Andries, I have been reading your information and the presentation is very good. I do appreciate it all being in one place. No criticism intended by my comment that it was posted before. It is going to take me a while to absorb this. I hope to get into it this afternoon when the rains (and snow?) come. Right now it is nice outside and need to make the most of it.
Andries, I notice that your are calculating with a 500g bullet when the section title says 450 or 480g. A bit confusing to me. What I know for certain is that at 60 yards a Peregrine VR2 480g bullet at muzzle velocity of 2235fps will absolutely penetrate through both shoulders of a large Cape Buffalo to be lost in space, heh. Better not be another one behind the one you shoot. There is spare power there.
Relative Penetration Values for Dangerous Game Cartridges
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 can be 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 x 4m/sec = 12 kg.m/sek. Say 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, which again has 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 backstop 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 only 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:
7. Thus: Impact Impulse: (impact momentum divided by time = 13N.s รท 1/500 = 6,507 Newton.
8. 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 less than that of the 9,3x62 but the thermodynamics are close enough make it a potential contender to be allowed to also 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:
1. 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.
2. To possess the 6,240 N. impact impulse the bullet must possess (6 240x1/500) = 12.5 Kg.m/sec momentum.
3. To obtain that momentum value a bullet of 18.5 g (284 gr) impacting at 680 m/sec (2,230 ft/sec) is required.
4. That demands that the 18.5 g bullet must be launched at 723 m/s (2,370 ft/sec) from 50 yards away.
5. To achieve this a muzzle pressure of 7,800 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.
6. The specific heat required for this performance is known, and the Somchem propellant meeting this is S355 with a 102.5% compressed load density.
8. Relative Penetration force: (impact impulse รท frontal area) = (9 ,14 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
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 frontal 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 no matter where the animal is hit.
A list of relative penetration indices for non-dangerous game cartridges based on the Peregrine VRG-3/4 series bullets will follow.
Good to learn that the back is better, Gary! I had a smaller experience of a similar chronic issue for twenty years. Then I married a world leader in sports injury rehabilitation and she fixed me up doing a three year very conservative stretching and self manipulating regime and for the past thirty years I have been fine - as long as I keep on doing hard physical work to keep the vertebrae from spontaneous fusing.
Pardon the repeat text about the principles underneath my maths, but I thought it wise to set the basics in place again. Be with you soon, my friend.
Thanks Andries. I hope that you and yours were well and happy over the holiday season.
I had a better year than normal. As you are probably aware, I am limited by injuries as to the frequency and duration of my hunting trips. I live only 65 miles from the property. I have an old mobile home my son and I use as a camp. It is reasonably comfortable, but I usually just go for the day. Typically, if I spend a day hunting I am not able to move around much the next day. If I stay overnight, I may be there for a lot longer than I want and stuck in the camp, unable to hunt. So, I drive home after the hunt and go back when I get over the previous trip. Two days a week, a few days apart is about my new limit. Cold damp weather limits me to one day a week. But, that is much better than it has been and I am thankful for that. Bad days also interfere with my thought processes (which can be debilitating and embarrassing), hence my request for the penetration information. It is caused by the medication though I use as little as I can get by with.
I had a spinal stimulator implanted this summer that helps with lower back pain. That is why I was able to hunt more this season. It is an electrical stimulator that I charge up with a proximity charger before I retire at night. It does nothing for the neck and upper back damage, but I do get a few pain free days here and there. Pain free days have been non-existent for the past 40 years or so.
I am hoping to do some hog hunting since deer season closed. Last year I was unable to do any, but this year I should be able to get out on occasion. I also have some test load worked up for my old tang safety Ruger ultra light in 270 Win. I retired that gun many years ago, frustrated with terminal performance of the bullets. Barnes bullets have performed so well for me the past few years in my other rifles, I decided to bring the old 270 back to life and give her a go. Bullets aside, I always loved that rifle.
I do appreciate the information on penetration. I know that much of what you sent was already available to me on this site. Thanks again. I will get back to you later on with some ideas after I have had a while to work on this.
Andries, please reproduce your penetration formula for me. I had enjoyed playing around with it. It has been a while since I looked at my notes and have apparently lost them all. Thanks.
Yea. I was impressed. He had retired mil sniper written all over him, but he never said a word about that. He was a nice and humble person, complementing me on the good looking 458 Win Mag. He told me he was going on an antelope hunt with his new rifle.
