Full Trajectory Calculator for Rifles, Muzzleloaders, and Bows

    In my previous posts I detailed the physics behind calculating the trajectory of an arrow shot from a bow under real conditions experienced by an archer.   The most important detail in calculating correct trajectory calculations is knowing the correct drag coefficient for the arrow and including the effect of drag on arrow trajectories.   In this post I expand these calculations to include typical hunting rifles, muzzle loaders, and bows with examples for each.  I have tried to make this post a bit more theory light and application heavy.  By comparing the trajectories of three different projectiles (arrows, 30 caliber bullets, and 50 caliber muzzle loader rounds) if becomes easier to understand the similarities and differences of these three weapons.

    In each case I begin with a complete trajectory calculation implemented in Microsoft Excel as described in my previous trajectory post.  To calculate the trajectory of any object you need to know four things:  1) the launch speed, 2) the object mass (weight),  3) the launch angle, and 4) the drag coefficient for the object.  Using a baseball analogy, we need to know how hard we are throwing the ball, the weight of the ball, the direction of the pitch, and if we have any spit or dirt on the outside of the ball.

Launch Velocity:   The launch velocity is the speed that the arrow or bullet leaves the bow or gun.   The launch velocity for arrows is highly dependent on the setup of the bow and arrow and really needs to measured directly.   Future posts will evaluate several different tools for measuring arrow launch speed, but for now I would recommend asking your local pro shop to measure your arrow speed or invest $99 in a Shooting F-1 Chronograph.  The same chronograph will also work for rifles or you may also use the ballistics tables provided for most rifle ammunition.  

Projectile Mass:  
The projectile mass is the amount of stuff being shot toward the target.   Mass and weight are often confused.   Weight is the force of an object being attracted toward the Earth by gravity.   A bullet has mass on the Earth and in space, but in space the bullet would be "weightless".  As a practical matter we measure mass using a balance that compares the weights (force) of two objects on Earth (balances don't work in space).   When the balance is level the weight of each object is the same.  Since gravity on Earth is almost constant, two objects on Earth with the same weight also have the same mass.  In the ballistics world the unit of mass is the grain (gr).  A grain is a unit of mass equal to a single seed of cereal grain.  A 150 grain bullet would have the same mass as a 150 grains of wheat.  One grain is equal to 0.064798 grams or 0.002285 ounces.

Direction:   Direction is simply the elevation of the arrow or gun barrel above or below the plane of the Earth measured in degrees.   The launch angle is determined by the sights on the gun or bow using your eye(s).  It is important to remember that your eyes are almost always above the arrow or rifle barrel.  This is why most trajectory calculations start 1.5 inches low for rifles and 3.5 inches low for bows.

Drag:  The drag coefficient describes the effect of air resistance on the arrow or bullet.  The more aerodynamic the object the smaller the drag coefficient.  It is possible to calculate the drag coefficient for a projectile based on shape of the object and our knowledge of fluid dynamics, however, we usually measure the drag coefficient empirically by measuring the drop of our arrow/bullet at several distances and calculating drag from a fit to the ballistics data.   The rest of this post will walk you through these calculations.  For references I have listed the drag coefficients for several projectiles.  Notice that the rifle bullet has about half the drag of the arrow or 50 caliber muzzle loader bullet.

 Table 1.    
 Projectile             Mass (gr/grams)     Diameter (inches/meters) k (1/meter)
Cabela's Stalker  Extreme, 29" arrow 244/25.5 0.3125/0.0079 0.00310
Hornday 300 Win Mag 180 gr SST Interlock rifle bullet 180/11.7 0.30/0.0076 0.00078
Powerbelt Aeortip, 100 gr charge, 28" barrel, muzzleloader bullet 223/14.5 0.50/0.0127 0.00284

Ballistics Data.   Ballistics data is the measured trajectory of a bullet or arrow at specific distances.   Table 2 and Figure 2 shows the ballistics data for a 50 caliber muzzleloader bullet.  Notice that the bullet starts below the target, arches upward to 70 meters, is level with the target at 131 meters (150 yards) and then falls below the target after 131 meters.   In this case the gun was sighted in at 150 yards and then fired at targets with distances from 25 to 250 yards.   The blue symbols are the actual shot positions.   The blue line is the calculated trajectory with drag, and the red line is the calculated trajectory not correcting for drag.  Drag makes a big difference at distances past 50 yards.

Table 2. Ballistics data for PowerBelt Aerotip 50 caliber muzzleloader bullet
shot from a 28" barrel with a 100 gr powder charge.

Range (yards) Shot Drop (inches)
0 -1.5
25 0.9
50 2.58
75 3.45
100 3.41
125 2.32
150 0
175 -3.83
200 -9.02
225 -16.15
250 -24.83

Figure 2.   Ballistics plot for PowerBelt Aerotip 50 caliber bullet.

    Ballistics data for rifles and muzzleloaders can be downloaded from the web by searching on your specific bullet.  The data is pretty good because most rifle barrels and the loads are very uniform.   However, each bow and arrow combination is different so you must collect your own ballistics data for each bow and arrow combination.   This is easily done by stacking two targets on top of each other and placing a small aiming point 12 inches below the top of the top target.   This will be the sighting point for your 30 yard pin.   Take a couple of shots at the aiming point from 30 yards to confirm that you bow is "zeroed in" at 30 yards.  Now move to 10 yards from the target and shoot at the aiming point with the 30 yard pin.   Your shot should be high by 4 to 5 inches.   Repeat shots from 20, 30, 40, and 50 yards always using the 30 yard pin.   Shots longer than 30 yards will be low.  Record the distance above or below the aiming point for each shot distance to generate the ballistics data for your  bow.   Figure 3 shows the results from my ballistics trials with a 62 pound draw weight PSE stinger shooting 244 gr arrows with Blazer vanes.  

