Project Appleseed

This topic grew out of an earlier thread involving requests for sight radius measurements for a possible table of sight adjustment data.  Interested parties may find it here:  http://appleseedinfo.org/smf/index.php?topic=5087.0

As mentioned in that thread, just posting the sight adjustment table simply provides "fish" for Instructors rather than "teaching them to fish".  Since I didn't find a "how to" posted elsewhere I volunteered to write something.  As you will see it got a little out of hand.   To prevent information overload I don't teach this to Appleseeders but I do believe Instructors should know the principles.

In IMC we tell Appleseeders "go from the known to the unknown".  Measure the distance of the center of your group in inches on the target, convert the information to MOA, then adjust your sights, in clicks, to move the center of the group to the desired point of impact.  Piece of cake; except sometimes the last part: figuring out how many MOA each "click" is worth.

It can be done empirically.  In the earlier thread Slim described a good way to empirically find or confirm the clicks-to-MOA relationship:

"tell people to zero then do 10 clicks, shoot a group, measure, do the math backwards and write it down."

There's no doubt you need to confirm any clicks-to-MOA relationship by shooting but I'd rather have an estimate before I shoot: it saves me both time and ammunition. 

When we calculate a MOA change during IMC it's for making a change in the angle between the line-of-sight and the axis of the bore, either vertically (elevation), or horizontally (windage), or both.  (For our purposes the bullet trajectory is considered a constant (i.e. repeatable) curve diverging from the axis of the bore at the muzzle.  If we change the angle between the line-of-sight and the axis of the bore, the trajectory, and the bullet strike on target, is shifted by the same MOA change. )

Determining the relationship between MOA and clicks can be broken down into a three-step procedure.

   1) Calculate how far the sight must be moved, in inches, to effect a one-MOA change.  I'll call this number 'specific displacement' and the units are inches-per-MOA ("/MOA).

   2)  Determine how to effect that change.  Is the sight to be simply drifted with a punch?  Is there a sliding plate adjustment with division marks? Are there offset screws or an adjustment knob with detents? Measure or calculate the separation between the marks or the distance the sight travels per unit of movement.  These values will probably be either "inches/division" or "inches/turn". 

   3) Combine the values from Steps 1) and 2) to arrive at the desired answer for your sight: MOA/turn, MOA/division, MOA/detent, etc. 

First, some definitions.  A line-of-sight is defined as a straight line from the rear sight, to the front sight, to a point on the target.  The distance between the rear sight and the front sight is the sight radius (sr).  The distance from the front sight to the target we'll call distance (L). 

If we physically displace place the rear sight (either elevation or windage) a distance (d), we create a new line-of-sight, intersecting the old line-of-sight at the front sight at angle A.  Since the opposite angles of intersecting lines are equal, the line of sight on the target is displaced distance (t), at the same angle A, opposite to the direction of the rear sight displacement.

Using some trig:

     rear sight displacement distance (d) = Tan A x sr. 

     (at the target) line-of-sight displacement distance (t) = Tan A x L.

Rearranging and solving for the common function Tan A

     Tan A = d/sr   and Tan A = t/L so d/sr = t/L

Solving for d, d= sr x t/L

The neat thing about this relationship is that we can measure one of these variables directly and infer two of the others from our knowledge of MOA and arrive at the "specific displacement" value for Step 1). 

     We'll measure the sight radius with a tape measure. (sr)

     We know that one MOA covers 1" at 100 yards so we'll set t = 1"/MOA

     Since we've specified (t) at 100 yards, then we must set L to the same distance and we'll also convert it to consistent units.

     L =100 yds x 36"/yd = 3600"

Example Problem #1

We have a standard Ruger 10/22 Carbine (18.5" barrel) with factory sights.  How far do we need to move the rear sight for each MOA we want to move the bullet impact on the target?

Step 1)  Measure the sight radius: 15.0".

