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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.]
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.
Practical Ballistics for Hunters and Shooters
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
* 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
Practical Ballistics for Hunters and Shooters
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
S DzSight Zero Drop at
Height Range Zeroinches yards inches
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.
if zeroed at U yards. Sights 1.5 inches above bore line.
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!
if zeroed at U yards. Sights 1.5 inches above bore line.
), and Calculated F0 below.
Practical Ballistics for Hunters and Shooters
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
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.
S Dz SSight Zero Drop at Sight
Height Range Zero Heightinches yards inches inches
2.6 25 0.15 4340 2.6
Vo Fo N Vo2,700 4,350 0.50 2,700
Ballistic Table Ballistic Table Flight
Range Path Drop Fa N Range25 0.0 0.15 4340 0.50 25
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
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.)
*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.
Practical Ballistics for Hunters and Shooters
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.
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.
S Dz SSight Zero Drop at Sight
Height Range Zero Heightinches yards inches inches
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
V0 ≡ Muzzle Velocity (fps)
F0 ≡ Initial Retard Coefficient (feet)
Hm ≡ Height Maximum (inches)
SH ≡ √(1+S/Hm)
Zn ≡ Near Zero Range (yards)
* Pejsa point-blank range8
Rz
Ballistic Table Ballistic Table Flight
Range Path Drop Fa N Range25 -0.2 0.15 4340 0.50 2529 0.0 0.20 4339 0.50 29
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.
← 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.
Practical Ballistics for Hunters and Shooters
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)
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)
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)
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.
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.
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.
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