practical ballistics for hunters and shooters

Practical Ballistics for Hunters and Shooters

First, if you REALLY want to understand external ballistics (or just want to know what that means)

To accurately predict a bullet's trajectory, you only need to know two things:(1) its initial velocity, and

The rate at which air drag slows down—or retards—a bullet is proportional to a Retard Coefficient: F.

Bullet muzzle velocities are usually measured in feet per second, and the unit for F is feet.

You can easily calculate F from the data found on your box of ammo or on the manufacturer's website.For example, looking at a box of Federal Fusion .30-'06 Springfield (product #: F3006FS3) ammuntion I found the following information:

Muzzle 100 yds 200 yds 300 yds 400 ydsVelocity (fps) 2700 2520 2350 2190 2030Energy (ft-lbs) 2915 2540 2205 1905 1640Height of Bullet Trajectory in inches above or below line of sigtht

Note: The sight height is listedAverage Range U -4.0 -14.3 as 1.5 inches above the bore.

F Pejsa References: (pp.16, 63, 67-68, and 124.)Given:

2700 2520 4350 R ≡ Range (yards)

To convert the manufacturer's range R to feet we multiply by 3. Average velocity (Va) is (2700+2520)÷2.Change in Velocity (dV) is 2700-2520. Therefore, (300 x 2610)÷180 = 4350 ft. (See above.)

With this information, we can predict a bullet's trajectory. [Go one to Page 2.]

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you should buy a copy of New Exact Small Arms Ballistics by Arthur J. Pejsa.

(2) the rate at which air drag slows it down.1

One percent (1%) of F is the distance in which a bullet loses 1% of it's speed to drag.2

if zeroed at U yards. Sights 1.5 inches above bore line.

Estimate F0 From Manufacturer's Velocity Data

MuzzleVelocity

(fps)

100 YardVelocity

(fps)

100yard

V0 ≡ Muzzle Velocity (fps)

V100 ≡ Velocity @ 100 yds (fps)

F ≡ Retard Coefficient (feet)

We calculate F from a bullet's change in velocity (dV) at an average velocity (Va) over a given range (in feet).3

1As paraphrased from Arthur J. Pejsa, New Exact Small Arms Ballistics (Saint Paul: Catalyst Graphics, 2008), p.65. 2As paraphrased from Pejsa, p.65. 3Pejsa, pp.16, 67-68.

Pejsa References: (pp.16, 63, 67-68, and 124.)Then:

100

2700 Manufacturer's chronograph data

2520 Manufacturer's chronograph data

4350 F=3*R*Va/dV (See footnote3)

(Saint Paul: Catalyst Graphics, 2008), p.65.


So, now we can start to calculate a bullet's trajectory using the formulas

On page one, we determined the following:

Research indicated that the value of F for different types of bullets had different loss rates.

Range R is in yards and N=0.5 for Spitzers.

We now have all the information we need to start using the Pejsa drop formula (in its simplest form).

Again, if you REALLY want to understand the drop formula and its derivation

derives time of flight (t) as a function of initial velocity and air drag. The resultant drop formula accurately predicts the combined effects of gravity, velocity, and air drag at a given range.

The next step in calculating a bullet's trajectory is to calculate bullet drop at various ranges.We'll do that on the next page.

derived in: New Exact Small Arms Ballistics by Arthur J. Pejsa.

(1) V0 the initial velocity which was 2700 fps

(2) F0 the initial rate at which air drag slows down the bullet which was 4350 ft.

F is the fractional loss in Velocity per foot of travel.4

Most hunting bullets—like the Fusion round on Page 1—are Spitzers.5

Spitzers have a loss rate of N ft per foot of travel that is N = 0.50.6

Pejsa calculated the average F by-bullet-type-by-range and called that value Fa.

Fa=F0-0.80*N*R

The Pejsa drop formula is: D = (41.68 / V0 / ( (1/R) - (1/Fa) ) )2

you should buy a copy of New Exact Small Arms Ballistics by Arthur J. Pejsa.

In the book, the author explains Newton's formula for distance fallen D = ½gt2, then he

4As paraphrased from Arthur J. Pejsa, New Exact Small Arms Ballistics (Saint Paul: Catalyst Graphics, 2008), p.16. 5A Spitzer bullet is an aerodynamic, spire point bullet.6Pejsa. p. 17.

(Saint Paul: Catalyst Graphics, 2008), p.16.

Using the information and calculations from Pages 1-2, we can calculate the bullet dropat 100 yard intervals as follows:

Rangefps fps yards

2,700 2,520 100 Fo = 4350

Vo Fo N2,700 4,350 0.50

Range Drop Fa N Drop*100 2.50 4310 0.50 2.50200 10.49 4270 0.50 10.50300 24.85 4230 0.50 24.86400 46.60 4190 0.50 46.64500 77.02 4150 0.50 77.10600 117.63 4110 0.50 117.79

700 170.32 4070 0.50 170.64800 237.42 4030 0.50 238.02900 321.84 3990 0.50 322.94

1000 427.25 3950 0.50 429.19

* Calculated using the Pejsa Ballistic program as described in

As you can see, even in its simplest form the Pejsa formula provides resultsthat match more complex ballistic programs out to at least 600 yards. This highligts the Relavtive Importance—or lack thereof—of "sources of error"such as Altitude, air Pressure, and ambient Temperature (at ranges below 600 yards).

But we haven't addressed a VERY IMPORTANT factor: Sight Height.As Shooters, we want to know where a bullet is going to strike in relation to where we are aiming. The is known as the Flight Path—or simply Path.

Let's look at the Federal Fusion data again. Muzzle 100 yds 200 ydsNote: The sight height is listed as 1.5 inches above the bore. Velocity (fps) 2700 2520 2350

Energy (ft-lbs) 2913 2540 2206Height of Bullet Trajectory in inches above or below line of sigtht

they listed the height of the bullet trajectory above or below the Line Of Sight. In other words, they gave us the Path of the bullet. Average Range U -4.0

The manufacturer's data listed to the right tells us: (1) The rifle was sighted in at 100 yards, and (2) The centerline of the scope was 1.5-inches above the bore.

Using this information, we can calculate the resulting flight path.

S DzSight Zero Drop at

Height Range Zeroinches yards inches

1.5 100 2.50 4310 0.50

Vo Fo N2,700 4,350 0.50

V0 V100

Arthur J. Pejsa, Modern Practical Ballistics (Minneapolis: Kenwood Publishing 1991).