Wow.. That also means he can shoot! That is 80% of an accurate rifle. Pity the rifle will not qualify for the Hunting Rifle Group Shooting Competitions - being custom made - but he should enter surely.
Was at the range a few weeks back and met a very large fellow there and we got to talking before the range opened. He had been on Safari for large game, no dangerous and was telling me how much fun he had in Africa. He had used his 338 Mag. His new rifle sported a new German first planar focus scope. He had built the 308 rifle himself using an ordered custom house Remington 700 short action receiver. He had bought the barrel system equipment because he is a machinist. The barrel was very high end USA made. The receiver was cut with a spiral bolt that ran in a spiral track. I saw him push it home with only his thumb and index finger. He bore sighted it and proceeded to move out to 100 yards. Before he finished I saw him put three bullets in he same hole at 100 yards, dead center on his target. I don't think he used 12 rounds.
You certainly can buy a Musgrave here but it can only be sent to your FFL dealer. There is a delay while the SA Police clears the export permit. Look at the page of the Musgraves. The factory is 3 hrs drive from me.
Displacement of muzzle less important than tangential VELOCITY of muzzle. I can guarantee this CAN be a LARGE factor in bullet placement because I have seen it myself. Perhaps not with a target rifle or a really high end accurized rifle because those are painstakingly built to minimize it. The actual wave form of the pressure in the barrel made by different powder and / or different bullet cause a different landing point. This is most noticeable with heavy bullets and thin barrels. You cannot avoid the harmonic vibration because it is a physical LAW of string mathematics. As far as stress relief of the steel, it must be at a molecular level. I am really impressed that all the rifles in South Africa are cryo treated! Can I buy one there and bring it home?
Many times I have sighted in a rifle with one bullet, only to switch manufactures with a slightly different bullet and it now hits several inches to the right or left. The 35 Whelen for example. 225g Sierra boat tails 1/2 MOA centered 1" high at 100 yards. Switch powder and bullets to 225g Barnes and they land consistently 4 inches right at 100 yards. It's a different wave on the barrel.
Plus I am beginning to see a pattern of SA pride here. That's fine too :)
Barrels of South African built rifles and those made by the local barrel manufacturer Truvello, as well as rifles from Europe, including barrels from Austria and Finland are already stress relieved. No lapping, no breaking in, no bedding required.
Many theories have been proposed regarding barrel movement during bullet transit and the resonance of the barrel after the bullet has left the barrel. (Despite what is often read in gun writing, barrel resonance has zero influence on the bullet as it is a reactive, vibratory whipping after the bullet had left the muzzle).There certainly is no symmetric pattern in even the most accurate barrels.
Somchem has high speed video with laser measured results of many rifles tested for accuracy. Here is the measured muzzle displacement of a locally manufactured barrel at the moment of bullet exit:
10 micron equals one 100th of a millimetre, which is less than one half of a thousandth of an inch. When considering that 1 MOA represents 1/64th of an inch at the muzzle, the maximum displacement of the tested rifle muzzle of 1/700th of an inch indicates what can be termed as an accurate piece.
Maybe this explains the results of the SA Hunting Rifle Group Shooting Competitions with commercial hunting rifles using PMP commercial ammunition as seen HERE.
Added to this complexity is the motion of the muzzle as the bullet exits. The tube oscillates, the thicker the tube the higher the frequency a lower the magnitude of motion. As the barrel heats up, if the steel has any cold stresses, they will be relieved progressively by the heat and the shots will walk. I have seen the high speed videos. The bullet wobbles and then stabilizes. The more the wobble the more energy is lost and the more accuracy is affected. The super long boat tail bullets stabilize faster. The bullet path has a lateral component of motion created by muzzle motion. The trick is to have the SAME muzzle motion with each shot regardless of how hot the barrel gets.
An example of this is my Ruger 3006 which walked it's shots. I had the receiver trued and the entire metal assembly cryo-treated. This involves and extreme heating and extreme cold cycle. This removed the stressed in the barrel and assembly. Now this rifle is 1/2 MOA and the shots do not walk. Re-fitting the stock and the scope required some adjustments though, because of slight dimensional changes.
