In my previous posts I detailed the physics behind calculating the trajectory of an arrow from a bow in real conditions experienced by an archer.
The most important details in calculating correct trajectory calculations are knowing the correct drag coefficient for the arrow and including the effect of drag on the arrow trajectory.
In this post I expand these calculations to include specific hunting rifles, muzzleloaders, 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 (arrow, 30 caliber bullet, and 50 caliber muzzleloader round) it becomes easier to understand the similarities and differences between these three weapons.
To calculate the trajectory of any object you need to know four things: 1) the projection speed, 2) the object’s mass (weight), 3) the angle of trajectory, and 4) the drag coefficient for the object.
Using the 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.
The launch velocity for arrows is highly dependent on the bow and arrow setup and needs to be measured directly.
Future posts will be evaluating several different tools to measure arrow launch speed, but for now I would recommend asking your local pro shop to measure your arrow speed or investing $99.
Shooting F-1 Chronograph, The same chronograph will also work for rifles or you can also use the ballistic tables provided for most rifle ammunition.
Projectile mass is the amount of material being shot towards the target.
mass and weight are often confused.
Weight is the force of an object being attracted to the earth by gravity.
A bullet has mass on Earth and in space, but the bullet would be “weightless” in space.
As a practical matter we measure mass using a balance that compares the weight (force) of two objects on Earth (balance doesn’t work in space).
When the equilibrium is flat, each object has the same weight.
Since gravity on Earth is almost constant, two objects of equal weight on Earth also have the same mass.
The unit of mass in the ballistic world is Grains (gr), Grain is a unit of mass equal to one seed of grain.
The mass of a 150 grain bullet would be equal to that of 150 grains of wheat.
One grain is equal to 0.064798 grams or 0.002285 ounces.
direction: The direction is simply the height of the arrow or gun barrel above or below the surface of the earth, measured in degrees only.
The projection angle is determined using your eye(s) on the gun or bow.
It is important to remember that your eyes are almost always at the top of the arrow or rifle barrel.
This is why most trajectory calculations start at 1.5 inches lower for rifles and 3.5 inches lower 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 our knowledge of the object’s shape and fluid dynamics, however, we usually calculate the drag coefficient by measuring the drop of its arrow/bullet over several distances and calculating the drag from one measure empirically.
Fit to ballistic data.
The rest of this post will walk you through these calculations.
For references I have listed the drag coefficients for several projectiles.
Note that a rifle bullet has about half the drag of an arrow, or a 50 caliber muzzleloader bullet.
|projectile||Mass (gr/gram)||Diameter (inch/meter)||K (1/m)|
|cabella hunter extreme, 29″ arrow||244/25.5||0.3125/0.0079||0.00310|
|Hornaday 300 Win Mag 180 Gr Sst Interlock Rifle Bullet||180/11.7||0.30/0.0076||0.00078|
|Powerbelt aortip, 100 gr charge, 28″ barrel, muzzleloader bullet||223/14.5||0.50/0.0127||0.00284|
ballistic data, Ballistic data is the measured trajectory of a bullet or arrow over a specific distance.
Table 2 and Figure 2 show ballistic data for the 50 caliber muzzleloader bullet.
Note that the bullet begins below the target, recedes upwards to 70 m, is 131 m (150 yards) with the target and then falls below the target after 131 m.
In this case the gun was sighted at a range of 150 yards and then fired at the target 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 that does not correct for drag.
Drag makes a huge difference in the last 50 yards.
Ballistic Data for the PowerBelt Aerotip 50 Caliber Muzzleloader Bullet
Shot from a 28″ barrel with a 100g powder charge.
|Range (yards)||shot drop (inch)|
Ballistic plot for the Powerbelt AeroTip 50 caliber bullet.
ballistic data The rifle and muzzleloader can be downloaded from the web by searching for your specific bullet.
The data is great because most rifle barrels and loads are very similar.