Federal considered BOTH the sight height AND the bullet drop, and if zeroed at U yards. Sights 1.5 inches above bore line

Rz


For most Hunters and Shooters, we can calculate a Ballistic Table on this page.

First, Look-up the manufacturer's velocity data for your rounds. Example:

Muzzle 100 yds 200 yds 300 yds 400 ydsVelocity (fps) 2700 2520 2350 2190 2030Energy (ft-lbs) 2913 2540 2206 1907 1640Height of Bullet Trajectory in inches above or below line of sigtht

Average Range U -4.0 -14.3

2700 2520 4350



1.5 200 10.49 4270 0.50

Vo Fo N2,700 4,350 0.50

Here is the Ballistic Table for your rounds.Flight

Range Path Drop Fa N Path*100 2.0 2.50 4310 0.50 2.0200 0.0 10.49 4270 0.50 0.0300 -8.4 24.85 4230 0.50 -8.4400 -24.1 46.60 4190 0.50 -23.9500 -48.5 77.02 4150 0.50 600 -83.1 117.63 4110 0.50

But, are you ready to zero your rifle?Let's look at the effects of sight height and zero range on the next page.


Next, Estimate F0 From Manufacturer's Velocity Data

MuzzleVelocity

(fps)

100 YardVelocity

(fps) F0

100yard F=3*R*Va/dV

Enter your Sight Height (S), desired Zero Range (Rz), Muzzle Velocity (V0), and Calculated F0 below.

Rz

Note: I found this data panel on another box of Fusion F3006FS3 ammunition.Muzzle 100 yds 200 yds 300 yds 400 yds

Velocity (fps) 2700 2521 2349 2185 2026Energy (ft-lbs) 2913 2540 2206 1907 1640Height of Bullet Trajectory in inches above or below line of sigtht

Average Range 2.0 U -8.4 -23.9

* Manufacturer's data.

As you can see, out to 400 yards, the basic Pejsa formula provides resultsthat closely match the manufacturer's data for a 200 yard zero.

The calculated data is never off more than 2/10ths of an inch!


), and Calculated F0 below.


The sight height and zero range greatly affect bullet flight path. (How "flat" your trajectory will be.)

Let's use the previous example and compare a 200 yard zero to a 25 yard zero.

S Dz SSight Zero Drop at Sight

Height Range Zero Heightinches yards inches inches

1.5 200 10.49 4270 1.5

Vo Fo N Vo2,700 4,350 0.50 2,700

Ballistic Table Ballistic Table Flight

Range Path Drop Fa N Range25 -0.2 0.15 4340 0.50 25

100 2.0 2.50 4310 0.50 100200 0.0 10.49 4270 0.50 200224 -1.4 13.26 4261 0.50 224275 -5.6 20.61 4240 0.50 275300 -8.4 24.85 4230 0.50 300

A 25 yard zero with this round AND THIS SIGHT HEIGHT will be the same as a 224 yard zero. This is very convenient, because many 25-yard ranges are available. Plus, a deer hunter could hold "on the fur" out to 275 yards.

Don't believe me? Let's change the sight height to 2.6 inches and re-zero at 25-yards and 200-yards.



2.6 25 0.15 4340 2.6

Vo Fo N Vo2,700 4,350 0.50 2,700


Range Path Drop Fa N Range25 0.0 0.15 4340 0.50 25

100 5.9 2.50 4310 0.50 100200 8.9 10.49 4270 0.50 200224 8.7 13.26 4261 0.50 224250 8.1 16.81 4250 0.50 250300 5.6 24.85 4230 0.50 300

Now we get very different results by zeroing at 25-yards versus 200-yards.

With a sight height of 2.6-inches (e.g., an M16A2 rifle with carrying handle) a 25-yard zero

Rz

But, the Sight Height plays a bigger role in the Flight Path than the type of round.

Rz

would produce a very exaggerated Flight Path A 200-yard zero would result in a very flat Flight Path out to 250 yards. So, you could walk out 200 yards, set up a target and zero atthat range, or you could use the information we calculated here, set up a target at 25 yards, and intentionally sight in your rifle to shoot 1.1-inches low at that range.

Note: In the 1970's, we zeroed our M16A1's at 25-meters. The point of aim was the base of the bull, but the desired point of impact was 24mm (0.94inch*) below the point of aim. That was

25 meters = 27.34 yards and 250meters = 273.4 yards



2.6 273.40 15.22 2699

Vo Fo N3,250 2,808 0.50

Ballistic Table Flight

Range Path Drop Fa N27.34 -0.94 0.13 2797 0.50

100.00 2.15 1.77 2768 0.50 200.00 2.78 7.66 2728 0.50 223.50 2.22 9.75 2719 0.50 250.00 1.22 12.48 2708 0.50 273.40 0.00 15.22 2699 0.50 328.08 -4.21 22.99 2677 0.50

As you can see, the Field Manual was correct. So, for a given sight height, how do we choose a zero range that gives us a flat Flight Path?We'll explore our options on Page 6.

Government Printing Office, 1974), pp. 83-86.

supposed to be equivalent to a 250m (273.4 yards) battlesight zero.7 Let's check it out.

Rz

7U.S. Department of the Army, M16A1 Rifle And Rifle Marksmanship, Field Manual 23-9 (Washington, D.C.:U.S. Government

The sight height and zero range greatly affect bullet flight path. (How "flat" your trajectory will be.)

DzZero Drop at

Range Zeroyards inches25 0.15 4340

Fo N4,350 0.50

Ballistic Table FlightPath Drop Fa N

0.0 0.15 4340 0.50 2.6 2.50 4310 0.50 1.2 10.49 4270 0.50 0.0 13.26 4261 0.50

-4.0 20.61 4240 0.50 -6.5 24.85 4230 0.50

This is very convenient, because many 25-yard ranges are available. Plus, a deer hunter could hold "on the fur" out to 275 yards.

Don't believe me? Let's change the sight height to 2.6 inches and re-zero at 25-yards and 200-yards.

DzZero Drop at

Range Zeroyards inches200 10.49 4270

Fo N4,350 0.50


-1.1 0.15 4340 0.50 1.4 2.50 4310 0.50 0.0 10.49 4270 0.50

-1.2 13.26 4261 0.50 -3.0 16.81 4250 0.50 -7.8 24.85 4230 0.50

Rz

Rz

*That's 3.4 MOA in case you're curious. Each 1.4cm square on the DA Form 3016 target represented 2MOA.