As for terminal performance we constantly get the dichotomy of clean pass through verses impact shock and massive damage. If you are counting on a pretty and effective kill you lean toward hard, bone penetrating shoulder or skull shots. If you are satisfied with a slower kill or a neck shot, then you lean toward the softer expanding type bullets. That argument continues here in the US certainly, and leans toward the softer. With the heavy boned, dangerous African game you can choose the hard go through the shoulder, or skull, or the wait for a better, softer path to the heart from forward or slightly behind. The shoulder/skull option needs a stable bullet through thick bone. I guess that sums up the requirement.
I assume Gerard referred to the โgas cloudโ at the moment of muzzle exit in the sense of the big influence the escaping gas over the bullet has on the repeated accuracy of bullets. It is both a function of bullet design and of perfect concentricity and edge condition of the crown. It is an interesting exceedingly momentary moment in the short life of a fired bullet.
Inside the bore the terms subsonic or supersonic do not apply because these refer to the physics of a body moving in free air causing airflow over its surfaces. So, as the bulletโs nose moves out of the muzzle (at say Mach 2 in the case of the .458 Win Mag) the free air suddenly creates a cone of highly compressed air around it, extending rearwards at an acute angle.
Immediate Aerodynamics - Bullet Front End
Two extreme cases exist: The apex of this angle is either attached to the bullet at a sharp point at the nose if the bullet nose is sharp and conical as in the Impala bullets, or the cone angle stretches backwards from a bulbous half-moon shock boundary ahead of the bullet nose and unattached to it if it is a typical old style .458 round nose 500gr bullet. This unattached shock is the cause of immediate and massive drag pressure on the bullet. Air faster than Mach 1 does not like any shape with curved surfaces in its way
Similarly, anywhere there is a measurable radius (ogive) to the bullet front end such detached shock waves form with significant increase in aerodynamic drag. Any nose shape where there are sharp corners with insignificant radii changing into a flat surface conical shape makes for less drag in the supersonic airflow.
Immediate Aerodynamics - Bullet Rear End
More to the point in this discussion on muzzle departure: the design of the bullet base is of more importance to the shooter than the nose shape regarding bullet stability and eventual accuracy. The high speed-high density gas column from the muzzle overtaking the bullet is the significant constant which influences the many variables in bullet base design and manufacture.
Flat base bullets: The moment a flat based bullet slips the muzzle at Mach 2 it gets engulfed and overtaken from behind by the bore-length column of gas containing a density relative to typically 7,000 psi (same as the pressure in an ocean 15 million feet down) and exerts considerable force onto the bullet from behind. This momentary force is much, much higher than the Mach 2 free air aerodynamic moments on the bullet from the front.
Apart from the mechanical displacement on the bullet tail by less than a 100% perfect 90o crown face or other minute nicks in the crown mechanically influencing the concentric departure, the smallest inconsistency in the bullet base allows the force of the outblow to put a lateral displacement into the bullet tail. Gyroscopic precession causes this displacement to be self-perpetuating. Fortunately, machined flat base bullets are invariably of perfect geometry.
Boat tail bullets: A boat tail bullet gets rammed from behind by this force even while the rear end is still inside the bore, exacerbating any crown interference. Furthermore the angled surface of the boat tail presenting a positive angle of attack to the ramming gasflow from behind sets the scene for an appreciable lateral displacement of the tail end should there be the slightest inconsistency in the radial concentricity of the changeover from bore contact surface into the boat tail cone, and further on behind this radial.
Reverse Airflow
As the compressed barrel gas bursts onto the bullet from behind the following happens momentarily:
The free stream air over the bullet is reversed and the dynamics are as if it is flying backwards into very dense water.
Very strong conical shock waves form on the bullet, coned forward.
Then as the hot gas pressure disperses into a cloud around the bullet there is another moment of zero air flow where the bullet with forward centre of mass receives no aerodynamic stability input, whereafter the normal Mach 2 stability interactions of aerodynamic and inertial moments resume and the bullet starts slowing down due to aerodynamic drag. This slow down lowers the kineses which progressively lowers the scalar joule value of kinetic energy contained in the bullet, translating its potential heat into real joules of heat energy in the bullet.
From here on the design of form and its interactions with the dynamics of supersonic flight drive the study of its ballistics.
When considering the principle of form follows function it is evident that bullet design in the US has for many many years followed the premise that flight through air was its main function. A โflat shootingโ form has been the driver, and not the performance into and through the very dense substances of skin, sinew, flesh and bone. The designs of the Barnes and the four makes of premium bullets from South Africa presented in the category Big Beasts, Best Bullets have one function in mind: in-animal-performance.
