Hit probability

[mman]

There is another calculator made with bfx functions:


Calculator is mainly for comparing calibres and projectiles for various applications but there is also lots of graphs about dispersions, max ranges etc... Internet is full of every kind of ballistic comparisons but many times most of the information does not describe right features for the current application. Another problem is that it's very hard to determine which values are important and what is the practical difference in group size or changes to hit the target. The key idea was to generate output values which would directly describe what is the practical difference between two projectiles.

User can compare four different projectiles at the same time. Green cells are for input values. "Update field" button updates sheet (macros must be enabled).

PRECISION AND ACCURACY

MAX RANGE ESTIMATION ERROR is an indicator of how straight bullet flies. Value depends on target size.

MAX WIND ESTIMATION ERROR is a wind drift indicator and depends on target size.

GROUP SIZE is a statistical value for group size, depending on dispersions. This is the most important value for target and BR shooters (known distance and sighters).

HIT PROBABILITY is a statistical value for changes to hit the target at the first shot. This is the most important value for hunters and tactical shooters (unknown distance and no sighters). One might argue mathematics behind these, but after all it's valid for comparison purpose anyway.


EFFECTIVE RANGE

Usually projectile is effective when it flies stable from the muzzle to the target. When rifling twist is adequate and bullet is supersonic for the whole trajectory it should be good to go for target shooting. Hunters are sometimes interested in terminal velocity due to bullets terminal performance low limit. That is one boundary for effective max range. Another one is energy which is also listed on effective range field. Then there is also OGW formula and couple indicator of recoil. Other sheets are for dispersion and effective range charts..      

Btw. calculator is made for screens with full HD resolution and excel 2007. There might be incompatibilities with other versions. Report me if you find errors or bugs.            

[admin]

Impressive!

[mman]

Robert,

What you think about dispersions from shooter's holding error? Is it linear or time of flight depended? In other words, is the group size angular or does it grow unlinearly along the range due to this error. Bryan Litz suggests in his book that shooter can introduce time of flight depended dispersions. This is important detail for group dispersion and hit probability models.

[admin]

... shooter's holding error? Is it linear or time of flight depended? In other words, is the group size angular or does it grow unlinearly along the range due to this error.

If so then my first guess would be that as drop and wind deflections vary quadratically with time, errors will also vary quadratically with time and more or less with distance. I have to think about it... And play a bit with my simulation that does not suffer from these issues.

However, in your spreadsheet you added the dispersions quadratically. Although many dispersions are indeed independend, at least two of them are a bit correlated. If a bullet leaves the muzzle at a higher than average speed, it will hit the target higher and will be deflected less by wind. It is a rather minor effect, barely noticable, but never the less present in my simulations (in the WhatDoIShootToday spreadsheet): bullets that are deflected less hit the target on average higher and vice versa.

Bryan Litz suggests in his book that shooter can introduce time of flight depended dispersions. This is important detail for group dispersion and hit probability models.

It would help if you could give me the page number, then I do not have to think to hard... 

[mman]

However, in your spreadsheet you added the dispersions quadratically.

True. That was easy way to sum them up. Then I just used normal distribution to stress center values. I'm not saying it's just equivalent with reality but at least it's enough for comparison. If you have something better in mind I'm still interested.

Although many dispersions are indeed independend, at least two of them are a bit correlated. If a bullet leaves the muzzle at a higher than average speed, it will hit the target higher and will be deflected less by wind. It is a rather minor effect, barely noticable, but never the less present in my simulations (in the WhatDoIShootToday spreadsheet): bullets that are deflected less hit the target on average higher and vice versa.

True. BC variation goes the same way. I have excluded those as small errors. Regarding muzzle velocity variation there is at least as important effect that's very hard to predict. That's muzzle velocity variation's dependence on zero range. This effect is introduced by barrel vertical vibration and barrel time. This is the reason why best benchrest rifles can shoot 1000 yard groups with nearly no vertical. More information about this here: http://www.varmintal.com/aeste.htm and ladder test which makes use of this effect: http://www.6mmbr.com/laddertest.html

It would help if you could give me the page number, then I do not have to think to hard...  

