What happens after the Maximum Pressure Moment (MPM)? As was stated in the discussion on internal ballistics (the initial thermodynamic phase immediately after ignition) the maximum pressure inside the chamber of our 30-06 rifle occurs when the bullet has reached a position about 1.4" in front of the case mouth. This does not mean that gas generation and expansion is now complete - far from it. It only means that as more gas is generated and expanded this pressurisation happens inside a combustion chamber the volume of which is expanding very rapidly. As the gas is generated it expands into an increasing space and the pressure starts dropping from say 62,000 psi at MPM to about 6,800 - 7,000 at the muzzle departure moment (MDM). This phase is what causes the following: The final (muzzle) pressure at the moment of bullet departure determines the final value of the bullet's muzzle velocity. The time difference for bullet acceleration from the MPM to MDM determines the impulse of the momentum value at bullet departure and partly determines the measurable recoil. The value difference between P.max at MPM and P.min at MDM determines the felt recoil and also is the physical determining value for the final muzzle pressure and therefor muzzle velocity - but this is only part of that equation- as -the base physical reason for the value of P.min at MDM is the actual volume of gas generated within the parameters of 1-3 above. Associated with the thermodynamics touched on above are the inertial dynamics caused by the bullet accelerating down the bore - mechanical issues which have to do with the interactive forces between the bullet and the barrel, the barrel and the action, the barreled action and the stock, and finally the complete rifle and the shooter holding it. The bullet engages the rifling. The bullet is forced from behind into the rifling - which normally is spiralling clockwise viewed from the rear - and as the lands engrave the bullet there is a radial force from the lands onto the bullet, forcing it to spin to the right. / displacementThe counter force by the bullet is an anti clockwise torqueing action on the lands and therefore into the whole barrel. This force increases as the bullet encounters less and less barrel stiffness moving away from the breech. Typically in a 1:12" twist rate barrel the bullet does two complete rotations before it leaves the muzzle, exerting an anti clockwise elastic torqueing deformation / displacement to the more flexible muzzle end of the barrel. High speed tracing of a laser beam emanating from the crown indicates that the nett movement of a 24" standard profile barrel with a clockwise twist is a typical 2-3 microns to the left with corresponding shift in impact points of the bullets. Vertical separation of the bore centre point at bullet departure appears to be neglibly minute. As the barrel heats up during repeated firing the increase in temperature exacerbates any post rifling-cutting residual metallurgical stresses in the metal and the displacement pattern of the barrel will change with every shot as the barrel heats up - even if the barrel does not make any contact with the stock channel. The standard displacement is repeated with every shot and in barrels that have been stress relieved (as all European and South African made barrels) the difference between five shots will not be more than about two microns (two thousands of a mm at the muzzle. This relates to an impact displacement of .25 mm at 100 yards which is so small it can be disregarded. Any barrel that has residual stresses from the very stressful rifling forming process will display random displacements orders of magnitude bigger than this. This is why the South African Hunting Rifle Group Shooting Competitions where only bona fide hunting rifles and PMP factory ammunition is used has the top ten shooters obtaining between 4mm and 5mm groups, and number 25 from the top typically at 6.5mm at 100 yards over a standard sandbag rest in front and no rest at the rear. The bullet leaves the crown. The torqueing, bending force on the barrel is ongoing as the bullet acellerates forward and continuously and progressively increasing the amplitude of displacement of the barrel against its elastic rigidity. The moment the bullet leaves the crown the displacing force is removed and the barrel springs back due to its own mechanical elasticity and typically does two vibratory oscillations before settling in its original position. This is a pure, very short term, elastic, damped vibration and has no influence at all on the bullet which by then is happily on its way to the heart of the animal it was aimed at. In American gun media the term barrel harmonics is often used and how this affects the bullet during departure - as if it is an outside influence on the bullet. The term " resonance " is already incorrect at it refers to a part of any structure which vibrates in harmony with an outside vibratory input due to the particular elastic similarity to the input vibration - the simbiotic harmonics of the input vibration and the reactive response by the resonating part. The terms resonance and harmonics by definition are reactive and outside of mechanical connection between the input energy and the reactive vibration. It does not apply to a barrel and a bullet. The term barrel harmonics which supposedly can affect the departure point of the bullet is incorrect terminology. Any post departure vibration of the barrel as explained above is due to the barrel's elastic return after having been mechanically displaced by the bullet. A harmonic or resonance is a form of auto-stimulation to to a vibratory input and not related to a barrel and a bullet. Behind the bullet, and sealed off from the outside atmosphere exists the inside atmosphere which is at a temperature of about 1,500 degrees centigrade and having a pressure of about 7,000 psi. In front of the bullet existed