5.4 Changing the Zero Range
Some of the questions most often asked in letters and telephone calls from shooters, concern what happens to the bullet trajectory when the zero range is changed to values not used in the Ballistics Tables. To help answer these questions, Sierra has prepared the Ballistics Tables in this Manual with four zero range selections for each bullet. We have tried to pick an appropriate selection of zero ranges for each bullet, depending on its primary purpose and on the muzzle velocities of the cartridges in which it is loaded. If you glance ahead to the Ballistics Tables, you will notice that bullet path data are listed for most rifle bullets at zero ranges of 100, 200, 300, and 400 yards, and at 25, 50, 75, and 100 yards for handgun bullets. In the Silhouette Ballistics section the zero ranges for both rifle and handgun bullets coincide with the ranges to the targets.
Clearly, it is not possible to present bullet trajectory data in the Ballistics Tables for all possible choices of zero range. Furthermore, the bullet path depends not only on zero range, but on the height of the gun sights above the bore centerline, and thus on whether the shooter uses iron sights or telescope sights ( Figure 5.4-1 illustrates these concepts. The “bullet path height” is the bullet’s distance above or below the shooter’s line of sight , rather than the level line between muzzle and target.) For the Ballistics Tables in this Manual, Sierra’s choices for sight height and zero range are reasonable for a large number of shooters, but it is clear that many shooters need trajectory data for a different set of these conditions.
Shooters generally have one or more of three typical questions. The first usually goes something like the following example:
“I load the .30 caliber 165 grain spitzer boat tail bullet at 2700 fps muzzle velocity in my .30-06, and my sight height is 1.8 inches. I want to set my zero range at 250 yards, but I need to use a 100 yard target to do it. How high should I set my rifle to shoot at 100 yards to be zeroed in at 250 yards?”
The second question the is:
“After I zero in at 250 yards, where does my rifle shoot at other ranges?”
The third question occurs less frequently, but it is still important. In this example it would be:
“My rifle is now zeroed in at 100 yards. How many ‘clicks’ do I need on my telescope sight to change the zero to 250 yards?”
The answers to these questions require some numerical calculations. Fortunately, there are simple equations for both the bullet patch height and the sight change to adjust the zero range, and these equations work for both rifles and handguns. To use these equations, you must know bullet drop as a function of range for your cartridge and load, and drop is almost always listed in ballistics tables. You must also know your sight height, and you can measure this easily. If you use a telescope sight, it is the distance between the centerline of the bore and the centerline of the telescope (to within a very small error). If you use iron sights, it is exactly the distance between the bore centerline and the top of the front sight.
The equation for bullet path height at any range from the muzzle is the following:
where y b (R) is bullet path height in inches at the range R (positive values for bullet above line of sight, and negative values for bullet below line of sight); R is the range from the muzzle at which y bis to be calculated, measured in yards; y(R) is the bullet drop (always negative) at the range R , measured in inches, and is taken from the drop column in the ballistics table; h s is the sight height above the bore centerline in inches (always positive); R z is the zero range in yards; y z is the drop in inches at the zero range R z (always negative), and it also is taken from the drop figures in the ballistics tables.
Figure 5.4-1 will help define the meanings of the terms used above.
This formula is good for ranges greater than R z as well as for ranges less than the zero range. The accuracy of the formula is very good for practical shooting purposes. The calculated bullet path height for level fire will be accurate to a small fraction of an inch for pistol bullets to ranges of 400 or 500 yards, and for rifle bullets to ranges exceeding 1000 yards.
The calculations for sight adjustments to change the zero range make use of the angle A shown inFigure 5.4-1 . This is the angle between the bore centerline and the shooter’s line of sight. If the gun is zeroed in at some initial zero range R z1 , the first value of this angle is given by:
where the variables h s , y z1 , and R z1 have the same meanings and physical units as in the definitions above, and the second subscript (1) just reminds us to use values corresponding to the first zero range. The numerical factor (95.493) causes the angle value to be expressed in minutes of angle (MOA).