, Field Manual 23-9 (Washington, D.C.:U.S. Government

*That's 3.4 MOA in case you're curious. Each 1.4cm square on the DA Form 3016 target represented 2MOA.


Point Blank Range—A Practical Way to Choose a Zero Range

First, let's define what we mean by Point Blank Range.

Many Infantrymen, and other Riflemen, describe point-blank range as the distance where a bullet's trajectory intersects the line of sight. In other words, the bullet hits the point of aim.

Most Artillerymen/Red Legs/Gun Bunnies describe point-blank range as Close Enough so that missing the target is unlikely.

Dr. Pejsa describes Point Blank Range (PBR) as the maximum distance at which the path of the bullet remains within an acceptable error; for example, no more than 2-inches above or below the point of aim. Restated, point-blank range is the distance between a firearm and a target of a given size such that the bullet in flight is expected to strike the target without adjusting the elevation of the firearm.

For most deer hunters, 2-inches is an acceptable error, for the hunter would not have to hold over or under his target anywhere within the PBR. Using the Pejsa formulas, we can find a zero rangethat will match our desired PBR.

Given: Then:

2700

4350S ≡ Sight Height Above Bore (inches) 1.50

2.00M ≡ Midrange (yards) 114.8G ≡ Gravitational Constant (Pejsa) 41.68

1.32SQ ≡ SH/√(2) 0.94

28.6Z ≡ Zero Range (Far) (yards) 197.6PBR ≡ Point-Blank Range* (yards) 231.0

The Federal Fusion .30-'06 round used in the example should be capable of shooting within 2-inches above or below the point of aim out to a range of over 230 yards.

Let's use the information above to check the flight path.



1.5 28.6 0.20 4339 1.5

Vo Fo N Vo2,700 4,350 0.50 2,700

Enter your Sight Height (S), desired max Height above/below line-of-sight (Hm), Muzzle Velocity (V


F0 ≡ Initial Retard Coefficient (feet)

Hm ≡ Height Maximum (inches)

SH ≡ √(1+S/Hm)

Zn ≡ Near Zero Range (yards)

* Pejsa point-blank range8

Rz


Range Path Drop Fa N Range25 -0.2 0.15 4340 0.50 2529 0.0 0.20 4339 0.50 29

100 1.9 2.50 4310 0.50 100200 -0.1 10.49 4270 0.50 200230 -1.9 14.09 4258 0.50 230250 -3.5 16.81 4250 0.50 250

Now you can choose a zero range that works best for you.

Remember that Sight Height plays a bigger role in the Flight Path than the type of round, solet's look at the M193 round in an M16A1 rifle again using the PBR equations.

Given: Then:

3250

2368S ≡ Sight Height Above Bore (inches) 2.60

2.00M ≡ Midrange (yards) 146.5G ≡ Gravitational Constant (Pejsa) 41.68

1.52SQ ≡ SH/√(2) 1.07

52.0Z ≡ Zero Range (Far) (yards) 233.6PBR ≡ Point-Blank Range* (yards) 267.7

25 meters = 27.34 yards and 250meters = 273.4 yards



2.6 50.00 0.43 2348 ← We'll choose a 50-yard zero for convenience (vs. 52-yds listed above).

Vo Fo N3,250 2,368 0.50

Ballistic Table Flight

Range Path Drop Fa N27.34 -1.07 0.13 2357 0.50

100.00 1.66 1.80 2328 0.50 200.00 1.62 7.90 2288 0.50 223.50 0.84 10.10 2279 0.50 250.00 -0.44 12.98 2268 0.50 273.40 -1.95 15.91 2259 0.50 ← The bullet remains within ± 2-inches out to 250m.




SH ≡ √(1+S/Hm)


* Pejsa point-blank range8

Rz

328.08 -7.04 24.31 2237 0.50

So, for a given sight height, you can use the PBR calculations to choose a zero range that gives you a "flat" trajectory.

8Pejsa. p. 20-21

The Federal Fusion .30-'06 round used in the example should be capable of shooting within 2-inches

DzZero Drop at

Range Zeroyards inches197.6 10.23 4271

Fo N4,350 0.50

), Muzzle Velocity (V0), and Calculated F0 below.

Rz


-0.2 0.15 4340 0.50 0.0 0.20 4339 0.50 1.9 2.50 4310 0.50

-0.1 10.49 4270 0.50 -1.9 14.09 4258 0.50 -3.5 16.81 4250 0.50

← We'll choose a 50-yard zero for convenience (vs. 52-yds listed above).

← The bullet remains within ± 2-inches out to 250m.

So, for a given sight height, you can use the PBR calculations to choose a zero range that gives you a "flat" trajectory.


The Ballistics Tables

OK, you're ready to start using the Ballistics Tables on the next page.

The tables on the next page are based on the work of Arthur J. Pejsa. Bibliography:

As disclosed on Page 1, to accurately predict a bullet's trajectory, you only need to know two things:(1) its initial velocity, and(2) the rate at which air drag slows it down.

Naturally, you will have to enter that data. You will enter the following:

Initial Input Data Fusion 30-06 Spring. Bullet Name ← Enter a description you like.

BTSP, Skived Tip Bullet Type ← The bullet type listed here describes a Boat Tail Spitzer with a skived tip.180 Bullet Weight (grains) ← From the Manufacturer's data.

2700 Muzzle Velocity V0 (fps) ← From the Manufacturer's data. 2520 Velocity @ 100 yards ← From the Manufacturer's data. 2350 Velocity @ 200 yards ← From the Manufacturer's data. 1.50 Sight Height (inches) ← Height of sight or scope above the center-line of the bore. 0.25 Adjustment per Click (MOA) ← Elevation and Windage knobs on scopes are typically ¼-½ Minute Of Angle (MOA). 2.00 HM ≡ Acceptable Error ← Bullet impact point above or below the point of aim out to Point Blank Range desribed on Page 6.509 Altitude (feet) ← Your local elevation.

29.94 Pressure (in.Hg) ← From your local weather.62 Temperature (°F) ← From your local weather.

5 Wind speed (mph) ← From your local weather.3.0 Wind dir. (o'clock) ← From your local weather.

8 Incline/Decline Angle (°) ← Enter the Incline/Decline angle from the shooter to the target.* *Whether you are shooting uphill or downhill, you will shoot high by the amount calculated. If you want to know why, read one of the books. For example, if you zero my .30-06 at 300yards, and you shoot up a 60° incline, your round at 300 yards will hit 12.37" high.

