirically determinable penetration distances. Fortunately, known and repeatable constants also exist to assist the safety conscious DG hunter to calculate a bullet's *penetration force* - if not its penetrating distance. Comparing this to the accepted minimum of the 9.3x62 he can set a norm for himself before choosing a suitable calibre to hunt his first Cape buffalo with. Equally important - to choose the proper bullet for the job.

A number of suitable equations exist in linear physics by which to determine comparable indices or values of penetration force. The value of these become apparent when compared to an accepted minimum-norm, which in South Africa is the 9.3x62 using a 286 gr solid bullet of premium quality. *It can never be stressed too much*: *No matter the hunter's choice of cartridge he should not be satisfied with using anything but the best premium quality bullet available.* That in fact holds true for non dangerous game as well. More and more outfitters in South Africa are demanding this.

The discussion to follow is about forces of *mass*, acceleration (more pointedly: *deceleration*), *momentum* and *impulse*. Much is made of bullet momentum but it only becomes a force to be reckoned with (in practice and in mathematics!) once it is integrated with real time and becomes an *impulse*. The reader will see that *kinetic energy* does not feature in the calculations. Being a scalar entity it has no association with force or penetration or killing ability of a bullet and therefor can not enter the discussion. It simply has no value in gun talk.

*Impulse* is defined as the force applied by a bullet's momentum onto the area of the beast against which it impacts within the time frame that the impact lasts. *Force = mass x acceleration* (or deceleration). Therefore: the change in a bullet's momentum (during its deceleration from say 2,300 ft/sec (700 m/sec) until it is stopped against the opposite skin) determines the force that the receiving mass exerts on the bullet. In accordance with Newton's Third Law this also is the force exerted by the bullet on the animal.

*Momentum* is the mass of the bullet multiplied by its velocity *while in undisturbed motion*. For ease of calculating and understanding, the International Standard of Units will be used; so, mass (kg) x velocity (metres/sec) = momentum which is expressed as kg.m/sec). American readers will quickly see the beauty of using the international way.

*Impulse* is die change in momentum between time of impact and bullet arrest, expressed in Newton.seconds. The examples below will define these entities:

In a ten pin bowling alley a 3 kg ball is rolled at 4 metres/sec. Die momentum of the ball is 3 kg x4m/sec = 12 kg.m/sek. The ball is stopped within one second by a big sponge backstop placed at the end of the alley.

**This pliant backstop exerted a counter force against the ball of 12 kg.m/sec, also called 12 Newton.sec (one kg.m/sec = one Newton.sec).**According to Newton's 3rd law, the impulse by the ball on the backstop was also 12 Newton.sec.Then a 2ft. thick solid log backstop is put in place which stops the ball, again having 12 kg.m/sec momentum,

*in one tenth of a second.*The counter force to have achieved this feat in one tenth of the time had to be 10x stronger, which is 12 N.sec divided by 1/10 sec = 120 Newton. The impulse against the ball thus was 120 Newton while the ball still had its 12 kg.m/sec momentum.**Newton's 3d law states that the ball also struck the backstop with an impulse force of 120 Newton,**even though it only possessed 12 Newton.second momentum.Then some fool fires a .358" (9.1mm) bullet weighing 17 gram (262gr) into the log, impacting at 700 metres/sec (2,300 ft/sec). Bullet momentum is (0.017 kg x 700m/sec) = 12 kg.m/sec or 12 Newton.sec. By chance the bullet happens to have the same momentum as the 3 kg ball ๐. It enters the log for about a foot and comes to a stop within 1/500th of a second. To accomplish this feat the log executed an impulse counter-force of 12 Newton.sec divided by 1/500th of a second = 6,000 Newton on the bullet.

**Newton's 3d law states that the bullet also struck the backstop with an impulse force of 6,000 N,**even though it only possessed 12 N.s momentum.

