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50cal bullet design...

groper

Sergeant
Full Member
Minuteman
Sep 2, 2008
212
0
australia
Id like to offer a free 1075grain .50cal VLD bullet design, you have my permission to use it however you wish. The design weight is accurate via 3D CAD calculation using copper as the material with a density of 8.95g/cm^3. Machined from Brass @ 8.47g/cm^3, it would be a bit lighter and the stability is also worse - i wouldnt recommend it.

So here it is...
50calfreebie.jpg




The following BC and stability calculations were performed with the McDrag Projectile drag and static stability calculator as per Robert McCoy, former ballistics researcher, Aberdeen proving grounds.

Required twist is 1:11.3 @ 2791 fps for a stability factor of 1.5, however a 1:12 twist should be adequate, and still provides a stability factor of 1.3 in std atmosphere conditions at sea level.

G1 BC`s are calculated as follows;
1.793 @ 3349 fps
1.669 @ 2791 fps
1.615 @ 2232 fps

G7 BC`s as follows;
.846 @ 3349 fps
.834 @ 2791 fps
.811 @ 2232 fps

Assuming we shoot this projectile at a muzzle velocity of 2600fps, we can assume the velocity regime of this projectile over the course of a 1000yd flight to be between 2600fps and 2075 fps @ 1000yds, remembering that this is based purely of the calculated BCs.

By averaging the BC`s across this flight regime, we get an average G7 of .816 and this equates to an average G1 BC of 1.621. These would be the numbers youd be interested in to initially build a trajectory from in your initial testing, and revise it to a real world BC from there.

I have no idea whether the dynamic stability of this projectile is adequate, so proceed at your own risk and have fun whilst your doing it. If someone actually does get this machined up, please let us know how it goes... Enjoy.
 
Re: 50cal bullet design...

Can you give the diameters and lengths of the driving bands. I have access to cnc lathes and I am thinking of taking this design one step further. My thought is to make the projectile then, turn it around and bore out from the back and cut out a lot of the weight. Then, either plug the back or fill it with a lightweight polymer. Unfortunatly my .50 cal is a 1:15 twist. But if I can lighten up the projectile it should be possible to push it fast enough for your calculated speeds. Now I would just need a faster twist barrell.
 
Re: 50cal bullet design...

I left the dimensions of those out as you may have different bore/groove than standard. However, the length of each band and groove of the projectile is .079 - the front groove shown is a bit longer in the drawing, you can vary these dimensions and it wont have dramatic effects. The depth of the grooves is .001 under your bore dimension (.499). The first part of the ogive forms the first band. However you may want to increase the diameter of the last band nearest the boat tail to slightly over bore size (.511) in order to create a good gas seal at the rear of the projectile once its tracking true in the bore.

Also, you may want to try reducing the diameter of the ogive to .507 - .509ish to modify the engraving pressures and seating depth sensitivity.
 
Re: 50cal bullet design...

This will not stabilize from a 1:15" twist.

Trust me.
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: Noel Carlson</div><div class="ubbcode-body">This will not stabilize from a 1:15" twist.

Trust me. </div></div>

Correct, the stability factor from a 1:15 twist is only 0.8 - will not work, no chance.
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: Dylanss</div><div class="ubbcode-body"> My thought is to make the projectile then, turn it around and bore out from the back and cut out a lot of the weight. Then, either plug the back or fill it with a lightweight polymer. </div></div>

Unfortunately if you do this, and reduce the weight, youll loose BC in the process - so it defeats the design. For example reducing the weight to 900grains by boring out the back of it and keeping the same form factor, the average G1 BC will go down to ~1.354 - but you will still require the fast twist barrel.

It would be better to design a lighter/shorter projectile that didnt need to be "lightened" and save the hassle... Being shorter and lighter, it would also become more stable and you wouldnt need such a fast twist rate barrel to stabilize it. Eventually, you will end up with a design similar to other commercially available projectiles that stabilize in a 1:15twist... surprised?
 
Re: 50cal bullet design...

you wont push it to 2600 fps out of a bmg case, mid to upper 900 grainers maybe.
 