Yea. I like that last paragraph, pretty funny. I would never try the beefier of the big five with a 45/70 myself! I did finally reach 2353 with the 450GSC Flat I am hand inserting into the 458 Winny. Not sure about the crimped MAX in the mag yet. Somewhere between 2338 and 2348.
As for the cloud of hot gas in front of the muzzle, shoot enough smoke poles and you get a good feeling for that :) One thing I learned with those, put more than 90 grains of loose powder in a Hawkin, it burns the 90 grains and just spits the rest out and messes with your accuracy. To put more than 90 you need 30 to 50 grain pellets and a really hot primer.
As I have said many times before - buffalo hunting is mostly a walk in the park - sometimes a long walk as forum member Rick can testify to. The moment the bullet diameter (read 'frontal area') goes wider than .375" the profile drag rise through sinew, muscle and bone is such that the effort to impart the necessary impulse (momentum onto frontal area) AND stay nose ahead becomes a challenge. Propellant volume is required - the reason why I have fallen in love with the .416 Rigby.
All this is the bottom line why Michael is working towards minimum 2,350 ft/sec with the 458450FN bullet.
You are right Andries. I was referring to cylinder shapes vs. round mushrooms, of the same diameter, in terminal ballistics not external ballistics.
The three main 'ballistics' that everyone talks about are internal, external and terminal ballistics. There are two areas that are little known in between each of those three called intermediate ballistics (in the cloud ahead of the muzzle) and transitional ballistics (when the bullet transitions from flight to entering the target).
BTW. I read the entire article linked above and it ends with: "The 45-70 is a quintessential American cartridge, big medicine for moose, bison, elk and brown bear. But the African hunter may well come across some crusty old dugga boy who has never heard the tales told around a Western campfire, is no cousin to the mean-spirited but thin-skinned grizzly, and who will take one contemptuous look at that lever-operated stick and utter the chilling Cape buffalo equivalent of โGo ahead, make my day.โ "
I do not think there is any dispute that a pointed tip bullet has better aerodynamics than flat tips. The best supersonic aerodynamic nose shape in fact is a true cone with sharp corners to the parallel surface of the shank. In the final instance the hunter wants the bullet which performs best inside the animal and if that means giving up on some aerodynamic elegance that is O.K.
In practice even the aerodynamics have a very small influence. I see no practical difference in impact point at 150 yards between the Peregrine 168gr semi flat point VRG-3 bullets from my .303 and the spitzer PMP 174gr. Penetration of the Peregrines through wildebeest shoulder is better due to better terminal dynamics.
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Andries, I have been reading your information and the presentation is very good. I do appreciate it all being in one place. No criticism intended by my comment that it was posted before. It is going to take me a while to absorb this. I hope to get into it this afternoon when the rains (and snow?) come. Right now it is nice outside and need to make the most of it.
Andries, I have an intense weight training program that is working. Just very slow. Typical physical therapy has been a joke so I took it on myself.
Andries, I notice that your are calculating with a 500g bullet when the section title says 450 or 480g. A bit confusing to me. What I know for certain is that at 60 yards a Peregrine VR2 480g bullet at muzzle velocity of 2235fps will absolutely penetrate through both shoulders of a large Cape Buffalo to be lost in space, heh. Better not be another one behind the one you shoot. There is spare power there.
Relative Penetration Values for Dangerous Game Cartridges
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 can be 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 x 4m/sec = 12 kg.m/sek. Say 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, which again has 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 backstop 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 only 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:
1. Bullet mass: 18.6g (286gr).
2. Bullet retained diameter: 9.3 mm .
3. Bullet retained frontal area: 68 sq.mm
4. Impact velocity at 50 yards: 700 m/sec.
5. Impact momentum: 13 kg.m/sec (13 N.s.)
6. Time of change in momentum: 1/500th sec.
7. Thus: Impact Impulse: (impact momentum divided by time = 13N.s รท 1/500 = 6,507 Newton.
8. 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 less than that of the 9,3x62 but the thermodynamics are close enough make it a potential contender to be allowed to also 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:
1. 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.