Now, if the zero range is to be changed to a new value R z2 , the new angle, A 2 will be given by:
Then, the sight adjustment needed to change the zero range is the difference between these angles.
When you calculate this difference, you get both the number of minutes of angle and the direction of the change that you must use. If the difference is positive, you must move the bullet upward on the target, and if the difference is negative, you must move the bullet downward on the target. Telescope sights are usually set so one “click” is 1/4 MOA, that is, it moves the bullet 1/4 inch at 100 yards. With iron sights it might not be as simple, since the MOA change caused by one “click” depends on the distance between front and rear sights on the gun.
The calculations described by these equations can be done by hand, and they are especially easy if you have an electronic calculator. It is necessary to manipulate negative numbers in the calculations, which might be a little unfamiliar to you if you’re not a whiz at math. The following example calculations will show how the method works and how the negative and positive numbers combine. If you follow the method in the example, you can perform these calculations for your own pet loads even if you aren’t a whiz at math!
Cartridge: .30-06 with Sierra .308 diameter 165 grain spitzer boat tail bullet
Muzzle velocity: 2700 fps
Zero range: 250 yards (change from 100 yards)
Sight height: 1.80 inches (telescope sight, high mounts)
Calculate: (1) Bullet path height at 100 yards to use for zeroing the rifle in at 250 yards.
(2) Bullet path height at 200, 300, 400, and 500 yards.
(3) Sight change required to move the zero range from 100 to 250 yards.
Ballistics Tables for reference: Sierra Rifle Reloading Manual
To start these calculations, we first note that R z = 250 yards, and h s = 180 inches, but when we look in the Ballistics Tables to find y z , we find that no value is given for 250 yards. This is not an unusual situation, and that was one of the reasons for choosing this example. There are many times when we need numbers that do not appear directly in the tables, and we have to determine them from numbers that do appear. In our case here, the Ballistics Table for the .30 caliber 165 grain SBT lists drop at 100 yard intervals. The easiest way to determine the drop at 250 yards is to use a piece of rectangular graph paper, plot the values of drop at 0, 100, 200, 300, and 400 yards, and then pass a smooth curve through all these points. If you do this carefully, you can read the drop at 250 yards from this curve to within an accuracy of about 1/10 inch, which is sufficient for most purposes. In this case the drop at 250 yards is about 16.6l inches, so y z = -16.6 inches.
(1) At R = 100 yards, y (100 = -2.37 inches from the Ballistics Table, and the bullet path height is calculated from the first formula above as follows:
= 3.19 inches
If you set your sights to shoot 3.2 inches high at 100 yards, your rifle will be zeroed in at 250 yards.
(2) The calculations proceed just as above, except that values for the desired ranges are used. Note that the values of Rz , y z , and h s do not change, which shortens some of the numerical calculations.
At R = 200 yards, y (200) = -10.25 inches, and the bullet path equation gives:
At R = 300 yards, y (300) = -24.56 inches, and
= (-26.36) + 22.08
= -4.28 inches (minus sign denotes bullet below line of sight)
At R = 400 yards, y (400) = -46.37 inches, and :
= (48.17) + 29.44
= -18.72 inches (below line of sight)
At R = 500 yards, y (500) = -77.03 inches, and:
= (-78.83) + 36.8
= -42.03 inches(below line of sight)
(3) To calculate the sight adjustment to be used to change the zero range, we must first calculate the two angles A1 and A2 . In the first case,
R z1 = 100 yards
y z1 = -2.37 inches
h s = 1.80 inches
= 3.98 MOA
For the second angle,
R z2 = 250 yards
y z2 = -16.6 inches
h s = 1.80 inches
= 7.03 MOA
The sight adjustment, then is the difference between these angles:
Sight Change = 7.03 – 3.98
= 3.05 MOA
If your telescope sight is graduated in 1/4 MOA clicks, you need to move the setting 12 clicks in the direction which raises the bullet impact point on the 100 yard target.