Re: 50cal bullet design...

Ah, so this might be too heavy to be practical in .510cal? funny that... But thats ok, if you launch it at a slower velocity, simply expect to have the real world BC revised downwards accordingly...
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: groper</div><div class="ubbcode-body">
Required twist is 1:11.3 @ 2791 fps for a stability factor of 1.5, however a 1:12 twist should be adequate, and still provides a stability factor of 1.3 in std atmosphere conditions at sea level.

G1 BC`s are calculated as follows;
1.793 @ 3349 fps
1.669 @ 2791 fps
1.615 @ 2232 fps

G7 BC`s as follows;
.846 @ 3349 fps
.834 @ 2791 fps
.811 @ 2232 fps
</div></div>

I would be interested to know which program(s) you use for BC estimations- if you
dont mind sharing that info. TIA.
 
Re: 50cal bullet design...

As per OP...
<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: groper</div><div class="ubbcode-body">
The following BC and stability calculations were performed with the McDrag Projectile drag and static stability calculator as per Robert McCoy, former ballistics researcher, Aberdeen proving grounds.
</div></div>
 
Re: 50cal bullet design...

Groper,

How does the McDrag handle the driving bands now? I'm only familiar with McDrag until about 1990's releases and at that point, McDrag would have substantially incorrect answers for bullets with driving bands because of simplifications within the code.

The code published in Mr. McCoy's book c. 1999 is not capable of handling driving bands so any BC projections based upon that code will be notably higher than actual tests would support.

To those interested in the bullet:

The front driving band/ogive integration on there is supposedly patented by GS Customs, though I have not managed to locate the patent in an international database. I have absolutely no business interest with Gerard Schultz's business but if it is indeed patented, this design may land you in hot water. Just an FYI.
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: groper</div><div class="ubbcode-body">As per OP...
<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: groper</div><div class="ubbcode-body">
The following BC and stability calculations were performed with the McDrag Projectile drag and static stability calculator as per Robert McCoy, former ballistics researcher, Aberdeen proving grounds.
</div></div> </div></div>

Thanks...
blush.gif
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: bohem</div><div class="ubbcode-body">Groper,

How does the McDrag handle the driving bands now? I'm only familiar with McDrag until about 1990's releases and at that point, McDrag would have substantially incorrect answers for bullets with driving bands because of simplifications within the code.

The code published in Mr. McCoy's book c. 1999 is not capable of handling driving bands so any BC projections based upon that code will be notably higher than actual tests would support.

To those interested in the bullet:

The front driving band/ogive integration on there is supposedly patented by GS Customs, though I have not managed to locate the patent in an international database. I have absolutely no business interest with Gerard Schultz's business but if it is indeed patented, this design may land you in hot water. Just an FYI.
</div></div>

I didnt allow for the driving bands in McDrag Bohem, they are only calculated BC`s afterall using a rather primitive method, so take them with a grain of salt...

I was not aware of GSC patent specifics, i too cannot find the details of it... However, there have been many other bullets in the past that use the ogive as the first band, predator projectiles springs to mind immediately, and then there is the barnes TSX bullets also... so if there is a patent on this, it obviously doesnt hold water...
2007-01-23_174816_HMbul_0905_B.jpg

 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: bohem</div><div class="ubbcode-body">Groper,

How does the McDrag handle the driving bands now? I'm only familiar with McDrag until about 1990's releases and at that point, McDrag would have substantially incorrect answers for bullets with driving bands because of simplifications within the code.

The code published in Mr. McCoy's book c. 1999 is not capable of handling driving bands so any BC projections based upon that code will be notably higher than actual tests would support.

To those interested in the bullet:

The front driving band/ogive integration on there is supposedly patented by GS Customs, though I have not managed to locate the patent in an international database. I have absolutely no business interest with Gerard Schultz's business but if it is indeed patented, this design may land you in hot water. Just an FYI.
</div></div>



I very much doubt attempting to patent driving bands would meet with much success? US Patents often get through with spurious claims however an EU Patent is anoter matter altogether.
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: Emouse</div><div class="ubbcode-body"><div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: bohem</div><div class="ubbcode-body">Groper,

How does the McDrag handle the driving bands now? I'm only familiar with McDrag until about 1990's releases and at that point, McDrag would have substantially incorrect answers for bullets with driving bands because of simplifications within the code.