2. To possess the 6,240 N. impact impulse the bullet must possess (6 240x1/500) = 12.5 Kg.m/sec momentum.
3. To obtain that momentum value a bullet of 18.5 g (284 gr) impacting at 680 m/sec (2,230 ft/sec) is required.
4. That demands that the 18.5 g bullet must be launched at 723 m/s (2,370 ft/sec) from 50 yards away.
5. To achieve this a muzzle pressure of 7,800 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.
6. 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:
1. Bullet Mass: 19.5g (300gr).
2. Bullet retained diameter: 9.52 mm.
3. Bullet retained frontal area: 71.3 sq.mm.
4. Impact velocity = 781 m/sec.
5. Impact momentum: 15,2 kg.m/sec.
6. Time of change in momentum: 1/500th/ sec.
7. Impact impulse: (impact momentum รท impact time) = 15,2 รท1/500 = 7,614 Newton.
8. 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:
1. Bullet Mass: 32.5g (500gr).
2. Bullet retained diameter: 11,6 mm.
3. Bullet retained frontal area: 105 sq.mm.
4. Impact velocity: 671 m/sec.
5. Impact momentum: 21,8 kg.m/sec.
6. Time of change in momentum: 1/500th/sec.
7. Impact impulse: (impact momentum รท impact time) = (21,8 รท 1/500) = 10,900 Newton.
8. Relative Penetration Force: (impact impulse รท frontal area) = (10 900 รท 105) = 104 Newton per sq.mm.
.416 Rigby, 400 gr Peregrine VRG-2:
1. Bullet Mass: 29g (400gr).
2. Bullet retained diameter: 10,6 mm.
3. Bullet retained frontal area: 88 sq.mm.
4. Impact velocity: 700 m/sec.
5. Impact momentum: 18.2 kg.m/sec.
6. Time of change in momentum: 1/500th/sec.
7. Impact impulse: (impact momentum รท impact time) = (18.2 รท 1/500) = 9,100 Newton.
8. Relative Penetration Force: (impact impulse รท frontal area) = (9 100 รท 88) = 103 Newton per sq. mm.
.458 Lott, 450 gr GS Custom FN:
1. Bullet Mass: 29.3g (500gr).
2. Bullet retained diameter: 11,6 mm.
3. Bullet retained frontal area: 105 sq.mm.
4. Impact velocity: 690 m/sec.
5. Impact momentum: 20,2 kg.m/sec.
6. Time of change in momentum: 1/500th/sec.
7. Impact impulse: (impact momentum รท impact time) = 20.2 รท 1/500 = 10,100 Newton.
8. 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:
1. Bullet Mass: 29.3g (500gr).
2. Bullet retained diameter: 11,6 mm.
3. Bullet retained frontal area: 105 sq.mm.
4. Impact velocity: 630 m/sec.
5. Impact momentum: 18,4 kg.m/sec.
6. Time of change in momentum: 1/500th/sec.
7. Impact impulse: (impact momentum รท impact time) = 18,4 รท 1/500 = 9,214 Newton/
8. Relative Penetration force: (impact impulse รท frontal area) = (9 ,14 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
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 frontal 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 no matter where the animal is hit.
A list of relative penetration indices for non-dangerous game cartridges based on the Peregrine VRG-3/4 series bullets will follow.
Good to learn that the back is better, Gary! I had a smaller experience of a similar chronic issue for twenty years. Then I married a world leader in sports injury rehabilitation and she fixed me up doing a three year very conservative stretching and self manipulating regime and for the past thirty years I have been fine - as long as I keep on doing hard physical work to keep the vertebrae from spontaneous fusing.
Pardon the repeat text about the principles underneath my maths, but I thought it wise to set the basics in place again. Be with you soon, my friend.
Thanks Andries. I hope that you and yours were well and happy over the holiday season.
I had a better year than normal. As you are probably aware, I am limited by injuries as to the frequency and duration of my hunting trips. I live only 65 miles from the property. I have an old mobile home my son and I use as a camp. It is reasonably comfortable, but I usually just go for the day. Typically, if I spend a day hunting I am not able to move around much the next day. If I stay overnight, I may be there for a lot longer than I want and stuck in the camp, unable to hunt. So, I drive home after the hunt and go back when I get over the previous trip. Two days a week, a few days apart is about my new limit. Cold damp weather limits me to one day a week. But, that is much better than it has been and I am thankful for that. Bad days also interfere with my thought processes (which can be debilitating and embarrassing), hence my request for the penetration information. It is caused by the medication though I use as little as I can get by with.