the earth's atmosphere at a pressure of about 14 psi. The bullet forces this column of air out at about Mach 2.5 and this creates a supersonic shock wave in the air in front of the muzzle slightly ahead of the bullet. Behind this shock wave, at that incremental moment before departure there exists a temporary vacuum in front of the muzzle into which the bullet acellerates, being pushed from behind and then gets overtaken by the gas pressure which expands into this partial vacuum at a speed in excess of Mach 3. So straight after having left the braking friction of the rifling the bullet still gains speed for a short while. This moment of freeing itself from the rifle is crucial to accuracy and will be discussed in this section of intermediate ballistics. This moment of freeing itself from the rifle is crucial to accuracy. Accepting that every part of the barreled action and interfacing with the stock is linear and concentric ( some models of the Ruger Mk. 1 and earlier Ruger M77s were so horribly eccentric that it was not worth to spend any money on gunsmithing ), and that the barrel had been properly stress relieved, the only remaining causes for inherent inaccuracy in any rifle / bullet combination are the following two issues: The perfect concentricity and squareness of the bore crown with the bore centreline. The perfect squareness to the shank and concentricity of the base of the bullet. The minutest deformity in the bore crown will cause uneven release of the base of the bullet, some wobbling, a lot of profile drag and poor airtime performance. Similarly, the minutest inaccuracy in the base of the bullet - and this particularly is the risk with boat tail bullets - will cause similar uneven release from the crown, but more so a marked effect by the dense atmospheric pressurised gas overtaking and shoving the bullet from behind just as it leaves the crown. This wave of very dense gas enveloping the bullet from the rear, for a short but significant time that apart from influencing the attitude of an inaccurately designed bullet, causes it to fly without any airflow from the front over it. First the bullet has a slight rear to front airflow as it flies behind the shockwave front of the expelled air from the muzzle; then there is a momentary true vacuum around it, and then there is the fast, flaming hot, high pressure gas surrounding and passing it from behind. Only once this interaction between the gas and the environment atmosphere has dissipated does the bullet enter a condition of free stream airflow over it and then its design aerodynamics come into effect. Next issue for discussion will be about recoil - what causes it, how total recoil is calculated and the difference between calculated and felt recoil, and the factors influencing the latter.
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 😁. 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 ÷ 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 ÷ 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 ÷ 1/500 = 6 507 N Penetration force: (impact impulse ÷ frontal area) = (6,507 N ÷ 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 ÷ impact time) = 15,2 ÷1/500 = 7 614 N. Relative Penetration force: (impact impulse ÷ frontal area) = 7 614 ÷ 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 ÷ impact time) = (21,8 ÷ 1/500) = 10 900 Newton. Relative Penetration force: (impact impulse ÷ frontal area) = (10 900 ÷ 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 ÷ impact time) = (18.2 ÷ 1/500) = 9 100 Newton. Relative Penetration force: (impact impulse ÷ frontal area) = (9 100 ÷ 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 ÷ impact time) = 20.2 ÷ 1/500 = 10 100 Newton. Relative Penetration force: (impact impulse ÷ frontal area) = (10 100 ÷ 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 ÷ impact time) = 18,4 ÷ 1/500 = 9 214 N.s Relative Penetration force: (impact impulse ÷ frontal area) = (9 214 ÷ 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.
Bullet Shape Like the design of supersonic fighter jets, bullet shape will always be a compromise. The optimum wing and fuselage design for a Mach 2+ fighter is old established knowledge but the unavoidable bottom line is that it will spend 90% of its airborne time at subsonic speeds. The very low speed for take-off and landing already determines that a perfect supersonic wing design will stall on getting airborne so we do not see any fighter with true supersonic wing designs. Similarly, despite a hunting bullet spending 100% of its flight time above Mach 2 the object of its existence is not flight but impacting and penetrating very dense bone and meat to mechanically cut through the animal’s heart and sever the oxygen flow to the brain so that neurological function is lost. This section is about the special bullet shape needed for special performance. Special performance when killing thick skinned, heavy boned, densely muscled dangerous game relates to penetration through these obstructions - and ONLY that. For the very thick skinned Cape buffalo and elephant the importance of a high BC bullet is so low on the priority scale that it can be ignored. Nobody shoots an elephant or a buffalo or even a giraffe further than 50 yards. Special performance for a special need. Bullet manufacturers in the USA have for scores of years defined the ”special performance” regime for hunting bullets as their low drag ability in flight. This, by marketing spin has re-actively become the #1 priority for most hunters. In South Africa the highest priority for most hunters is the terminal moment, so their demand on bullet manufacturers is to define the region for special performance as that 1,000th of a second after impact. The same reason why heavier, slower bullets are in demand here. Bullet design in the USA is manufacturer and marketing-spin driven, while in Africa it is driven by hunter requirements for on-game