The code published in Mr. McCoy's book c. 1999 is not capable of handling driving bands so any BC projections based upon that code will be notably higher than actual tests would support.

To those interested in the bullet:

The front driving band/ogive integration on there is supposedly patented by GS Customs, though I have not managed to locate the patent in an international database. I have absolutely no business interest with Gerard Schultz's business but if it is indeed patented, this design may land you in hot water. Just an FYI.
</div></div>



I very much doubt attempting to patent driving bands would meet with much success? US Patents often get through with spurious claims however an EU Patent is anoter matter altogether. </div></div>

I would really like to read that patent disclosure and I've spent several hours searching for the patent(s). It might be under a SA patent, but I don't know how to go through that database. Google Patents searches the USPO disclosures very well and there's some decent stuff for the EU stuff online as well.

If it's only a South African patent, that would explain how Barnes can get around it without trouble, just don't sell those bullets in SA they can't be touched.
 
Re: 50cal bullet design...

Any idea how much the B.C. would be impacted by using a Von Karman ogive in place of the simple radius?
 
Re: 50cal bullet design...

In truth, i know there are many improvements that could be made to this projectile offered above. With Nose profile changes, whether von karman or a simply a more agressive secant ogive would give a lower drag profile, then there is boat tail changes aswell... At the end of the day, all these changes add up to very little. Worth going after? - most definately... Enough to make to bullet seem magic? - Hell No...

The design offered, illustrates that even an average design, can have pretty damn high "calculated BC`s", due in the most part because its extremely heavy, not because its a good well thought out design...
 
Re: 50cal bullet design...

Is that longhand for: "I'm not really sure"?
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: Trigger29</div><div class="ubbcode-body">Is that longhand for: "I'm not really sure"? </div></div>

Think about what your asking, the answer to your question is not an easy one because it cannot be answered without first defining many caveats to place upon it, including but not limited to;

A von karman profile of the same aspect ratio? or can we change this aspect too since we are changing the nose anyway?

A change in nose profile alters the volume = alters the mass = also alters BC, so you cant change one without the effecting the other.... So what are we measuring, the loss in BC from the loss in mass, or the gain in BC from a more efficient profile? So what do we do, increase the length to offset the loss in mass, or keep it the same length and loose the mass when comparing the 2 different profiles?

You question would also have to be answered, at what instantaneous speed? - the wave drag component of total drag changes with velocity, as does the base drag and viscous or "skin friction" components, as does the way in which BC is modelled.

How would the BC change compared to what? -The theoretical modeled BC`s in McDrag or the total "cumulative real world average BC" over the entire flight regime of ?what? distance? - which is the number we are really interested in isnt it? The latter is unfortunately much more difficult to answer...

So without going into great detail, ill simply put it like this;
The total zero yaw drag on a secant ogive, 7 degree boat tail projectile (typical in dimensions to our VLD projectiles) in the average velocity regime of 2200fps can be broken down into 3 components. Approximately 33% Wave drag, 18% viscous or skin friction drag and 49% base drag. The von karman ogive only increases efficiency in the wave drag component of the nose section, and does so by about 10% (which varies depending on the velocity), (maximum 3% of total) However its not clear how much of the wave drag is coming from the boat tail and therefore its very difficult to quantify its effects on the total drag. It really is a complex problem best answered by an experienced ballistic engineer. Even then, it would be a very difficult problem for them to solve computationally, and the answers not easily understood by the majority of people reading this forum.
It would be easier, to simply test the 2 different versions over dopplar radar, or even a well executed physical long range firing test performed by an experienced ballistician, comparing the 2 exact same bullets, except for the different nose profiles - which would also give you a real world difference instead of a computed value. I can guarantee you however, that the difference is not spectacular in terms of you would barely notice it at anything but EXTREMELY long range, well into the transonic region and beyond - the von karman sees its greatest efficiency gains in the entire transonic region and immediate supersonic region above this upto about 2000fps.




