Adjusting for Distance & Conditions¶
Operation of Sights¶
Whether sights are telescopic, iron, red dot, LASER or work by some other method, their purpose is the same: to be lined up so that there is a straight line (which I will call the line of sight) from the gun to the desired point on the target, which is called the point of aim. (The point of aim should be chosen so that it is easy to identify: for circular targets whose black circle extends to the 7 ring and which is too far away for the bull marking to be seen, you may want the point of aim to be the bottom of the black or you may stick a white patch on the bull.)
If your shoots form a group 2” beneath the bull, you could make them hit the bull by choosing a point of aim that is 2” higher (possibly by putting an aiming mark there on the target). If you want to move the group while using the same point of aim, you can adjust the sights. It is important to remember that the sight adjustments alter the angle between the line of sight and the gun barrel, and that this angle may be expressed as a gradient. There are knobs to alter the vertical and horizontal angles and a common instruction on them is 1 click = 1/4” at 100 yards. You need to turn the knob in the direction that raises the barrel, but the number of clicks you should pass depends on how far the target is away. A gradient that goes up 1/4” in 100 yards will go up 1/2” at 200 yards, 1” at 400 yards, and so on.
Suppose the target is at 300 yards;
for 1 click the gradient would go up by 3/4”;
to achieve a rise of 2”, you need 2/(3/4) = 8/3 = 2 2/3 clicks, so use the nearest whole number: 3 clicks;
check by working out the rise for 3 clicks at 300 yards: 3 clicks at 100 yards is a 3/4” rise, so 3 clicks at 300 yards is a rise of 3x3/4” = 9/4” = 2 1/4”, which is about right.
On some sights, the instruction talks about angle rather than a gradient; the angle is usually expressed in minutes of arc, which may be abbreviated to MOA. For practical purposes 1MOA = 1” at 100 yards. If 1 click = 1/5 MOA, then 1 click = 1/5” at 100 yards. The above calculation for these sights goes similarly.
Suppose the target is at 300 yards;
for 1 click the gradient would go up by 3/5”;
to achieve a rise of 2”, you need 2/(3/5) = 10/3 = 3 1/3 clicks, so use the nearest whole number: 3 clicks;
check by working out the rise for 3 clicks at 300 yards: 3 clicks at 100 yards is a 3/5” rise, so 3 clicks at 300 yards is a rise of 3x3/5” = 9/5” = 1 4/5”, which is about right.
Note that the same sort of calculation has to be done for the horizontal gradient to account for sideways error, so initial adjustment for the movement due to wind (calculated in Wind below) needs to take target distance into account in the same way.
Using Published Charts for Initial Sight Adjustment after changing Target Distance or Ammunition¶
Most ammunition manufacturers publish tables showing the amount their ammunition drops after different distances. You can use these tables to make an initial change of sights to account for the difference in drop between that for the distance your rifle is already sighted-in at and the new distance. The change in angle (in MOA) corresponding to the drop is sometimes given.
Similar tables for the effect of the wind are also given. Changes from when the sights were last set should be dealt with similarly.
Using Experience: Adjusting for Small Changes¶
The calculation of the flight of a bullet from accurate physical theories is too complicated for anyone to do. In practice measurement of actual flights are taken and less accurate theories (that permit simpler calculations) can be used to estimate what happens for flights under conditions close to those used when the measurements were taken. Some of these calculations are simple enough to be done on the range and others are useful as a rough check to ensure that you have not used an incorrect method of calculation which might introduce a large error.
Forces¶
There are 3 forces on a flying bullet:
gravity pushing the bullet down;
wind pushing the bullet left or right; and
air resistance, slowing the bullet down by pushing against its direction of travel.
The effect of a force is to change the speed of the bullet in the direction of the force and the amount of the change depends on how long the force acts for; 2x the time means 2x the change of speed, 3x the time means 3x the change of speed and so on.
Wind¶
A bullet starts with a speed of 0 from left to right and its time of flight depends on the distance to the target: 2x the distance means roughly 2x the time, and so on. If you increase the distance by 2x, then the force of the wind pushes the bullet for 2x the time, increasing the final speed and the average speed by 2x. But the bullet is moving sideways for 2x the time at 2x the average speed, so actually moves sideways by 4x the distance (as 2x2=4). If you increase the distance by 3x, the sideways movement increases 9x (as 3x3=9), and so on. (However, if you increase the distance by 2x, the sight adjustment (which is angular) only increases by 2x; if you increase the distance by 3x, the sight adjustment only increases by 3x, and so on; see Operation of Sights)
But the force of the wind can vary with its speed. The force of the wind is produced by the molecules of air hitting the side of the bullet. If the air molecules move 2x as fast, they hit with 2x more force, but if the air is moving 2x as fast, then 2x the number of molecules will hit the bullet. So 2x the wind speed means moving sideways by 4x the distance (as 2x2=4).