It's page of 184 in the second edition of his book.

I think that's chapter 11 in the first edition. Here he speaks about the same thing:
"For this example, we’ll consider a 30-06 rifle shooting 185 grain bullets at an average muzzle velocity of 2850 fps. The rifle will be modeled as being capable of 1” groups at 100 yards on average. For this current analysis, we’ll consider the impact zone at 500 yards. According to the principles of bullet dispersion given in Chapter 11, we can extrapolate the group size from 100 yards to 500 yards based on the bullets time of flight. The bullets time of flight at 100 yards is 0.1084 seconds, and at 500 yards, it’s 0.6154 seconds. Since the rifle is capable of grouping into 1 inch at 100 yards, we can expect the dispersion to produce a 500 yard group of: 1 inch times 0.6154/0.1084 = 5.7 inches minimum." http://www.longrangehunting.com/articles/applied-ballistics-long-range-shooting-bryan-litz-book-1.php

[admin]

Regarding the

... shooter's holding error? Is it linear or time of flight depended? In other words, is the group size angular or does it grow unlinearly along the range due to this error.

I have read the passage of Litz (I have the first edition, and indeed it is chapter 11) on page 185 there is an paragraph about the natural point of aim. Brian reasons that a body could inflect random movement of the muzzle during recoil (barrel time). Any line of sight perpendicular (lateral) velocity component translates (because of the non-linear relations ship between flight time and distance) into a non-linear dispersion. Certainly this is a non linear, time dependent contribution to the shooters accuracy. I think Brian has a point here.

One of the others accuracy affecting factor of the shooter is his aiming ability. This will not produce a lateral velocity. This is an angular contribution.

The inability to read the wind results in an aiming error. However, if the ability to read the wind is a constant, regardless the distance to target, the effect of it increases with travel time and gives to a non linear contribution to the accuracy

Thanks for the link to the barrel vibrations article.

[admin]

Personally, I think my aiming ability is much better than my ability to maintain a natural point of aim, which is not bad either.... The funny thing is that my conscious mind wants to control the aiming and neglects the natural point of aim.

[admin]

Interesting issue, I never thought about is. It tells why a indoor 300m range is more exiting than a 50m one with a reduced target.

[mman]

I have read the passage of Litz (I have the first edition, and indeed it is chapter 11) on page 185 there is an paragraph about the natural point of aim. Brian reasons that a body could inflect random movement of the muzzle during recoil (barrel time). Any line of sight perpendicular (lateral) velocity component translates (because of the non-linear relations ship between flight time and distance) into a non-linear dispersion. Certainly this is a non linear, time dependent contribution to the shooters accuracy. I think Brian has a point here.

One of the others accuracy affecting factor of the shooter is his aiming ability. This will not produce a lateral velocity. This is an angular contribution.

Regarding the linear aiming error, I agree with bryan but with this: "Any line of sight perpendicular (lateral) velocity component translates (because of the non-linear relations ship between flight time and distance) into a non-linear dispersion." is what I must disagree. The way I see it is that there is no way how initial velocity of bullet alone could lead to unlinear movement. Let's think about this...

I'll quote you the newton's first law:

"Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it."
 
I ask you this: What is the force which makes the dispersion to curve along the trajectory? Let's think about this another way. Bullet has a velocity vector when it leaves the muzzle. Vector can always be divided to components but drag affects to all of those components equally. That leads to conclusion that "lateral" velocity component slows down also. So the velocity component due to launch dynamics leads still only to linear dispersion, not TOF dispersion. At least that's the way I see it...

[admin]

I ask you this: What is the force which makes the dispersion to curve along the trajectory? Let's think about this another way. Bullet has a velocity vector when it leaves the muzzle. Vector can always be divided to components but drag affects to all of those components equally. That leads to conclusion that "lateral" velocity component slows down also. So the velocity component due to launch dynamics leads still only to linear dispersion, not TOF dispersion. At least that's the way I see it...