And calculate specific displacement:

d=sr x t/L = 15.0 inches x (1"/MOA)/3600 inches = 0.0042"/MOA

Step 2)

For windage adjustment on these sights you have to drift the rear sight.  There are no marks or divisions for windage adjustment on the example rifle.

The elevation adjustment is a bit more scientific.  The Ruger factory rear sight has a screw-secured sliding plate, with some very small division marks.

For this I needed a dial caliper.  On the example rifle four divisions spanned a total distance of 0.084" so each division averages 0.084"/4 divisions=0.021"/division. 

Step 3)

For windage adjustments, I remember that a sheet of plain paper (like a Liability Form) is .004" thick.  My final adjustment value is therefore "1 MOA/sheet of paper".   In practice, I put a reference mark on the barrel and rear sight junction with a Sharpy, take a breath, and carefully whack the sights with a hammer and brass punch until I've offset the sights in the appropriate direction the appropriate distance.  (And I ask the Appleseeder before smacking the sight "Is this your rifle, or did you borrow it from someone?" Beating on a rifle is only done after I'm really sure it's needed)

For elevation adjustment, dividing the "/division from Step 2) by the "/MOA specific-displacement from Step 1) results in the answer:

(0.021"/division)/(0.0042"/MOA) = 5 MOA/division. 

And this is in line with what I've observed at Appleseeds.

Example Problem 2

We have a Marlin 795 with factory sights.  How many MOA does each rear-sight elevation-step move the bullet impact?

Step 1) Again, measure the sight radius:  15 3/8"

And calculate specific displacement:

d=sr x t/L = 15.375 inches x (1"/MOA)/3600 inches = 0.0043"/MOA.

Step 2)

Measure the height of each rear-sight-elevator step with a dial caliper and calculate the incremental difference between each step:  they're 0.025-.026"/per step on my rifle. (call it 0.0255"/step) 

Step 3)

Dividing the "/step from Step 2) by the "/MOA specific-displacement from Step 1) results in the answer:

  (0.0255"/step)/(0.0043"/MOA) = 6 MOA/step


Next topic.  What about actual "clicks"?

With the exception of the M1-M14/M1A elevation adjustment, which uses a custom milled gear arrangement, the precision open sights I've looked into use some form of standardized commercial machine thread.  These may be either Standard, reported as threads-per-inch (tpi), or metric, reported as pitch in millimeters.  Pitch is the distance between the crest of one thread to another.  I've found the most accurate way to measure pitch is with a thread pitch gauge as found in tap and die sets.  An alternative is to use a "screw checker" plate available from Brownells and other sources, but when they get worn the results aren't as definitive as the pitch gauge.  The bad news is that you usually have to dismantle the sight to access this threaded piece.  It is not something I do casually.       

Example Problem 3

We have a Marlin 795 with Tech Sights.  How many MOA does each click move the bullet impact on the target?

Step 1) The sight radius measures 21.5" on my rifle.

Solving for specific displacement:

d=sr x t/L = 21.5 inches x (1"/MOA)/3600 inches = 0.0060"/MOA.

Step 2)

Tech Sights use a screw adjustment with detents. 

The elevation screw on my Tech Sights measured 36 tpi.  We need to calculate the pitch of this thread, i.e. how far does the thread move in one turn?

Pitch = 1"/36 turns = 0.0278"/turn

Step 3)

Again combining the results of Steps 1) and 2)

(0.0278"/turn)/(0.0060"/MOA) = 4.6 MOA/turn.

The elevation screw has five detents (clicks) per revolution, so

(4.6 MOA/turn)/(5 clicks/turn)= 0.9 MOA/click.

Example Problem 4

We have an AK with stock sights.  How many MOA does each turn of the front sight move the bullet impact on the target?

This isn't a trick question but the theoretical math for the total distance to the target is a slightly different for front sight calculations.  Fortunately the difference isn't significant for our purposes but I'll do it anyway so somebody doesn't bring it up later.