**Frontal, or "impact", or "affected" area **

The frontal surface of the ball that struck the log had an area of 650 square millimeters which absorbed die impulse counter-force of 120 N.sec against the ball. The ball certainly did not penetrate the backstop because that 120 Newton impulse was spread over an area of 650 square millimetres. The

*effective penetration force*was a low 120 รท 650**= 0.19 N per sq. mm.**The frontal surface of the bullet that struck the log has an area of 65 square millimeters which absorbed die impulse counter-force of 6 000 Newton. The effective penetration force by the bullet was 6,000 รท 65

**= 92 Newton per sq. mm.**

Therefore: the relative penetration ability (not measurable penetration distance) of a bullet with a known and constant mass and frontal area is a function of its impulse over time onto a known frontal area. That humerus bone and the tough tendons around it, and the dense muscle of a buffalo exert a counter force against the bullet which will modify its momentum and in many cases stop it completely. This typically happens in about 1/500th of a second.

**To calculate the relative penetration force of the official minimum cartridge and bullet weight for dangerous game in South Africa, the 9.3x62:**

Bullet mass: 18.6g (286gr)

Bullet retained diameter: 9.3 mm

Bullet retained frontal area: 68 sq.mm

Impact velocity at 50 yards: 700 m/sec

Impact momentum: 13 kg.m/sec (13 N.s.)

Time of change in momentum: 1/500th sec.

**Impact impulse:**(impact momentum divided by time = 13N.s รท 1/500 =**6 507 N****Penetration force:**(impact impulse รท frontal area) = (6,507 N รท 68 mm2) =**96 Newton per sq. mm.**This is the minimum allowed for dangerous game in S.A.

This bullet at this penetration impulse from the 9,3x62 is known to break through the humerus bone of a Cape buffalo, break through a rib, slice open the heart, possibly break through an opposite rib but often does not break the opposite humerus or shoulder joint. This performance is the very reason why it is approved as the minimum cartridge and calibre bullet for South African dangerous game.

**Comparing other cartridges to this minimum standard**

.**35 Whelen **(9.1x63): The case capacity of the Whelen is slightly less than that of the 9,3x62 but the thermodynamics are close enough make it a potential contender to be allowed to replace the latter as the minimum calibre allowed to hunt dangerous game with. To determine whether this completely unknown cartridge in South Africa would equal the 9.3x62 as the minumum suitable for dangerous game I back-engineered the above equations:

**To achieve 96 Newton penetration force**onto the 65 sq.mm frontal area of the Whelen the bullet needs to have a**6 240 Newton impact impulse.**To possess the 6 240 N. impact impulse the bullet must possess (6 240x1/500) = 12.5 Kg.m/sec momentum.

To obtain that momentum value a bullet of 18.5 g (284 gr) impacting at 680 m/sec (2,230 ft/sec) is required.

That demands that the 18.5 g bullet must be launched at 723 m/s (2,370 ft/sec) from 50 yards away.

To achieve this a muzzle pressure of 6 800 psi is required, which means a propellant which gives 57,800 psi peak pressure when the bullet has moved 1,26 inches from the case mouth must be used. The specific heat required for this performance is known, and the Somchem propellant meeting this is S355 with a 102.5% compressed load density.

__.375 H&H, 300 gr Peregrine VRG-2__:

Bullet Mass: 19.5g (300gr)

Bullet retained diameter: 9.52 mm

Bullet retained frontal area: 71.3 sq.mm

Impact velocity: 781 m/sek

Impact momentum: 15,2 kg.m/sek

Time of change in momentum: 1/500th/ sec.

**Impact impulse:**(impact momentum รท impact time) = 15,2 รท1/500 =**7 614 N.****Relative Penetration force:**(impact impulse รท frontal area) = 7 614 รท 71.3 =**107 Newton per sq.mm.**

__.458 Lott, 500 gr Peregrine VRG-2__:

Bullet Mass: 32.5g (500gr)

Bullet retained diameter: 11,6 mm

Bullet retained frontal area: 105 sq.mm

Impact velocity: 671 m/sec

Impact momentum: 21,8 kg.m/sec

Time of change in momentum: 1/500th/sec.

