More On Range Estimation And Angle Fire

[echo3mike]

Sierra's website has a new article on angle firing. The bottom line for scope adjustments seems to be that you'll correct for angle fire by taking the scope elevation for the slope range X cos slope angle. This method seems to be more accurate than using the scope adjustments for the horizontal range calculated from the slope range X cos slope angle.


And a couple of articles on ranging methods, other than using an LRF or mil-dots. It's probably a good idea to have some knowledge of these methods...batteries die unexpectedly and anyone who's ever used a mil dot scope has found that they're prone to user error. One from Dean Michaelis / SP uses the mil dot reticle and the scope elevation to acquire the target range. Another from Kevin Mussack / SH has several options for ranging targets. (To scroll the article, use the up/down arrows located on the bottom left corner of the article.) Of particular note are the methods using topo maps and GPS systems, tools that many hunters and hikers use frequently.

FWIW,
S.

 

[InTheBlack]

Interesting but always complicated. Funny, I have NEVER seen any explanation of sloped fire which bothers to define the term "slant range distance." I presume it is simply the straight line distance between muzzle and target. Therefore I don't see the need to use the term at all, since that distance is the same at any muzzle angle.

And I wish he had done the example calculations for a round ball instead of some projectile that needs stabilization. That would remove the archane guestimation inherent in stability & drag formulas from the results.

Tables 5-7 all show growth in the error from top left to bottom right, and the three methods have different _magnitudes_ of error at every point; however the big question is whether the way in which the error grows along this diagonal is the _same_ for all three or not. If they do, then they all probably have the same systemic error and you should use the one whose magnitude is least. But if not, then maybe each gives the truest result for different regions of the mapped space. And maybe they all totally bollux up some regions.

Another thing is that nobody ever explains WHY sloped fire shoots high. Here's the way I think of it:

understanding bullet drop and why an inclined shot will impact high

On a piece of paper, place a point that represents the muzzle of the rifle. Aim the barrel horizontally; the bore is truly level.

Draw a horizontal line from the bore to a point 100 yards away. Make the line 10 inches on paper. Call this the "extended bore line."

Now draw a vertical line "hanging" downwards 2 inches long. This represents how far the bullet fell due to gravity. It is out of scale, but it doesn't mattter for understanding the principal. Call this line the "drop line." The end of the drop line is the POI. There is a 90 degree angle between the extended bore line and the drop line.

Now complete this triangle by drawing the hypotenuse from the muzzle to the POI. The hypotenuse is your LOS. Assume for now the front sight is at the very tip of the muzzle; otherwise the barrel would physically block your view.

Although we say that we have "sighted in at 100 yards" and that we know the "bullet drop at 100 yards" the actual bullet traversed a curved path inside the triangle. Calculating the actual distance the bullet flew along this curve would be difficult. But we never need to do so, because we will think of the problem of trajectory in a particular way.

We say that the range is 100 yards, but the actual range is the length of the hypotenuse. However, the difference is so miniscule that we can ignore it. Three feet of drop makes the hypotenuse less than 2/10 of an INCH longer than 100 yards. A .308 Sierra Match King will drop about 3 yards at 500 yards. That hypotenuse is about 3/10s of an inch longer than 500 yards.

The distance the bullet falls depends on the density of the air (which we can assume is constant) and how much time has elapsed since it left the muzzle.

Elapsed time is proportional to the actual distance along that curved path. HOWEVER, our ballistic charts have tabulated the drop distance for various intervals along the horizontal line. This is more convenient. We are only interested in the 0 to 100 yard interval in this example.

But what happens when we shoot upwards? Pivot your piece of paper to angle the extended bore line upwards. Our laser rangefinder says that the target is still 100 yards away. But we will refer to this as "100 yards slant range" just to remind ourselves that it will turn out differently from the horizontal shot.