When the front sight is adjusted with the rear sight fixed the line of sight is displaced at the front sight and at the target in the same direction.  The terms remain the same.  However, the new line of sight intersects the old line of sight only at the rear sight at angle A. 

In this new relationship, the tangent of angle A still equals the distance the front sight moves divided by the sight radius.  The tangent of angle A is also equal to the distance the line of sight moves on the target divided by the total distance from the rear sight to the target which is now the sight radius plus the distance L.  Mathematically:

    Tan A = d/sr and Tan A = t/(L+sr), so d/sr = t/(L+sr)

Solving for d,  d = sr x (t/(L+sr))

Step 1)  Measure the sight radius: 15.0"

Solving for specific displacement:

d = 15.0 inches x (1"/MOA/(3600 inches + 15 inches)) = .0041"/MOA

If I'd ignored the 15" difference in total distance the answer would be .0042"/MOA.  In this case it's not enough to worry about.

Step 2)

The AK front sight is a threaded post and is screw-adjusted for elevation.  The thread is measured as 0.8 mm pitch.  Converting to standard units:

(0.8 mm/turn)/(25.4 mm/inch) = 0.0315 "/turn

Step 3)

Again combining the results of Steps 1) and 2)

(0.0315 "/turn)/(0.0041"/MOA) = 7.7 MOA/turn.


Some final thoughts.
   
The inquisitive reader can extrapolate these methods to reverse-calculate a number values from published information on various rifles.  This has some pitfalls.  One of the problems I've run across while researching sight adjustment numbers is that published information seems to be rounded off.  This is not a problem if you're going to use the published information for an equivalent rifle, but it can lead to some pretty significant errors if you're combining an estimate of sight radius with a rounded off MOA/click value and a guessed screw pitch.  (Ask me how I know this.)   And you'll get better results if you save the rounding for the end of your calculations. 

Also be observant: I wasn't, at first, and spent time trying to reconcile some of my results.  Some numbers are reported as MOA; some are reported as inches- or centimeters-of-bullet-movement at a particular distance.  In USMC MCRP 3-01A (03/2001) the USMC notes that one click of M16A2 front sight elevation change will move the bullet 3.5 cm at 100 meters or 1-1/4" at 100 yards, keeping units in consistent systems (cm-meters, inch-yard.)  The Army (apparently) uses only metric distances and in FM3-22.9 (Aug 2008) reports 3.5 cm at 100 meters or 1-3/8" at 100 meters for the same rifle.  The 1-3/8" bullet movement converts back to 1.25" at 100 yards but if you had assumed that 100 meters was "close enough" to 100 yards (as I did) and that this is a 1-3/8 MOA adjustment you're going to be off a bit (as I was).  And if more precision is required, remember 1 MOA isn't 1"/100 yards it's a bit over 1.047"/100 yards.

And finally, to keep the post on topic, consider PM'ing me with questions, especially about specific rifles.  Please remember this post is about "how to fish".

12/27/11 first edit for a (really glaring) math-conversion typo error, for clarity, and some revisions to the last paragraphs for full disclosure.   ::)
12/29/11 added dates to the referenced USMC and US Army manuals.  An earlier version of FM 3-22.9 (Sept 2006) didn't contain the listed information.
01/04/12 Correct spelling error; edit one line for clarity.   

Setting windage rather precisely...

Tooling to push dovetailed parts helps avoid "minute of whack" using a.hammer and drift

Measuring displacement per click. Averaging several a plus...

A dial indicator allows measuring sight movement per click; avoids getting at adjusting screws to measure thread pitch...

Your dial indicator mount is a very elegant piece of equipment - very handy for any sight measurement.  O0 

IMHO You would do better to loose the trig and go with similar triangles. Much easier concept to get across to those who don't do trig.

Quote from: douglas34474 on November 05, 2013, 03:17:45 AM
IMHO You would do better to loose the trig and go with similar triangles. Much easier concept to get across to those who don't do trig.