**Impact impulse:**(impact momentum รท impact time) = (21,8 รท 1/500) =**10 900 Newton.****Relative Penetration force:**(impact impulse รท frontal area) = (10 900 รท 105) =**104 Newton per sq.mm.**

**.416 Rigby, 400 gr Peregrine VRG-2:**

Bullet Mass: 29g (400gr)

Bullet retained diameter: 10,6 mm

Bullet retained frontal area: 88 sq.mm

Impact velocity: 700 m/sec

Impact momentum: 18.2 kg.m/sec

Time of change in momentum: 1/500th/sec.

I

**mpact impulse**: (impact momentum รท impact time) = (18.2 รท 1/500) =**9 100 Newton.****Relative Penetration force:**(impact impulse รท frontal area) = (9 100 รท 88) =**103 Newton per sq. mm.**

**.458 Lott, 450 gr GS Custom FN:**

Bullet Mass: 29.3g (500gr)

Bullet retained diameter: 11,6 mm

Bullet retained frontal area: 105 sq.mm

Impact velocity: 690 m/sec

Impact momentum: 20,2 kg.m/sec

Time of change in momentum: 1/500th/sec.

**Impact impulse:**(impact momentum รท impact time) = 20.2 รท 1/500 =**10 100 Newton.****Relative Penetration force:**(impact impulse รท frontal area) = (10 100 รท 105) =**96 Newton per sq. mm.**

**.458 Win Mag, 450 gr GS Custom FN:**

Bullet Mass: 29.3g (500gr)

Bullet retained diameter: 11,6 mm

Bullet retained frontal area: 105 sq.mm

Impact velocity: 630 m/sec

Impact momentum: 18,4 kg.m/sec

Time of change in momentum: 1/500th/sec.

**Impact impulse**: (impact momentum รท impact time) = 18,4 รท 1/500 =**9 214 N.s****Relative Penetration force:**(impact impulse รท frontal area) = (9 214 รท 105) =**88 Newton per sq. mm.**

These final figures - not surprisingly - are quite accurate penetration indices to compare cartridge, calibre, and bullet weight ability on Cape buffalo. Relative to the minimum allowed ability of the 9.3x62 and the .35 Whelen (the latter which I still need to prove in practice) the unblemished historical success of the .375 H&H and the .416 Rigby is underscored by the figures in the tables above. It is hard to improve on these two.

The .458 Lott is up there at the top with them of course; it must be noted though that once bullet diameter increases to .45" and wider, the mass and velocity needed for a sufficiently high penetration impulse puts it outside the realm of joyful shooting for recoil shy individuals.

No matter the proven ability of a cartridge, the shooter has to have the desire and pleasure to regularly practise with it to gain full confidence in his own ability to each and every time, without fail, put the bullet on the exact spot on a buffalo's skin to reach the top chambers of the heart. Nothing else is safe enough.

## Penetration Of Different Cartridges Using Expanding Bullets

## Adding to the reworked post above on solid bullets:

The premise of a penetration force exerted by a 9.3x62 solid bullet weighing 286gr and momentum value 13kg.m/sec being considerably higher than that of a 450gr from a .458 Win Mag having more than 18 kg.m/sec momentum of course has to do with the momentum focused onto the affected area surface area. All the above calculations were based on the frontal areas of non-expanding solid bullets.

The same relative calculations can be made for expanding bullets but only if the expansion is always uniform so that a new, post impact frontal area can be calculated. The only bullet in production that allows this is the Peregrine series with expansion plunger. It creates a round mushroom expansion form of 1.5x calibre for all practical impact velocities.

A list of relative penetration indices for non-dangerous game cartridges based on the Peregrine VRG-3/4 series bullets will follow.