Based on what you entered, the spreadsheet will calculate the following information.

Calculated Data 4350 F0 ≡ Retardation coeff.4459 F0a ≡ Adj. Retard. Coeff. ← Adjusted for Altitude, Temperature, and Air Pressure. 0.51 BC ≡ Ballistic Coeff.0.53 BCa ≡ Adj. Ballistic Coeff. ← Adjusted for Altitude, Temperature, and Air Pressure. 198 Z ≡ Zero Range (Far) (yards) ← Zero range in yards for calculated PBR. 231 PBR ≡ Point-Blank Range* (yards)

Please note that the spreadsheet will calculate a BC for your round. That information is only for you; that BC isn't used in any calculations. As described on Page 1 and above, all of the data is calculated from the velocity and environmental data you enter.

The spreadsheet will calculate a Point Blank Range for you, and it will display the Far Zero Range that achieves that PBR. You do not have to use that zero range, you can enter any desired zero range in the "Input Shooting Data" section that follows.

Next, you'll see the standards and constants used by the program.

Constants / Standards29.92 Standard Pressure (in.Hg)

Arthur J. Pejsa, New Exact Small Arms Ballistics (Stevens Point: Kenwood Publishing 2008). Arthur J. Pejsa, Modern Practical Ballistics (Minneapolis: Kenwood Publishing 1991).

1.047 Inches per MOA @ 100 yards

Finally, you can enter your desired zero range, a special range, the starting range for the table (0 is the default), and the increment.

Input Shooting Data 200 Zero Range (yards) ← Enter your desired zero range. I chose one close to the PBR far zero range shown above because I like a fairly flat trajectory.

25 Special Range (yards) ← This is a useful feature. Say you want a 200 yard zero but your local range is only 25 yards. You can use the Path data to sight in at 25 yards as described on Page 5.

0 Starting Range (yards)25 Increment (yards) ← Take a look at the Table. You'll see.

The first table displayed shows your Zero Range data and your Special Range data.

Range Path Elevation(yd) (inch) (MOA)

Zero Range Data: 200 0.00 0.00

Special Range Data: 25 -0.15 0.59

The spreadsheet will print your descriptive data next to the calculated ballistics table as shown below: Note how the range starts at 0, the Starting Range, and it increases in increments of 25 yards.

Fusion 30-06 Spring. Range Path Elevation ElevationBTSP, Skived Tip (yd) (inch) (MOA) (clicks)

180 grain Bullet 0 0.00 0.0 0.02700 fps Muzzle Velocity 25 -0.15 0.6 2.41.50 inch Sight Height 50 0.88 -1.7 -6.7200 yard Zero 75 1.60 -2.0 -8.2

100 1.99 -1.9 -7.6125 2.03 -1.6 -6.2150 1.73 -1.1 -4.4175 1.05 -0.6 -2.3200 0.00 0.0 0.0225 -1.45 0.6 2.5250 -3.30 1.3 5.0275 -5.58 1.9 7.8300 -8.30 2.6 10.6325 -11.47 3.4 13.5350 -15.12 4.1 16.5375 -19.26 4.9 19.6400 -23.91 5.7 22.8425 -29.09 6.5 26.2450 -34.83 7.4 29.6475 -41.14 8.3 33.1500 -48.04 9.2 36.7525 -55.57 10.1 40.4550 -63.75 11.1 44.3575 -72.60 12.1 48.2600 -82.16 13.1 52.3625 -92.45 14.1 56.5650 -103.51 15.2 60.8675 -115.37 16.3 65.3700 -128.06 17.5 69.9

You're ready. Move on to the next page.

As disclosed on Page 1, to accurately predict a bullet's trajectory, you only need to know two things:

← The bullet type listed here describes a Boat Tail Spitzer with a skived tip.

← Height of sight or scope above the center-line of the bore. ← Elevation and Windage knobs on scopes are typically ¼-½ Minute Of Angle (MOA). ← Bullet impact point above or below the point of aim out to Point Blank Range desribed on Page 6.

← Enter the Incline/Decline angle from the shooter to the target.* *Whether you are shooting uphill or downhill, you will shoot high by the amount calculated. If you want to know why, read one of the books. For example, if you zero my .30-06 at 300yards, and you shoot up a 60° incline, your round at 300 yards will hit 12.37" high.

← Adjusted for Altitude, Temperature, and Air Pressure.

← Adjusted for Altitude, Temperature, and Air Pressure. ← Zero range in yards for calculated PBR.

Please note that the spreadsheet will calculate a BC for your round. That information is only for you; that BC isn't used in any calculations. As described on Page 1 and above, all of the data is calculated from the velocity and environmental data you enter.

The spreadsheet will calculate a Point Blank Range for you, and it will display the Far Zero Range that achieves that PBR. You do not have to use that zero range, you can enter any desired zero range in the "Input Shooting Data" section that follows.

Finally, you can enter your desired zero range, a special range, the starting range for the table (0 is the default), and the increment.

← Enter your desired zero range. I chose one close to the PBR far zero range shown above because I like a fairly flat trajectory.← This is a useful feature. Say you want a 200 yard zero but your local range is only 25 yards. You can use the Path data to sight in at 25 yards as described on Page 5.

Windage Drop Speed Energy Time Fa N(MOA) (inch) (fps) (ft-lb) (s) (ft)0.67 10.47 2349 2205 0.24 4383 0.50

0.08 0.15 2655 2817 0.03 4449 0.50

The spreadsheet will print your descriptive data next to the calculated ballistics table as shown below:

Windage Windage Drop Speed Energy Time Fa N(MOA) (clicks) (inch) (fps) (ft-lb) (s) (ft)