I had a spinal stimulator implanted this summer that helps with lower back pain. That is why I was able to hunt more this season. It is an electrical stimulator that I charge up with a proximity charger before I retire at night. It does nothing for the neck and upper back damage, but I do get a few pain free days here and there. Pain free days have been non-existent for the past 40 years or so.
I am hoping to do some hog hunting since deer season closed. Last year I was unable to do any, but this year I should be able to get out on occasion. I also have some test load worked up for my old tang safety Ruger ultra light in 270 Win. I retired that gun many years ago, frustrated with terminal performance of the bullets. Barnes bullets have performed so well for me the past few years in my other rifles, I decided to bring the old 270 back to life and give her a go. Bullets aside, I always loved that rifle.
I do appreciate the information on penetration. I know that much of what you sent was already available to me on this site. Thanks again. I will get back to you later on with some ideas after I have had a while to work on this.
Andries, please reproduce your penetration formula for me. I had enjoyed playing around with it. It has been a while since I looked at my notes and have apparently lost them all. Thanks.
My kind of hunter, sure. Not "did I hit it?" kind of man.
Yea. I was impressed. He had retired mil sniper written all over him, but he never said a word about that. He was a nice and humble person, complementing me on the good looking 458 Win Mag. He told me he was going on an antelope hunt with his new rifle.
Wow.. That also means he can shoot! That is 80% of an accurate rifle. Pity the rifle will not qualify for the Hunting Rifle Group Shooting Competitions - being custom made - but he should enter surely.
Was at the range a few weeks back and met a very large fellow there and we got to talking before the range opened. He had been on Safari for large game, no dangerous and was telling me how much fun he had in Africa. He had used his 338 Mag. His new rifle sported a new German first planar focus scope. He had built the 308 rifle himself using an ordered custom house Remington 700 short action receiver. He had bought the barrel system equipment because he is a machinist. The barrel was very high end USA made. The receiver was cut with a spiral bolt that ran in a spiral track. I saw him push it home with only his thumb and index finger. He bore sighted it and proceeded to move out to 100 yards. Before he finished I saw him put three bullets in he same hole at 100 yards, dead center on his target. I don't think he used 12 rounds.
You certainly can buy a Musgrave here but it can only be sent to your FFL dealer. There is a delay while the SA Police clears the export permit. Look at the page of the Musgraves. The factory is 3 hrs drive from me.
Here: https://www.bullet-behavior.com/the-new-musgraves
Displacement of muzzle less important than tangential VELOCITY of muzzle. I can guarantee this CAN be a LARGE factor in bullet placement because I have seen it myself. Perhaps not with a target rifle or a really high end accurized rifle because those are painstakingly built to minimize it. The actual wave form of the pressure in the barrel made by different powder and / or different bullet cause a different landing point. This is most noticeable with heavy bullets and thin barrels. You cannot avoid the harmonic vibration because it is a physical LAW of string mathematics. As far as stress relief of the steel, it must be at a molecular level. I am really impressed that all the rifles in South Africa are cryo treated! Can I buy one there and bring it home?
Many times I have sighted in a rifle with one bullet, only to switch manufactures with a slightly different bullet and it now hits several inches to the right or left. The 35 Whelen for example. 225g Sierra boat tails 1/2 MOA centered 1" high at 100 yards. Switch powder and bullets to 225g Barnes and they land consistently 4 inches right at 100 yards. It's a different wave on the barrel.
Plus I am beginning to see a pattern of SA pride here. That's fine too :)
Barrels of South African built rifles and those made by the local barrel manufacturer Truvello, as well as rifles from Europe, including barrels from Austria and Finland are already stress relieved. No lapping, no breaking in, no bedding required.
Many theories have been proposed regarding barrel movement during bullet transit and the resonance of the barrel after the bullet has left the barrel. (Despite what is often read in gun writing, barrel resonance has zero influence on the bullet as it is a reactive, vibratory whipping after the bullet had left the muzzle).There certainly is no symmetric pattern in even the most accurate barrels.
Somchem has high speed video with laser measured results of many rifles tested for accuracy. Here is the measured muzzle displacement of a locally manufactured barrel at the moment of bullet exit:
Shot 1. Muzzle displaced 6.5 micron right, 1 micron down.
Shot 2. Muzzle displaced 5.8 micron right, 1 micron down.
Shot 3. Muzzle displaced 9.0 micron right, 1 micron down.