 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: groper</div><div class="ubbcode-body"><div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: Trigger29</div><div class="ubbcode-body">Is that longhand for: "I'm not really sure"? </div></div>

The von karman ogive only increases efficiency in the wave drag component of the nose section, and does so by about 10% (which varies depending on the velocity), (maximum 3% of total) </div></div>This is pretty much what I was looking for. Thank you. If programming ths bullet for your lathe, it seems a lot of work to program a Von Karman for very little gain.
 
Re: 50cal bullet design...

<div class="ubbcode-block"><div class="ubbcode-header">Quote:</div><div class="ubbcode-body">This is pretty much what I was looking for. Thank you. If programming ths bullet for your lathe, it seems a lot of work to program a Von Karman for very little gain. </div></div>

I guess this highlights the differences between various manufacturers...I personally beleive, that every last ounce of efficiency is worth going after, in every aspect of a projectiles geometry
smile.gif
 
Re: 50cal bullet design...

I never said it wasn't worth going after. My own bullets currently use a Von Karman. I actually asked the question because I calculated the b.c. of them using a secant ogive, then I changed it to the Von Karman, and they shoot WAY flatter than what I had calculated. Now I'm trying to figure out what I really have.
 
Re: 50cal bullet design...

Trigger,

At high velocity, the difference between a simple radius, and a VonKarman, is at best 4-5% in efficiency. That disparity expands dramatically below Mach 1.5 to ~14%.

This is what you are seeing in your actual shooting.
 
Re: 50cal bullet design...

umm, 14% of what? Just the nose cone efficiency or the total drag on the projectile, or the change in BC?

If just the nose cone, 14% compared to what simple radius? An agressive large radius secant ogive, or a small radius tangent ogive, and over what aspect ratio are we talking about?

Impossible to quantify the end result Noel unless you know the exact 2 geometries your comparing, which we dont. And it doesnt explain the differences he is seeing unless we know how much difference he is seeing, and to what range he has shot them?

In my expereince, theres some small gains to be had from the von karman, but i would not go so far as to say "they shoot WAY flatter" compared to a radius nose of the same aspect ratio...
 
Re: 50cal bullet design...

Groper,

There is only one way to make a sensible <span style="font-style: italic">nose</span> comparison; A VonKarman contrasted with a tangent ogive of equal volume. Too many other variables enter otherwise... which actually may be the basis of your 10% figure (assuming you have isolated the nose only).

When your drag numbers for the balance of the projectile are added, the distinction is further obscured.
 
Re: 50cal bullet design...

My head hurts after reading this post! I will continue to follow it though, very interesting! I would like to launch a 900 plus grain projectile out of my Barrett 99!
 
I know this thread is old but I made some today
<a href="http://s464.photobucket.com/user/Dylanss180/media/20_50/IMAG0140_zpsfea07d7f.jpg.html" target="_blank"><img src="http://i464.photobucket.com/albums/rr9/Dylanss180/20_50/IMAG0140_zpsfea07d7f.jpg" border="0" alt=" photo IMAG0140_zpsfea07d7f.jpg"/></a>

I am a while out before trying them out.
 
I take it these are for your necked down 20mm project, Dylan? Definitely look a bit better than the pic of your previous attempt. Good to see you making progress and remaining dedicated, especially to a post you made over 2 years ago.
 
Yeah, these bullets are for this project. I will try shooting them after I finish test firing the gun with normal 50 cal bullets.
 
So after all the interest in this thread how did they shoot???
 
Hey Sry to bring this old post up again but this seems to be the only post i ve seen that actually has dimensions for a bullet so i model this same bullet in solidworks and everything is exactly the same but i cant seem to find any loading data for ~1050g bullet so i cut off the back to make it 800g again i am not a ballistics expert but i don't see why this wouldn't work. Any one have any thoughts i was going to have a trial run tomrrow on this to see if i can get the correct tolerances out of the cnc lathe i got a hold of.

here is the pic of the new bullet




here is the lathe
Sherline CNC lathe systems

this may not work but fun never the less...

my option 2 is to buy some commercial VLD 50 bmg bullets and model those and see if i get the same results
 
I am sure it will work but the boat tail has some necessity in the design. I am not a ballistics expert either but i know the shape will affect the flight. Let us know how they come out on the sherline. I was looking into those but never went for it just because I was afraid of the accuracy of the machine.
 