0.0 0.0 0.00 2700 2914 0.00 4459 0.50 0.1 0.3 0.15 2655 2817 0.03 4449 0.50 0.2 0.6 0.61 2610 2722 0.06 4440 0.50 0.2 1.0 1.39 2565 2630 0.09 4430 0.50 0.3 1.3 2.49 2521 2541 0.11 4421 0.50 0.4 1.6 3.94 2478 2454 0.14 4411 0.50 0.5 2.0 5.75 2434 2368 0.18 4402 0.50 0.6 2.3 7.92 2391 2286 0.21 4392 0.50 0.7 2.7 10.47 2349 2205 0.24 4383 0.50 0.8 3.1 13.41 2307 2127 0.27 4373 0.50 0.9 3.4 16.76 2265 2050 0.30 4363 0.50 1.0 3.8 20.53 2224 1976 0.34 4353 0.50 1.0 4.2 24.75 2183 1904 0.37 4344 0.50 1.1 4.6 29.42 2142 1833 0.41 4334 0.50 1.2 5.0 34.56 2102 1765 0.44 4324 0.50 1.3 5.4 40.20 2062 1699 0.48 4314 0.50 1.5 5.8 46.34 2022 1634 0.51 4304 0.50 1.6 6.2 53.02 1983 1572 0.55 4294 0.50 1.7 6.7 60.25 1944 1511 0.59 4284 0.50 1.8 7.1 68.05 1906 1452 0.63 4274 0.50 1.9 7.6 76.46 1868 1395 0.67 4264 0.50 2.0 8.0 85.48 1830 1339 0.71 4254 0.50 2.1 8.5 95.15 1793 1285 0.75 4243 0.50 2.2 9.0 105.50 1756 1233 0.79 4233 0.50 2.4 9.4 116.56 1720 1182 0.84 4223 0.50 2.5 9.9 128.35 1684 1133 0.88 4213 0.50 2.6 10.5 140.90 1648 1086 0.92 4202 0.50 2.7 11.0 154.25 1613 1040 0.97 4192 0.50 2.9 11.5 168.44 1578 995 1.02 4182 0.50

InclineError Elevation(inch) (MOA)0.09 -0.04

0.01 -0.04

InclineError Elevation(inch) (MOA)

0.00 0.000.01 -0.040.02 -0.040.03 -0.040.05 -0.040.06 -0.040.07 -0.040.08 -0.040.09 -0.040.10 -0.040.12 -0.040.13 -0.040.14 -0.040.15 -0.040.16 -0.040.17 -0.040.19 -0.040.20 -0.040.21 -0.040.22 -0.040.23 -0.040.24 -0.040.26 -0.040.27 -0.040.28 -0.040.29 -0.040.30 -0.040.31 -0.040.33 -0.04

Initial Input Data Fusion 30-06 Spring. Bullet Name

BTSP, Skived Tip Bullet Type180 Bullet Weight (grains)

2700 2520 Velocity @ 100 yards2350 Velocity @ 200 yards1.50 Sight Height (inches) ← Height of sight or scope above the center-line of the bore. 0.25 Adjustment per Click (MOA) ← Elevation and Windage knobs on scopes are typically ¼-½ MOA.

2.00 ← Bullet impact point above or below the point of aim out to Point Blank Range. 509 Altitude (feet)

29.94 Pressure (in.Hg)62 Temperature (°F)

5 Wind speed (mph)3.0 Wind dir. (o'clock)

8 Incline/Decline Angle (°) ← Resulting Incline/Decline Error is exact for 60° and within 1% of Drop for angles from 1° to 64°.

Calculated Data

4350

4459 ← Adjusted for Altitude, Temperature, and Air Pressure. 0.51 BC ≡ Ballistic Coeff.

0.53 ← Adjusted for Altitude, Temperature, and Air Pressure. 198 Z ≡ Zero Range (Far) (yards) ← Zero range in yards for calculated PBR. 231 PBR ≡ Point-Blank Range* (yards)

Constants / Standards29.92 Standard Pressure (in.Hg) 1357 End Zone 1 Velocity (fps)

1.047 Inches per MOA @ 100 yards 1174 End Zone 2 Velocity (fps) 1017 End Zone 3 Velocity (fps)

0 Zone 4

Input Shooting Data 200 Zero Range (yards)

25 Special Range (yards)0 Starting Range (yards)

25 Increment (yards)

Range Path Elevation Windage(yd) (inch) (MOA) (MOA)

Zero Range Data: 200 0.00 0.00 0.67

Special Range Data: 25 -0.15 0.59 0.08

Fusion 30-06 Spring. Range Path Elevation ElevationBTSP, Skived Tip (yd) (inch) (MOA) (clicks)

180 grain Bullet 0 -1.50 0.0 02700 fps Muzzle Velocity 25 -0.15 0.6 21.50 inch Sight Height 50 0.88 -1.7 -7200 yard Zero 75 1.60 -2.0 -8

Muzzle Velocity V0 (fps)

HM ≡ Acceptable Error

F0 ≡ Retardation coeff.

F0a ≡ Adj. Retard. Coeff.

BCa ≡ Adj. Ballistic Coeff.

100 1.99 -1.9 -8125 2.03 -1.6 -6150 1.73 -1.1 -4175 1.05 -0.6 -2200 0.00 0.0 0225 -1.45 0.6 2250 -3.30 1.3 5275 -5.58 1.9 8300 -8.30 2.6 11325 -11.47 3.4 13350 -15.12 4.1 17375 -19.26 4.9 20400 -23.91 5.7 23425 -29.09 6.5 26450 -34.83 7.4 30475 -41.14 8.3 33500 -48.04 9.2 37525 -55.57 10.1 40550 -63.75 11.1 44575 -72.60 12.1 48600 -82.16 13.1 52625 -92.45 14.1 57650 -103.51 15.2 61675 -115.37 16.3 65700 -128.06 17.5 70

NOTE: The above-listed table is designed for bullet speeds above the minimum Zone 1 velocity listed above. To calculate bullet drop at extended ranges, go to the Long Range Table page and enter your maximum range.

Given: Then: M ≡ Midrange (yards) 114.8 115.0G ≡ Gravitational Constant (Pejsa) 41.68 41.68

2700 2700

4350 4459S ≡ Sight Height Above Bore (inches) 1.50 1.50

2.00 2.00

4378 4487

1.32 1.32SQ ≡ SH/√(2) 0.94 0.94

28.6 28.6Z ≡ Zero Range (Far) (yards) 197.7 198.1PBR ≡ Point-Blank Range* (yards) 231.1 231.5 * Gunnery point-blank range * * Formula from Arthur J. Pejsa, Modern Practical Ballistics (Minneapolis: Kenwood Publishing 1991), p.31.