You mean just don't call a tangent a tangent, draw triangles showing the sights, target, desired POI shift, and tell folks "you're just keeping it in proportion"?  Wow, I haven't done this math in years!

To determine a 1 MOA adjustment, you're shooting at 100 yards and you need to move the POI 1"', 100 yards = 3600 inches, so you need to move the POI 1/3600 of the distance.  Since you see from the drawing (rest assured I've drawn on a target backer for an authentic Appleseed 'feel') that these triangles are similar, their sides will be in proportion.  So in this case, for a rear sight move, you need to move the rear sight in that same proportion to the sight radius as the desired change in point of impact to range, 1/3600.  So 1/3600 * 15" sight radius = .0042".

Or, for a front sight move add the SR to the original distance to target, so, 1/3615 * 15" = .0041"?  (Different triangles, if I were trying to teach this I would probably start with this because it is perhaps more intuitive to a non geometry person how these triangles are similar...the "sight" triangle just shaving the end off the "target" triangle rather than convincing folks that opposed angles make for similar triangles.)

For a shorthand version...physical sight displacement needed to make a 1 MOA rear sight adjustment = (sight radius / distance to target)
         Ditto for a 1 MOA Front sight adjustment = (sight radius / (sight radius + distance to target))

(Measurements have to be in inches for this version.)

Thanks for the original post and for bumping it, I learned something this morning before coffee!

In the spirit of KISS  I like Slim's method best.  Remind them they need to do it for elevation and windage as they may move a different amount per click.  Have them to shoot the square to make sure the sights move consistently.

Then get this information attached to the rifle.

Yep....KISS.... For 99% just getting across how 1/4 or 1/2 MOA clicks work is tough enough. For those of us that work with math a lot, that's one thing but we could spend an hour getting into details and waste 59 minuets of it! :)....O.L.

I sure would've paid a lot more attention in school if I would've known I was doing sight corrections with this math. "We're never going to use this in the real world, are we?"  :slap:

IMC is one of those things we could, quite literally, get out the micrometer and measure, plugging stuff into scientific calculators to create tables and programs after spending hours doing trig functions. Or, we could conceptualize it in less than five minutes and send them on their way. I had someone at the last event tell me, after almost five years of attending events, this was the first time they'd actually "got it" when it came to IMC.

The more I get into these "advanced computations" the more I see the simple rules work just fine.

Thanks for putting all this out here. It's like watching MythBusters and seeing something proven. (Or Busted!) The beauty of math is it's not just a suggestion... it's the law!

You use the old "means divided by the extremes" or "extremes divided by the means."

Using your example, solving for X:

You need  to move the POI 1" at 100 yards and your rifle has a 15" SR.

1":3600" (1 is to 3600) as X:15 (X is to 15) X being the amount you have to move the sight, the unknown. You multiply 1x15 (the extremes) to get 15 then divide 15 by 3600 (the means) to get 0.0041", the amount you have to move the sight.

Or 3600:1 as 15:X. Again you would multiply 1x15 (the means) to get 15 then divide 15 by 3600 (the extremes) to get 0.0041"

This used to be elementary school math.

PS. I was talking to a math teacher with 18 years of teaching experience last night. When I mentioned MOA she looked at my like I had a third eye in the middle of my forehead! How many people on the firing line do you think are even going to know what a "tangent" is, let alone how to do trig?

I'm with douglas34474 on this one. There is no point in calculating the MOA of adjustment needed, just to turn it back into a linear value of how much the sight should move. Similar triangles/proportionality is a lot easier to use and to teach/learn for applications where you don't know the MOA/click of the sight or the sight doesn't even have clicks. I've used that method numerous times to adjust the sights on my own firearms, and can often do the calculations in my head. The limitation is usually making the measurement of movement, or getting the sight to move at all!

Even when I'm helping friends sight in their rifles with scopes, I just stick to "1 inch at 100 yds per click, 1/2" at 50, etc." rather than getting into the whole MOA thing. With new shooters I've found that even that is enough to make their brains full.