The problem with shooting heavy bullets like these, is the lack of powder available slow enough to optimize velocity. I have shot projectiles very similar to these (930 grains) out of a light class competition rifle (30#) and they certainly make the rifle twist and jump on ignition. I was only able to get into the 2400 fps range with VV20N29. We will need a canister grade powder slower than this to effectively make use of 900+ grain projectiles like this...
 
Yes, there is very few choices for slower powders that can work for something that heavy. I have found a few powders that are canon extruded powders but Alliant powder will not respond to my inquires about that powder. They won't even return my e mails.
 
If you start doing .375 designs, I'm interested...very interested...

Sent from my iPad using Tapatalk HD
Once I have my project done I could make some 375 for you. I don't have that caliber of gun but if you want to have me make you some different versions I could.
 
Are these Sherline lathes heavy enough to accurately turn these bullets? I'd be interested in buying one and learning how to use it if they were....

Wes


Sent from my iPad using Tapatalk HD
 
Hey Sry to bring this old post up again but this seems to be the only post i ve seen that actually has dimensions for a bullet so i model this same bullet in solidworks and everything is exactly the same but i cant seem to find any loading data for ~1050g bullet so i cut off the back to make it 800g again i am not a ballistics expert but i don't see why this wouldn't work. Any one have any thoughts i was going to have a trial run tomrrow on this to see if i can get the correct tolerances out of the cnc lathe i got a hold of.

here is the pic of the new bullet




here is the lathe
Sherline CNC lathe systems

this may not work but fun never the less...

my option 2 is to buy some commercial VLD 50 bmg bullets and model those and see if i get the same results

As soon as my laser scanner gets here from Kickstarter, i could scan any projectile you wanted.


Sent from my iPad using Tapatalk HD
 
Ya I am finding out that working a lathe is a little more complicated then a cnc machine given that the stock.. atleast for my aluminum proto typing isnt strong enough to only use one support from the back so I am in the middle of redoing the g-code to cut out the whole bullet except for the tip so that it has support from both sides until the very end.

Also it appears that is sherline lathe cant handle much more then .005 cutting depth (aluminum) so needless to say it takes awhile to cut this stuff out but I will post pics of my wood/ aluminum proto types
 
I would really like to get the models/dimensions of the barnes tac-x vld or the woodleigh version
 
I hope everyone realizes that this projectile will never stabilize for many reasons!
It may look good on paper but that's about all it would be good for
a paper weight

ok
flame suit on

Id like to offer a free 1075grain .50cal VLD bullet design, you have my permission to use it however you wish. The design weight is accurate via 3D CAD calculation using copper as the material with a density of 8.95g/cm^3. Machined from Brass @ 8.47g/cm^3, it would be a bit lighter and the stability is also worse - i wouldnt recommend it.

So here it is...
50calfreebie.jpg




The following BC and stability calculations were performed with the McDrag Projectile drag and static stability calculator as per Robert McCoy, former ballistics researcher, Aberdeen proving grounds.

Required twist is 1:11.3 @ 2791 fps for a stability factor of 1.5, however a 1:12 twist should be adequate, and still provides a stability factor of 1.3 in std atmosphere conditions at sea level.

G1 BC`s are calculated as follows;
1.793 @ 3349 fps
1.669 @ 2791 fps
1.615 @ 2232 fps

G7 BC`s as follows;
.846 @ 3349 fps
.834 @ 2791 fps
.811 @ 2232 fps

Assuming we shoot this projectile at a muzzle velocity of 2600fps, we can assume the velocity regime of this projectile over the course of a 1000yd flight to be between 2600fps and 2075 fps @ 1000yds, remembering that this is based purely of the calculated BCs.