These change factors combine by being multiplied together: if the wind in the afternoon is 2x what it was in the morning and the distance is 3x what it was in the morning, then the sideways movement is 36x what it was in the morning (as 4x9=36).
If the change is different over different parts of the range, the calculation can still be done, but the simplification of starting from 0 speed does not apply to parts other than the beginning. For example, if the wind is the same over the 1st 1/2 of the range and 2x greater over the 2nd 1/2, the calculation is as follows:
For the 1st 1/2:
the speed increase is 1/2 that of the original (whole) shoot, as the time of application of the force is 1/2 that of the original (whole) shoot, and the force is the same;
as the speed started at 0:
the final speed for the 1st 1/2 is then 1/2x that of the original (whole) shoot;
the average speed for the 1st 1/2 is then 1/2x that of the original (whole) shoot; and
the sideways movement for the 1st 1/2 is then 1/4x that of the original (whole) shoot (movement at 1/2x the average speed for 1/2x the time).
For the 2nd 1/2:
the speed increase is 2x that of the original (whole) shoot, as the time of application of the force is 1/2 that of the original (whole) shoot, and the force 4x that of the original (whole) shoot (and 1/2x4=2);
as the speed started at 1/2 the final speed of the original (whole) shoot:
the final speed for the 2nd 1/2 is 2 1/2x that of the original (whole) shoot (as 1/2+2=2 1/2);
the average speed for the 2nd 1/2 is then:
1 1/2x the final speed of the original (whole) shoot (as 1/2+2 1/2=3 and 3/2=1 1/2);
3x the average speed for the original (whole) shoot (as the average speed for the original (whole) shoot is 1/2x the final speed of the original (whole) shoot);
the sideways movement for the 2nd 1/2 is then 1 1/2x that of the original (whole) shoot (movement at 3x the average speed for 1/2x the time).
The total sideways movement is 1 3/4x that of the original (whole) shoot (as 1/4+1 1/2=1 3/4).
Wind speed usually has to be judged by the behaviour of flags or similar things. When not shooting, the wind speed corresponding to a picture of a flag can be reasonably accurately measured by flying a drone to the point on the bullet trajectory near the flag and measuring how fast it is blown when on hover, either using on-drone GPS or timing its crossing of an angle by observing it through a scope (see Operation of Sights).
If ranges do not co-operate with the above method, set up a flag somewhere else.
Gravity¶
A bullet starts with a speed of roughly 0 downwards, so paragraph 1 of Wind applies in that direction, but only roughly. The slowing of the bullet and its not quite horizontal initial trajectory are known more accurately than the wind speed, so the calculation under Wind is not used, other than as a rough check.
(Note that the barrel is beneath the line of sight, so it has to be pointed slightly upwards to compensate for this. At very short ranges, the bullet may still be rising when it hits the target; in this case, a small increase in range might mean that the barrel has to be lowered. Most fullbore non-gallery rifles use about the same sight setting for 25m and 100m: at the 25m target the bullet is still going up, and it had come back down to the same level by the time it reaches the 100m target.)
Air Resistance¶
The force on the bullet due to air resistance is the same as a stationary bullet would experience if it had a head wind of the same speed as the moving bullet. So paragraph 2 of Wind applies, but only roughly because of complications that can be ignored at speeds much lower than those of the air molecules (and the speed of sound, as sound is transmitted by the movement of the molecules). Notice that the difference in speed of a bullet over the 1st 100m is more than that over later gaps of 100m, as the speed is less and the force of air resistance is therefore reduced.
Effect of Mass (or Weight) of Bullet¶
The amount of force required to produce a given change in speed in a given time depends on the mass (or weight) of the object being pushed: 2x the force required for 2x the mass, and so on.
Effect on the Wind Calculation¶
No effect due to the following. Heavier bullets of the same calibre are slightly longer, but roughly the same shape. They are made of roughly the same material so any increase in mass is produced by the same amount of increase in length, volume and side-on area, but no increase in head-on area.
Roughly speaking a percentage increase in weight results in the same percentage decrease in the speed changing effect of the force, but also the same percentage increase in side-on area to be hit by the air molecules, which results in the same percentage increase in the force itself. The net change in the speed changing effect is therefore 0.
Effect on the Gravity Calculation¶
None as any percentage increase in mass produces the same increase in gravitational force.
Effect on the Air Resistance Calculation¶
Roughly speaking a percentage increase in weight results in the same percentage decrease in the speed changing effect of the force. The compensation for this in the calculation in Effect on the Wind Calculation is absent as the head-on area is unaltered (see Effect on the Wind Calculation).