So the interval is still the 0 to 100 yard distance. Therefore the bullet has traveled the same actual distance along its curved path, and gravity has had the same amount of time to pull it downwards.

The drop line must still point straight down (gravity is always straight down). So think of the drop line as being attached to the end of the extended bore line on a pivot. The angle must now be less than 90 degrees. With a blank piece of paper, trace your drawing, except change the drop line so that it is correctly pointing straight down.

You can imagine that the drop line will rotate around in a circle (until it hits the extended bore line when you point the muzzle straight up). In fact, you can do this on a piece of cardboard with a thumb tack and a piece of string.

KEY FACT #1: The end of the drop line is ALWAYS the POI, no matter what angle it is hanging at. Why? Because the drop distance was tabulated for intervals along the horizontal line, and the interval is still from 0 to 100 yards. So the length of the drop line is still correct; but you can see that its position on our diagram has moved.

KEY FACT #2: Your sights are still pointing along the LOS to where the drop line was hanging when it was at a 90 degree angle. The drop line has "rotated" and its end (which is the POI) is now ABOVE the LOS line. Therefore, your gun will shoot high.

The formulas for adjusting the LOS make use of a trigonometry value called a "cosine." It accounts for the circular motion of the drop line.

 

[4v50 Gary]

Good stuph. Thank you. I'm going to have to take time to digest it and it'll help me with an academic problem (Weitzel's Mill) that I've been wrestling with.

 

[InTheBlack]

This is a very understandable explanation which I just came across:

http://www.horusvision.com/sh_ballistics.cfm#jump3

>>>

Now suppose that all the conditions are the same as those described above
except that the tall vertical target is on higher ground at an uphill angle of 30
degrees. Since the vertical drop of the bullet is the same as before, the bullet
would strike the vertical target at a point 147 inches below the bullseye if we
fired with the sight adjusted for the sight-in range. However, as we look upward
at a 30-degree angle toward the target, the vertical line between the bullet hole
and the center of the bullseye would appear to be less that 147 inches long
because of the angle from which we are viewing it. We can calculate by
trigonometry that a vertical line 147 inches long would appear to be only about
127 inches (147*cos 30o) when viewed from a location 30 degrees below.
Therefore, we could hit the bullseye by (1) aiming 127 inches high or (2) by
making an elevation adjustment of about 18 moa (127/7)) on our sight.
>>>

Now the worst case would be a 45 degree angle. For a 30-06 match bullet, I figured that a horizontal shot at 500 yards would have a slant range of 707 yards at 45 degrees. The extra time to travel 207 yards is about 0.681 seconds, which means that gravity will make it fall another 74 inches.

Of course, viewing the drop from the 45 degree angle makes it look like it is shorter. If you think of the 74 as the hypotenuse of the triangle (like slant range), then you need to figure the "apparent" drop along the horizontal (side b of the triangle). And the trigonometric formula for this is:
length of h times cosine of 45 degrees = side b

I just needed to figure it out from my own perspective, which started with "hey, the bullet takes longer to get there, so how can you use the horizontal range?"

Now this turns out to be the "Improved Rifleman's Rule" as talked about in the article "Inclined Fire" on the exteriorballistics.com site.

All in all it was an educational way to spend my Sunday evening. And Night. And part of the day... Oh well it made me dredge up how to do Trig so that my math teacher's life wasn't in vain...

 

[InTheBlack]

Oh, and I forgot to include that I spent some time thinking about the Riflemans Rule vs the Improved Riflemans Rule; I just don't grok the way the author explained them, so I parsed them out and now I see the difference:

Rifleman's Rule
1 multiply slant range x cosine
2 goto table using adjusted range
3 Find come-up
4 use table come-up

Improved Rifleman's Rule
1 goto table using slant range
2 Find come-up
3 Multiply come-up x cosine
4 Use adjusted come-up

Four common factors plus one different factor:
goto table
slant range
cosine
table come-up
USE adjusted range OR USE adjusted come up