Shot 4. Muzzle displaced 8.0 micron right, 1 micron down.
Shot 5. Muzzle displaced 5.7 micron right, 1 micron down.
10 micron equals one 100th of a millimetre, which is less than one half of a thousandth of an inch. When considering that 1 MOA represents 1/64th of an inch at the muzzle, the maximum displacement of the tested rifle muzzle of 1/700th of an inch indicates what can be termed as an accurate piece.
Maybe this explains the results of the SA Hunting Rifle Group Shooting Competitions with commercial hunting rifles using PMP commercial ammunition as seen HERE.
Added to this complexity is the motion of the muzzle as the bullet exits. The tube oscillates, the thicker the tube the higher the frequency a lower the magnitude of motion. As the barrel heats up, if the steel has any cold stresses, they will be relieved progressively by the heat and the shots will walk. I have seen the high speed videos. The bullet wobbles and then stabilizes. The more the wobble the more energy is lost and the more accuracy is affected. The super long boat tail bullets stabilize faster. The bullet path has a lateral component of motion created by muzzle motion. The trick is to have the SAME muzzle motion with each shot regardless of how hot the barrel gets.
An example of this is my Ruger 3006 which walked it's shots. I had the receiver trued and the entire metal assembly cryo-treated. This involves and extreme heating and extreme cold cycle. This removed the stressed in the barrel and assembly. Now this rifle is 1/2 MOA and the shots do not walk. Re-fitting the stock and the scope required some adjustments though, because of slight dimensional changes.
As for terminal performance we constantly get the dichotomy of clean pass through verses impact shock and massive damage. If you are counting on a pretty and effective kill you lean toward hard, bone penetrating shoulder or skull shots. If you are satisfied with a slower kill or a neck shot, then you lean toward the softer expanding type bullets. That argument continues here in the US certainly, and leans toward the softer. With the heavy boned, dangerous African game you can choose the hard go through the shoulder, or skull, or the wait for a better, softer path to the heart from forward or slightly behind. The shoulder/skull option needs a stable bullet through thick bone. I guess that sums up the requirement.
I assume Gerard referred to the โgas cloudโ at the moment of muzzle exit in the sense of the big influence the escaping gas over the bullet has on the repeated accuracy of bullets. It is both a function of bullet design and of perfect concentricity and edge condition of the crown. It is an interesting exceedingly momentary moment in the short life of a fired bullet.
Inside the bore the terms subsonic or supersonic do not apply because these refer to the physics of a body moving in free air causing airflow over its surfaces. So, as the bulletโs nose moves out of the muzzle (at say Mach 2 in the case of the .458 Win Mag) the free air suddenly creates a cone of highly compressed air around it, extending rearwards at an acute angle.
Immediate Aerodynamics - Bullet Front End
Two extreme cases exist: The apex of this angle is either attached to the bullet at a sharp point at the nose if the bullet nose is sharp and conical as in the Impala bullets, or the cone angle stretches backwards from a bulbous half-moon shock boundary ahead of the bullet nose and unattached to it if it is a typical old style .458 round nose 500gr bullet. This unattached shock is the cause of immediate and massive drag pressure on the bullet. Air faster than Mach 1 does not like any shape with curved surfaces in its way
Similarly, anywhere there is a measurable radius (ogive) to the bullet front end such detached shock waves form with significant increase in aerodynamic drag. Any nose shape where there are sharp corners with insignificant radii changing into a flat surface conical shape makes for less drag in the supersonic airflow.
Immediate Aerodynamics - Bullet Rear End
More to the point in this discussion on muzzle departure: the design of the bullet base is of more importance to the shooter than the nose shape regarding bullet stability and eventual accuracy. The high speed-high density gas column from the muzzle overtaking the bullet is the significant constant which influences the many variables in bullet base design and manufacture.
Flat base bullets: The moment a flat based bullet slips the muzzle at Mach 2 it gets engulfed and overtaken from behind by the bore-length column of gas containing a density relative to typically 7,000 psi (same as the pressure in an ocean 15 million feet down) and exerts considerable force onto the bullet from behind. This momentary force is much, much higher than the Mach 2 free air aerodynamic moments on the bullet from the front.
Apart from the mechanical displacement on the bullet tail by less than a 100% perfect 90o crown face or other minute nicks in the crown mechanically influencing the concentric departure, the smallest inconsistency in the bullet base allows the force of the outblow to put a lateral displacement into the bullet tail. Gyroscopic precession causes this displacement to be self-perpetuating. Fortunately, machined flat base bullets are invariably of perfect geometry.