Given: Then: c ≡ Mayevski Constant 246.0 246.0

-0.45 -0.45

2700 2700V ≡ Velocity (fps) 2520 2350




F ≡ F0+15*√(Hm+S)**

SH ≡ √(1+S/Hm)


NIngall's ≡ Drag Curve Coefficient


585 1159

0.224 0.460

-1.07 -2.12

0.0005 0.0010

0.513

0.523

angle Ø radians sinØ cosØ8.0000 0.13963 0.13917 0.99027

r ≡ (c/N)*(V-N-V0-N) (feet)

T ≡ (c/(1+N))*(V-1-N-V0-1-N) (sec)

(V-N-V0-N)

(V-1-N-V0-1-N)

BC100 ≡ G1

BC200 ≡ G1

← Height of sight or scope above the center-line of the bore. ← Elevation and Windage knobs on scopes are typically ¼-½ MOA.

← Bullet impact point above or below the point of aim out to Point Blank Range.

← Resulting Incline/Decline Error is exact for 60° and within 1% of Drop for angles from 1° to 64°.

← Adjusted for Altitude, Temperature, and Air Pressure.

← Adjusted for Altitude, Temperature, and Air Pressure. ← Zero range in yards for calculated PBR.

End Zone 1 Velocity (fps) 0.50 N ≡ Zone 1End Zone 2 Velocity (fps) 0.00 N ≡ Zone 2End Zone 3 Velocity (fps) -4.00 N ≡ Zone 3

0.00 N ≡ Zone 4

Drop Speed Energy Time Fa N(inch) (fps) (ft-lb) (s) (ft)10.47 2349 2205 0.24 4383 0.50

0.15 2655 2817 0.03 4449 0.50

Windage Windage Drop Speed Energy Time Fa N(MOA) (clicks) (inch) (fps) (ft-lb) (s) (ft)

0.0 0 0.00 2700 2914 0.00 4459 0.50 0.1 0 0.15 2655 2817 0.03 4449 0.50 0.2 1 0.61 2610 2722 0.06 4440 0.50 0.2 1 1.39 2565 2630 0.09 4430 0.50

0.3 1 2.49 2521 2541 0.11 4421 0.50 0.4 2 3.94 2478 2454 0.14 4411 0.50 0.5 2 5.75 2434 2368 0.18 4402 0.50 0.6 2 7.92 2391 2286 0.21 4392 0.50 0.7 3 10.47 2349 2205 0.24 4383 0.50 0.8 3 13.41 2307 2127 0.27 4373 0.50 0.9 3 16.76 2265 2050 0.30 4363 0.50 1.0 4 20.53 2224 1976 0.34 4353 0.50 1.0 4 24.75 2183 1904 0.37 4344 0.50 1.1 5 29.42 2142 1833 0.41 4334 0.50 1.2 5 34.56 2102 1765 0.44 4324 0.50 1.3 5 40.20 2062 1699 0.48 4314 0.50 1.5 6 46.34 2022 1634 0.51 4304 0.50 1.6 6 53.02 1983 1572 0.55 4294 0.50 1.7 7 60.25 1944 1511 0.59 4284 0.50 1.8 7 68.05 1906 1452 0.63 4274 0.50 1.9 8 76.46 1868 1395 0.67 4264 0.50 2.0 8 85.48 1830 1339 0.71 4254 0.50 2.1 8 95.15 1793 1285 0.75 4243 0.50 2.2 9 105.50 1756 1233 0.79 4233 0.50 2.4 9 116.56 1720 1182 0.84 4223 0.50 2.5 10 128.35 1684 1133 0.88 4213 0.50 2.6 10 140.90 1648 1086 0.92 4202 0.50 2.7 11 154.25 1613 1040 0.97 4192 0.50 2.9 12 168.44 1578 995 1.02 4182 0.50

NOTE: The above-listed table is designed for bullet speeds above the minimum Zone 1 velocity listed above. To calculate bullet drop at extended ranges, go to the Long Range Table page and enter your maximum range.

* * Formula from Arthur J. Pejsa, Modern Practical Ballistics (Minneapolis: Kenwood Publishing 1991), p.31.

InclineError Elevation(inch) (MOA)0.09 -0.04

0.01 -0.04

InclineError Elevation(inch) (MOA)

0.00 0.000.01 -0.040.02 -0.040.03 -0.04

0.05 -0.040.06 -0.040.07 -0.040.08 -0.040.09 -0.040.10 -0.040.12 -0.040.13 -0.040.14 -0.040.15 -0.040.16 -0.040.17 -0.040.19 -0.040.20 -0.040.21 -0.040.22 -0.040.23 -0.040.24 -0.040.26 -0.040.27 -0.040.28 -0.040.29 -0.040.30 -0.040.31 -0.040.33 -0.04

Range Elevation Windage(yd) (MOA) (MOA)100 -1.9 0.3 125 -1.6 0.4 150 -1.1 0.5 175 -0.6 0.6 200 0.0 0.7 225 0.6 0.8 250 1.3 0.9 275 1.9 1.0 300 2.6 1.0 325 3.4 1.1 350 4.1 1.2 375 4.9 1.3 400 5.7 1.5 425 6.5 1.6 450 7.4 1.7 475 8.3 1.8 500 9.2 1.9

2700 fps— BC 0.51

Practical Exercise (pp.16, 63, 124)Given:

2700 2520 2350 4350 R ≡ Range (yards)

4329 V ≡ Velocity at Range (fps)

listed above are calculated based onbullet chronograph data (see p.16). c ≡ Mayevski Constant

BC ≡ Ballistic Coefficient

Practical Exercise (p.63) Given:

2700 2520 2350 4424 R ≡ Range (yards)N ≡ Pejsa's Drag Curve Coefficient

4473V ≡ Velocity (fps)

listed above are calculated based on

the Pejsa velocity formula.