Figure 3.  Arrows positions after shots at 10, 20 , 30, 40, and 50 yards using the 30 yard pin.  Each shot is labeled in black type face. Notice that the 20 yard shot is above the 10 yard shot due to the arc in the arrow trajectory.

Data for my bow taken from Figure 3.
 Distance     shot height above aiming point (inches)
 20 5
 30 0
 40 -12
 50 -30

Caution, the above experiment is not easy to perform.   Your groups must be small or shot errors will be larger than the change in arrow position due to range differences.

    So now it is time to use the ballistics data to compute aerodynamic drag.    Attached to the end of this post are three Excel spreadsheets with typical trajectories for a 30 caliber rifle, a 50 caliber muzzleloader, and a 60 pound compound bow.   Pick the file that is closest to your weapon.   Each file has complete directions listed on the directions tab.   Start by entering your specific ballistics data starting in row 54.  Set the target distance to the sighting distance for your weapon (bow = 30 yards, muzzleloader = 150 yards) and click on the green button to solve for the correct launch angle for our your target.   The trajectory should pass through zero at your sighting distance.   Now click on the blue button to adjust the aerodynamic drag coefficient to fit the trajectory of your bullet/arrow.  Repeat the process by clicking on the green button to improve the launch angle, blue button to refine the drag coefficient, and repeat this cycle two more times.  This process optimizes the launch angle and drag for your bow/rifle.  Save the file to a new name and don't mess with the drag coefficient unless you change ammunition.

    Lets look at my trajectories.  Figure 4 is the calculated trajectory for my bow shot at range of 30 yards.

Figure 4:  Arrow trajectory for a 62 pound bow. Notice the significant drop with distance.

Notice that the arrow climbs upward from launch, reaches maximum height at about 1/2 the distance to the target and then falls well below the target beyond 30 yards.   The steep downward curvature beyond 30 yards is due to the increasing downward velocity of the arrow due to gravity.   For archers this can be a big deal.   Take a look at figure 5, and notice the shot placement for shots taken at 25, 30, and 35 yards.

Figure 5. Shot placement for a deer at 25, 30, or 35 yards shot using the 30 yard pin.

    The middle green dot is the dead-on shot.  The upper dot is where the arrow would hit if the deer was really at 25 yards (a high, but effective shot), and the lower dot is where the arrow would hit if the deer was at 35 yards.  In both cases the archer was aiming using the 30 yard pin.  Because of the arrows increasing downward velocity due the acceleration of gravity a long shot has a much greater error, and likely a wounded deer.   It is always better to error on the long side of any range estimate (shoot the lower pin).

    It takes about 0.3  seconds for the downward velocity of any projectile to become large enough to cause significant shot errors due to target range errors.  Faster rifle bullets travel farther in the horizontal direction in 0.3 seconds than relatively slow arrows, while both projectiles drop the same distance in the vertical direction.

    Compare the bow to high performance hunting rifle (Figure 6).  Because the rifle bullet is moving 10 times faster than the arrow it gets to the target 10 times quicker.  This means that the bullet has much less time to drop and has a much flatter trajectory.    As a good rule of thumb, you can compare effective hunting ranges of weapons to projective velocities.   A bullet moving twice as fast will have twice the effective range.

Figure 6. 180 grain rifle bullet trajectory.  Notice the flat trajectory.

Figure 7. Shot placement for a deer at 170, 200, or 230 yards shot using the 200 yard sight.

    Comparing Figures 5 and 7 you can see why range estimation with a hunting rifle is much less critical.   A 30 yard uncertainty in range estimation has almost no effect, at least within 300 yards.  Very few deer in Maine are shot at distances over 200 yards.

    The trajectory calculations for a muzzleloader is somewhere in between the bow and the rifle.  Again, comparing projectile velocities of a rifle (2960 ft/s) to a muzzleloader (2000 ft/s) suggests that a muzzleloader in experienced hands should have good shot placement out to 200 yards.  The actual calculations attached below confirm this prediction.

    If you want to compute trajectories at any range simply enter the range into cell B11 and click on the green button to recalculate.   This will compute the correct launch angle for that range.   You may also change the short and long range estimates (cells I18 and I19) to see the effects of estimating target range incorrectly.   Finally, you may play with the hunter elevation above/below the target to convince yourself that only the horizontal distance to the target has a significant effect on shot placement.  In all cases, the hunter is relatively immune to target range estimates if the projectile hits the target in less than 0.3 seconds.  

Simple Effective Range Estimate (Multiply Launch Speed by 0.3 seconds)
 Weapon         Launch Speed (ft/s)     Effective Range (yrs)
 244 25
 muzzleloader 2000 200
 rifle 3000 300

Have fun playing with the trajectory calculations.
Whitney King,
Sep 6, 2011, 7:19 PM
Whitney King,
Sep 6, 2011, 7:14 PM
Whitney King,
Sep 6, 2011, 7:19 PM