Boat tail bullets: A boat tail bullet gets rammed from behind by this force even while the rear end is still inside the bore, exacerbating any crown interference. Furthermore the angled surface of the boat tail presenting a positive angle of attack to the ramming gasflow from behind sets the scene for an appreciable lateral displacement of the tail end should there be the slightest inconsistency in the radial concentricity of the changeover from bore contact surface into the boat tail cone, and further on behind this radial.
Reverse Airflow
As the compressed barrel gas bursts onto the bullet from behind the following happens momentarily:
The free stream air over the bullet is reversed and the dynamics are as if it is flying backwards into very dense water.
Very strong conical shock waves form on the bullet, coned forward.
Then as the hot gas pressure disperses into a cloud around the bullet there is another moment of zero air flow where the bullet with forward centre of mass receives no aerodynamic stability input, whereafter the normal Mach 2 stability interactions of aerodynamic and inertial moments resume and the bullet starts slowing down due to aerodynamic drag. This slow down lowers the kineses which progressively lowers the scalar joule value of kinetic energy contained in the bullet, translating its potential heat into real joules of heat energy in the bullet.
From here on the design of form and its interactions with the dynamics of supersonic flight drive the study of its ballistics.
When considering the principle of form follows function it is evident that bullet design in the US has for many many years followed the premise that flight through air was its main function. A โflat shootingโ form has been the driver, and not the performance into and through the very dense substances of skin, sinew, flesh and bone. The designs of the Barnes and the four makes of premium bullets from South Africa presented in the category Big Beasts, Best Bullets have one function in mind: in-animal-performance.
Yea. I like that last paragraph, pretty funny. I would never try the beefier of the big five with a 45/70 myself! I did finally reach 2353 with the 450GSC Flat I am hand inserting into the 458 Winny. Not sure about the crimped MAX in the mag yet. Somewhere between 2338 and 2348.
As for the cloud of hot gas in front of the muzzle, shoot enough smoke poles and you get a good feeling for that :) One thing I learned with those, put more than 90 grains of loose powder in a Hawkin, it burns the 90 grains and just spits the rest out and messes with your accuracy. To put more than 90 you need 30 to 50 grain pellets and a really hot primer.
โGo ahead, make my day...โ
As I have said many times before - buffalo hunting is mostly a walk in the park - sometimes a long walk as forum member Rick can testify to. The moment the bullet diameter (read 'frontal area') goes wider than .375" the profile drag rise through sinew, muscle and bone is such that the effort to impart the necessary impulse (momentum onto frontal area) AND stay nose ahead becomes a challenge. Propellant volume is required - the reason why I have fallen in love with the .416 Rigby.
All this is the bottom line why Michael is working towards minimum 2,350 ft/sec with the 458450FN bullet.
You are right Andries. I was referring to cylinder shapes vs. round mushrooms, of the same diameter, in terminal ballistics not external ballistics.
The three main 'ballistics' that everyone talks about are internal, external and terminal ballistics. There are two areas that are little known in between each of those three called intermediate ballistics (in the cloud ahead of the muzzle) and transitional ballistics (when the bullet transitions from flight to entering the target).
BTW. I read the entire article linked above and it ends with: "The 45-70 is a quintessential American cartridge, big medicine for moose, bison, elk and brown bear. But the African hunter may well come across some crusty old dugga boy who has never heard the tales told around a Western campfire, is no cousin to the mean-spirited but thin-skinned grizzly, and who will take one contemptuous look at that lever-operated stick and utter the chilling Cape buffalo equivalent of โGo ahead, make my day.โ "
I will go with that :-)
I do not think there is any dispute that a pointed tip bullet has better aerodynamics than flat tips. The best supersonic aerodynamic nose shape in fact is a true cone with sharp corners to the parallel surface of the shank. In the final instance the hunter wants the bullet which performs best inside the animal and if that means giving up on some aerodynamic elegance that is O.K.
In practice even the aerodynamics have a very small influence. I see no practical difference in impact point at 150 yards between the Peregrine 168gr semi flat point VRG-3 bullets from my .303 and the spitzer PMP 174gr. Penetration of the Peregrines through wildebeest shoulder is better due to better terminal dynamics.