4424 4473 8611 0.514

0.520

Ingalls Table (100 fps intervals) pp.32-33

V (fps) r (ft) T (sec) F (ft) N c3600 0.0 0.000 9801 -0.45 246 From Ingall's Table3500 274.4 0.077 9678 -0.45 2463400 553.1 0.158 9553 -0.45 246

2700 2644.7 0.849 8611 -0.45 246 Calculated RE: Pejsa p.31

Example:

Estimate F0 From Manufacturer's Velocity Data

MuzzleVelocity

(fps)

100 YardVelocity

(fps)

200 YardVelocity

(fps) F0

100yard


200yard


Note: The 100yard and 200yard F0's


Estimate F0 From Range and Velocity Data Using Pejsa Formula

MuzzleVelocity

(fps)

100 YardVelocity

(fps)

200 YardVelocity

(fps) F0

100yard

200yard V0 ≡ Muzzle Velocity (fps)

Note: The 100yard and 200yard F0's V/V0

(1-(V/V0)N

F0 ≡ 3RN/(1-(V/V0)N

Estimate BC From F0

F0-100

(feet)F0-200

(feet)F

(feet)Ballistic

Coefficient

100yard

200yard

3200 1124.3 0.331 9295 -0.45 246 Calculated RE: Pejsa p.31

Estimate BC From Velocity Data

3060

listed above are based on calculated


3060


MuzzleVelocity

(fps)

Note: The 100yard and 200yard BC's

Ingall's Table figures, where BC = r/(r2-r1).

MuzzleVelocity

(fps)

Note: The 100yard and 200yard BC's

Ingall's Table figures, where BC = r/(r2-r1).

Then:

100 200

2700 2700 Manufacturer's chronograph data

2520 2350 Manufacturer's chronograph data

4350 4329

4426 4478246.0 246.00.514 0.520

Then:

100 2000.50 0.50

2700 27002520 2350

0.9333 0.8704

0.034 0.067

4424 4473

r T9801 246 0.0 0.00 0.0000 0.0009678 246 274.4 -0.50 0.0002 0.0779552 246 553.1 -1.01 0.0004 0.158

4473 0.520 8611 246 2644.7 -4.84 0.0019 0.849

Note: From chronograph data (p.16), where F0=3*R*Va/dV

Note: From formula on p.63, where F0=BC*(c*V0-N)

F0 BCIngalls FCalculated cCalculated (V-N-V0-N) (V-1-N-V0

-1-N)

4629 0.498 9295 246 1124.3 -2.06 0.0007 0.331

Practical Exercise (pp.31-32) Given: Then:

2840 2630 0.449 c ≡ Mayevski Constant 246.0

-0.45

0.450 3060V ≡ Velocity (fps) 3060

0

0.000

0.00

0.0000


2800 2560 0.378 c ≡ Mayevski Constant 246.0

-0.45

0.384 3060V ≡ Velocity (fps) 3060

0

0.000

0.00

0.0000

100 YardVelocity

(fps)

200 YardVelocity

(fps)Ballistic

Coefficient

100yard




T ≡ (c/(1+N))*(V-1-N-V0-1-N) (sec)

Ingall's Table figures, where BC = r/(r2-r1). (V-N-V0-N)

(V-1-N-V0-1-N)

100 YardVelocity

(fps)

200 YardVelocity

(fps)Ballistic

Coefficient

100yard




T ≡ (c/(1+N))*(V-1-N-V0-1-N) (sec)


(V-1-N-V0-1-N)

246.0 246.0

-0.45 -0.45

3060 30602840 2630

668 1334

0.227 0.470

-1.22 -2.44

0.0005 0.0011

246.0 246.0

-0.45 -0.45

3060 30602800 2560

793 1562

0.271 0.558

-1.45 -2.86

0.0006 0.0012



2700 2520 2350 0.513 c ≡ Mayevski Constant

0.518V ≡ Velocity (fps)


Estimate BC by Pejsa Formula

Practical Exercise (p.124) Given: Then:

180 0.308 1 0.528 Wt ≡ Bullet Weight (grn)Di ≡ Caliber/Bullet Diameter (inch)SC ≡ Shape ClassSC ≡ Spitzer Boat TailSC ≡ Spitzer Flat BaseSC ≡ Semi SpitzerSC ≡ Flat NoseSC ≡ Round NoseBC ≡ Ballistic Coefficient

MuzzleVelocity

(fps)

100 YardVelocity

(fps)

200 YardVelocity

(fps)Ballistic

Coefficient

100yard



Note: The 100yard and 200yard BC's r ≡ (c/N)*(V-N-V0-N) (feet)

T ≡ (c/(1+N))*(V-1-N-V0-1-N) (sec)


(V-1-N-V0-1-N)

Weight(grn)

Caliber(inch)

ShapeClass

BallisticCoefficient

246.0 246.0 246.0

-0.45 -0.45 -0.45

2700 2700 27002700 2520 2350

0 585 1159

0.000 0.224 0.460

0.00 -1.07 -2.12

0.0000 0.0005 0.0010

1800.308

112344

0.528

1° = 60'one 1' = 0.01666° 0.01666667SIN(0.01666°) = 0.00029089

Yards inch/MOA mm/MOA25 0.262 6.650 0.524 13.375 0.785 19.9

100 1.047 26.6125 1.309 33.2150 1.571 39.9175 1.833 46.5200 2.094 53.2225 2.356 59.8250 2.618 66.5275 2.880 73.1300 3.142 79.8325 3.403 86.4350 3.665 93.1375 3.927 99.7400 4.189 106.4425 4.451 113.0450 4.712 119.7475 4.974 126.3500 5.236 133.0525 5.498 139.6550 5.760 146.3575 6.021 152.9600 6.283 159.6

27.34 0.286 7 Note: 25m = 27.34 yards

New Exact Small Arms Ballistics Copyright © 2008 by Arthur J. Pejsa. All rights reserved.

Practical Exercise (p.17) Given: Then: D ≡ Drop (inches) 8.62 38.98 100.52 208.30 387.39

G ≡ Gravitational Constant (Pejsa) 41.68 41.68 41.68 41.68 41.68R ≡ Range (yards) 200 400 600 800 1000

3000 3000 3000 3000 3000

Fa ≡ Mean Retard Coefficient (feet) 3720 3640 3560 3480 3400

3800 3800 3800 3800 3800N ≡ Loss Rate of F 0.40 0.40 0.40 0.40 0.40

Practical Exercise (p.18) Given: Then: D ≡ Drop (inches) 2.02 8.50 20.13 37.76 62.42G ≡ Gravitational Constant (Pejsa) 41.68 41.68 41.68 41.68 41.68R ≡ Range (yards) 100 200 300 400 500

3000 3000 3000 3000 3000Fa ≡ Mean Retard Coefficient (feet) 4301 4261 4221 4181 4141

4341 4341 4341 4341 4341N ≡ Loss Rate of F 0.40 0.40 0.40 0.40 0.40S ≡ Sight Height Above Bore (inches) 1.50 1.50 1.50 1.50 1.50