By averaging the BC`s across this flight regime, we get an average G7 of .816 and this equates to an average G1 BC of 1.621. These would be the numbers youd be interested in to initially build a trajectory from in your initial testing, and revise it to a real world BC from there.

I have no idea whether the dynamic stability of this projectile is adequate, so proceed at your own risk and have fun whilst your doing it. If someone actually does get this machined up, please let us know how it goes... Enjoy.
 
I hope everyone realizes that this projectile will never stabilize for many reasons!
It may look good on paper but that's about all it would be good for
a paper weight

ok
flame suit on

kd185

Can you explain why it won't work. I would like to be more educated in what the key things are needed in order to make one work.
 
kd185

Can you explain why it won't work. I would like to be more educated in what the key things are needed in order to make one work.


I agree please explain... I am reading that it bullet will start to wobble because of the length vs my twist rate 1:15 in the ar-50 but I have no idea why legth has anything to do wwith it and it seems like the center of mass and the center of pressure should be equal... but just a thought since I am a electrical engineer I dont have much expirance in ballistic design... but I do plan on buying a ballistics book and read more they are just quite expensive
 
many many reasons

1. no available match grade powders slow enough to launch a bullet that heavy (possible some military pull downs but nothing readily available, easily accessible or of the quality to match the quality of cnc brass solid)
2. a bmg case is not large enough to carry an adequate powder charge for a bullet that heavy (a charge of 250 grains of Reloader 50 doesn't fill it but there is not much room left)
3. nor can it get that bullet up to a speed fast enough to stabilize
4. you would need a longer barrel than the standard long barrels of 36" to get a full burn (yes i know M2 barrels are longer than 36" but not designed for this application)
5. a brass solid projectile in .510 cant get much heavier than 820 with out getting to long to stabilize or have a decent center of gravity
6. state arms in the 90's was one of the only company's to ever offer a barrel twist of anything besides 1 in 15 and those are few and far between these days (as far as i know of)
7. there is a noticable difference in recoil from a 750 to an 800 can you imagine what a 1075 would be like
8. spend a few minutes with any of these (the miller twist rule, the greenhill formula, a drag/twist calculator) and tell me what you find
9. read Applied Ballistics by Bryan Litz and tell me what you find

i dont claim to be an expert but i have 2 - 50 bmg rifles for the last 7 years
i reload everything i shot
i have shot in 12 FCSA 1,000 yard matches in the last 6 years
ive got over 1,500 bmg rounds down range
the most ive shot was 96 rounds in 1 day due to a make-up day in the rain
my best group to date was 7-1/8" 5 shot @ 1,000 yards off the ground with a bi-pod and a sand bag
nothing great but nothing to laugh at either

ive been following this thread since the beginning and i cant believe its gotten this far
i dont know the OP "groper"
and dont want to call anyone names
but it sounds like some has spent to much time in front of a cad-cam program and not enough time behind a trigger or a reloading bench
yes they have included a few trick features of match grade custom ELR bullets but again it looks good on paper or a monitor
ask Lehigh or TenX (2 of the best brass solids on the market in my opinion) why they dont make bullets much heavier than 800 grains in brass solids

as far as bullets go i have been researching this as i have just recently started swaging my own copper jacketed lead core BMG bullets with plans to shot my bullets in next years fcsa matches

rant over for now
 
Last edited:
I see your point and I am the one who brought this out of the grave and my orginal intention was never to replicate this exact bullet through my gun but use it as a starting point to make barnes/woodleigh like bullets by hand.

I personally am not able to shoot everyday and with an engineering back ground I think ballistics are quite interesting. And so I drove into this indever not because its cheaper and not because I am an internet troll with nothing better todo. But just to see if it can be done and I when I came accross this post I figured they were others that have done this and save me some headaches from their lessons learned. Learning is a hobby for me... being able to apply it to things that go bang is a plus

I have not read applied ballistics yet but has been a christmas request. But I am curious what is the purpose of the slots cut in the sides I heard somewhere it creates drag abd intern bullet stability and I guess the first one also acts as a crimp groove. You say the bullet doesnt stabilize for several hundred meters is there an equation for that?