performance - one shot through the heart killing ability. Bullet manufacturers respond to user demands here or they will go out of business. The best shapes for lowest aerodynamic drag at supersonic velocity cannot be applied due to the requirements of the special performance demand mentioned above (compare Figure1). Similarly, the requirements for low drag at subsonic speed that at first sight appears to be better suited to the special performance demand can also not be utilised because the massive supersonic drag on these designs causes unacceptable bullet slow down even in the first 50 yards (see Figure 2). Popular values for ballistic coefficients (BC) - which often do not follow actual supersonic aerodynamic drag principles, -are useless when a DG bullet is designed to meet its post impact special performance demands. Figure 1. ¾ Parabolic Cone. This is the best aerodynamic shape for Mach 2 - the speed of a 500 gr .458 Lott at 50 yards. Low shock wave drag is the design driver for the ¾ Parabolic Conical nose section. For in-animal performance no poorer shape can be imagined. Field experience has long ago showed that a slender nose, also being too long for magazine rifles, gets displaced (yawed) in virtually every instance by bone in buffalo or elephant. The centre of mass of such bullets are also too far to the rear. The biggest flaw of such a design is the massive in-flesh surface friction it experiences due to its into-tissue “creeping” which causes linear and rotational friction, slow-down, and loss of gyroscopic stability, wobble, and low penetration. Furthermore, slender nose shapes are too weak for the impulse counter-force they must endure at impact. Comparing bullets to jet fighters again: the only advantage this shape has for a Mach 2 jet fighter over a straight-edged cone is the lack of a shock wave where the cone flows into the fuselage; a shock cone here would cause unacceptable conditions for some systems, including the perspex canopy. On a bullet inside an animal with 800x the density of air in such a shock front from a cone shape angular transition from nose cone to shank is very beneficial. Figure 2. Elliptical. This nose shape is the best design for lowest drag for velocities below 850 ft/sec. For any hunting bullet it is an aerodynamic drag disaster at supersonic speeds as its critical drag rise Mach is already at M=0.80. Even a flat nose, conical shape has lesser shock wave drag at Mach 2. The velocity degradation on a bullet of such a nose shape is acute during the highest velocity regime which is out to 50 yards from the muzzle. This shape does put more mass into the front end of the bullet, making it stronger, but the long nose still is its Achilles Heel, so to speak, regarding in-tissue deflection - particularly by bone, causing yaw, followed by wobbling and immediate loss of penetration. Figure 3. Elliptically blunted cone. This blunt design was a feature of a Winchester FMJ (nickel coated) in the 1970s. It had stable penetration in soft skinned game due to the almost flat nose but the ogive design still caused it to slip into the animal tissue which caused wound channels of less than calibre size. This was the start for moving away from aerodynamic performance towards animal tissue performance. It was popular on thin skinned animals like eland due to the low amount of meat damage. It was effective on elephant frontal and side-on brain shots but not on heart shots on either elephant or Cape buffalo. Figure 4. Power Series. The midrange shape in the radii indicates the mean compromise between a straight cone and the elliptically blunted cone. It retains the low supersonic drag coefficient of the straight cone while allowing for a blunt nose to create fluid containing tissue to shear ahead of the bullet at the moment of imminent contact, preventing the onset of friction drag on the bullet immediately. This was the start of the cavitation principle. With a larger blunter nose and therefor flatter conical angle of attack this is close to what the Peregrine VRG-2/3 series have. Figure 5. Bi-conical. It was previously stated that a distinct shock front immediately in front of a flat nose bullet creates a tissue shearing action and the start of a cavitation bubble. Another shock front at the transition from nose cone to shank enhances the cavitation bubble and lessens skin friction drag tremendously. This is an acceptable and cheap rocket design. By removing the frontal cone at Ø1 the start of the perfect bullet for in-animal special performance has begun. As mentioned earlier bullet design is a compromise between aerodynamic and in-animal performance, and for the best performance to kill a Cape buffalo at 40-50 yards aerodynamics must take a back seat. Figure 9. Super Cavitation. This concept will immediately be understood by boat owners: when a propeller cavitates in the water the removal of the drag force on the blades cause an immediate and significant increase in rpm (angular velocity). For a bullet inside animal tissue this ability has huge benefits. Very promising empirical results on elephant have been obtained with this concept in South Africa. The flow separator is a separate disk centered on the mephlat which strengthens a shock front around the body by vapourising any fluid containing tissue. Penetration through tissue and bone is impressive. The main advantage is preventing friction-creating tissue to be in contact with the bullet shank. Keeping this - what is known in aerodynamic terms as the "wetted area" as small as possible, skin friction drag is limited to the flow separator and penetration is considerably enhanced per impulse force. The flat nose concept of the Peregrine and GS Custom bullets achieve this to a certain extent - not only inside an animal but they also have considerably less aerodynamic drag at Mach 2 than round nose dangerous game bullets. (Next: Cavitation and Super Cavitation bullets)