3.52 10.00 21.63 39.26 63.92Z ≡ Zero Range (yards) 200 200 200 200 200

4.78

5.00 10.00 15.00 20.00 25.00H ≡ Height Relative to Sights (inches) 1.48 0.00 -6.63 -19.26 -38.92Note: 1 MOA = 1" @ 95.5 yards

Practical Exercise 31-33) Given: Then: M ≡ Midrange (yards) 122.6 115.0 120.5 120.5 120.5G ≡ Gravitational Constant (Pejsa) 41.68 41.68 41.68 41.68 41.68

2900 2700 2900 2900 2900

4173 4473 3200 3200 3200S ≡ Sight Height Above Bore (inches) 1.50 1.50 1.50 1.50 1.50

2.00 2.00 2.00 2.00 2.00

4201 4501 3228 3228 3228

1.32 1.32 1.32 1.32 1.32SQ ≡ SH/√(2) 0.94 0.94 0.94 0.94 0.94

30.6 28.6 30.3 30.3 30.3Z ≡ Zero Range (Far) (yards) 210.6 198.1 205.7 205.7 205.7PBR ≡ Point-Blank Range* (yards) 245.9 231.6 239.6 239.6 239.6 * Gunnery point-blank range * * Formula from Arthur J. Pejsa, Modern Practical Ballistics (Minneapolis: Kenwood Publishing 1991 p.31.





DT ≡ Total Drop @ Range (inches)

SC ≡ Sight Correction (in/100 yards)

SC Effect




F ≡ F0+15*√(Hm+S)**

SH ≡ √(1+S/Hm)


Practical Exercise (pp.16, 63, 124)Given: Then: R ≡ Range (yards) 100 200

3060 30602840 2630

4023 3970 Note: Derived from manufacturer's chronograph data

4099 4099 Note: From formula on p.63

c ≡ Mayevski Constant 246.0 246.0BC ≡ Ballistic Coefficient 0.450 0.450

* * Formula from Arthur J. Pejsa, Modern Practical Ballistics (Minneapolis: Kenwood Publishing 1991 p.31.

V0 ≡ Muzzle Velocity (fps)V ≡ Muzzle Velocity (fps)



Note: Derived from manufacturer's chronograph data

Note: From formula on p.63

New Exact Small Arms Ballistics Copyright © 2008 by Arthur J. Pejsa. All rights reserved.

Mayevski-Ingalls Reference A Function

The A (deceleration/retardation) function for the Mayevski projectile is: c and N are defined in speed zones.For example: for 2600<V<3600 fps: N = -0.45 c = 246.0

Practical Exercise (pp.31-32) Given: Then: c ≡ Mayevski Constant 246.0 246.0 246.0 246.0 246.0N ≡ Drag Curve Coefficient -0.45 -0.45 -0.45 -0.45 -0.45

3600 3600 3600 3600 3600V ≡ Velocity (fps) 2600 3200 2500 2400 1100

2967 1124 3296 3633 9005

0.970 0.331 1.099 1.237 4.552F ≡ Ingall's Table Value @ V (feet) 8466 9295 8318 8166 5749

-5.43 -2.06 -6.03 -6.64 -16.47

0.0022 0.0007 0.0025 0.0028 0.0102

Practical Exercise (pp.16, 63, 124)Given: Then: R ≡ Range (yards) 100 200

3060 3060 Manufacturer's chronograph data2840 2630 Manufacturer's chronograph data

4023 3970

4099 4099c ≡ Mayevski Constant 246.0 246.0BC ≡ Ballistic Coefficient 0.450 0.450

A=V2+N/c



T ≡ (c/(1+N))*(V-1-N-V0-1-N) (sec)

(V-N-V0-N)

(V-1-N-V0-1-N)

V0 ≡ Muzzle Velocity (fps)V ≡ Muzzle Velocity (fps)

F0 ≡ Initial Retard Coefficient (feet) Note: From chronograph data (p.16)

F0 ≡ Initial Retard Coefficient (feet) Note: From formula on p.63

Ingalls Table (100 fps intervals) pp.32-33

V (fps) r (ft) T (sec) c3600 0.0 0.000 9801 -0.45 2463500 274.4 0.077 9678 -0.45 2463400 553.1 0.158 9553 -0.45 2463300 836.4 0.243 9425 -0.45 2463200 1124.4 0.331 9295 -0.45 246 4648 0.5003100 1417.4 0.424 9164 -0.45 2463000 1715.7 0.552 9029 -0.45 246 4370 0.4842900 2019.4 0.625 8893 -0.45 246

2800 2329.1 0.734 8753 -0.45 2462700 2644.8 0.849 8611 -0.45 246 4478 0.520

2600 2967.1 0.970 8472 -0.45 246

2500 3297.6 1.100 8379 -0.30 801.32400 3637.6 1.239 8277 -0.30 801.3

2300 3987.6 1.388 8172 -0.30 801.3

2200 4348.4 1.548 8063 -0.30 801.32100 4720.9 1.731 7952 -0.30 801.32000 5106.1 1.909 7838 -0.30 801.31900 5505.0 2.114 7717 -0.30 801.31800 5918.8 2.338 7596 -0.30 801.31700 6353.1 2.586 7599 0.00 7599

Manufacturer's chronograph data 1600 6813.8 2.865 7599 0.00 7599Manufacturer's chronograph data 1500 7304.2 3.182 7599 0.00 7599

1400 7828.5 3.544 7599 0.00 7599

1300 8403.8 3.971 8034 1.00 1.045E+071200 9078.6 4.511 9132 1.00 1.045E+071100 9986.6 5.304 11856 3.00 1.578E+131000 11294.7 6.544 15780 3.00 1.578E+13900 13149.0 8.512 18720 1.00 1.685E+07800 15489.1 11.275 21060 1.00 1.685E+07700 18342.2 15.090 21386 0.00 21386

4114 9110

Fr (ft) NMayevski F0 BCIngalls

chronograph data (p.16)

9801 2469678 2469552 2469425 2469295 2469163 2469029 2468892 246

8753 2468611 246

8466 246

8379 801.38277 801.3

8172 801.3

8063 801.37952 801.37836 801.57717 801.47592 801.77599 7599

7599 75997599 7599

7599 7599

8038 1.044E+078708 1.096E+07

11856 1.578E+1315780 1.578E+1318722 1.685E+0721063 1.685E+0721386 21386

FCalculated cCalculated

practical ballistics for hunters and shooters

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