I see your point. In absence of external and assymetrically applied forces (wind) the bullet flies in a straight line. With lateral muzzle velocity it just flies in another direction than the bore axis /line of sight is pointing. However, the dispersion is another quantity. The real question here is if the deflection due to dispersion of the lateral muzzle velocity scales with time. What is  propagtion of the quantity y1-y2 where y1 and y2 are the lateral displacements as a function of distance. Although this is easy to calculate with my 3DOF, the question how vy relates to vx (the velocity along the line of sight): vy^2-vx^2=v^2 = constant? If so then the lateral velocity dispersion will also translate into a dispersion of the launch direction, complicating a straightforward comparision.

[mman]

I just think it's Vx^2 + Vy^2 = V^2 as long as there no external forces (besides drag which is always parallel with V). If Vy is horisontal velocity component from launch dynamics and Vx is muzzle velocity component then V is just a little bit more with Vx than without it. And that is too little to worry about. Direction is different with Vy but that is just angular deflection.

[admin]

The simulations learned what I knew already - the difference in distance between points y on two straight lines at a distance x increases linear with x. With other words, dispersions scale with distance and are angular.

To sum up: an additional lateral horizontal muzzle velocity shifts in the horizontal plane the angle of the trajectory.If one place a target at a certain distance z the poi differences increase linearly with distance.

mman is right, it looks if Bryan makes an error here.

[mman]

Okay, it seems that this one is solved. Again one more thing towards higher level of understanding. Accordingly groups should grow pretty much linearly along shooting distance until REAL unlinear dispersions start to dominate. Main unlinear dispersion factors are wind, muzzle velocity variation, bc variation and cant. Cant is little different than others because effect is unlinear withing zero range but not withing shooting distance. Anyhow it's effect that makes it harder to shoot small groups for longer distances. Usually gun is zeroed at shooting distance so in my mind it can be called as unlinear dispersion factor.

All in all shooter should concentrate on linear dispersions at short range and on unlinear dispersions at long range. I think this conclusion was no surprise to dedicated competitive shooter.

[ThunderDownUnder]

Mman, what about gyroscopic drift? Along with any velocity changes there will also be gyroscopic drift changes. Can gyro drift be overlooked in your group dispersion probability calcs?

Ian

[mman]

Mman, what about gyroscopic drift? Along with any velocity changes there will also be gyroscopic drift changes. Can gyro drift be overlooked in your group dispersion probability calcs?
Ian

Ian,

Overall spin drift is something about 0.2 mils at 1000 meters. This is a considerable deflection when calculating trajectories but when we speak about dispersions this is a minor thing. That is because spin drift can vary something like +/- 2 % between shots due to muzzle velocity, bc and initial jaw variations. That's 0.008 Mils (or 8 mm at 1000 meters) of total dispersion from spin drift variation when shooting groups. So it's there but not so much to worry about. When it comes to hit probability, deflection always have to be corrected. Then there is a question how accurately spin drift can be calculated. My calculator assumes that there is no correction error in spin drift. But there is an input value for ballistic correction error which is for drop correction only. This is an indicator of difference between calculated trajectory and true trajectory. Drop correction is way biggest factor that must be corrected so all else is pretty minor compared to that. Good point anyway!

There is one more thing I would like to point out. When dispersions are summed up quadratically (a^2+b^2+c^2)^(1/2), minor dispersions fade out. For example if a is 100 and b is 100 then (a^2+b^2)^(1/2)=141.4. But when we add c = 10 which is minor compared to a and b we have (a^2+b^2+c^2)^(1/2)=141.8. In this case 10 mm dispersion affects only 0.4 mm to group size. So there there is no point to add minor variatios because they don't do much for the result. That is true for calculation AND in reality. You just can't affect very much to group size if you can count out spin drift variation (8 mm) when your wind variation is 100 mm.