However, each bow and arrow combination is different so you will need to collect your own ballistic 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 sight point for your 30 yard pin.
Take a few shots at the target point at 30 yards to confirm that your bow is “zero” at 30 yards.
Now go 10 yards from the target and shoot at the target point with a 30 yard pin.
Your shot should be 4 to 5 inches high.
Always repeat shots at 20, 30, 40 and 50 yards using the 30 yard pin.
Shots longer than 30 yards will be short.
Record the distance above or below the aiming point for each shot distance to generate ballistic data for your bow.
Figure 3 shows the results of my ballistic tests shooting 244 gr arrows with the 62 pound draw weight PSE Stinger Blazer vanes.
Figure 3.Arrow position after shots at 10, 20, 30, 40, and 50 yards using a 30 yard pin.
Each shot is labeled in black type face.
Note that a 20-yard shot is higher than a 10-yard shot due to the arc in the arrow trajectory.
The data for my bow is taken from picture 3.
|distance||Height of shot above target point (inches)|
Your groups must be smaller or the shot errors will be larger than the change in arrow position due to the range difference.
So now it’s time to use ballistic data to calculate aerodynamic drag.
Attached to the end of this post are three Excel spreadsheets with typical trajectories for a 30 caliber rifle, 50 caliber muzzleloader and 60 pound compound bow.
Choose the file that is closest to your weapon.
Each file contains complete instructions listed on the Directions tab.
Begin by entering your specific ballistic data starting on line 54.
Set the target distance for your weapon (bow = 30 yards, muzzleloader = 150 yards) and click the green button to have us sort out the correct launch angle for you.
The trajectory should pass through the void at your sight 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 the green button to improve the launch angle, the 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 until you change the ammunition.
Let’s look at my trajectory.
Figure 4 is the calculated trajectory for my bow shot at 30 yards.
Figure 4: Arrow trajectory for a 62 lb bow.
Note the significant drop with distance.
Note that the arrow ascends from launch, reaches maximum height at about 1/2 the distance from the target and then drops significantly below the target beyond 30 yards.
There is a downward curvature over 30 yards due to the downward velocity of the arrow due to gravity.
This can be a big deal for archers.
Take a look at Figure 5, and note the shot placement for shots taken at 25, 30 and 35 yards.
Figure 5.Shot placement at 25, 30, or 35 yards for a deer shot using a 30 yard pin.
The middle green dot is a dead-on shot.
The upper point is where the arrow would be shot if the deer was actually at 25 yards (a high, but effective shot), and the lower point is where the deer would be arrowed at 35 yards.
In both cases the archer was aiming using a 30-yard pin.
A long shot has a lot of error, and possibly an injured deer, due to the arrows moving downwards due to the acceleration of gravity.
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 be large enough that target range errors can result in significant shot errors.
Faster rifle bullets cover a greater distance in 0.3 seconds in the horizontal direction than relatively slower arrows, while both projectiles drop the same distance in the vertical direction.
Compare the bow to a high performance hunting rifle (Figure 6).Because a rifle bullet travels 10 times faster than an arrow, it reaches the target 10 times faster.
This means that the bullet has little time to fall and has a very flat trajectory.
As a good rule of thumb, you can compare the effective hunting ranges of weapons to projectile velocities.
The effective range of a bullet fired at twice the speed will be doubled.
Figure 6.180 grain rifle bullet trajectory.
Note the flat trajectory.
Figure 7.Shot placement at 170, 200, or 230 yards for a deer shot using a 200-yard sight.
Comparing Figures 5 and 7 you can see why estimating range is much less important with a hunting rifle.
An uncertainty of 30 yards in the range estimate has almost no effect, at least within 300 yards.
Very few deer in Maine are shot beyond 200 yards.
The trajectory calculation for a muzzleloader is somewhere between a bow and a rifle.
Again, comparing the projectile velocity of a rifle (2960 ft/s) to that of a muzzleloader (2000 ft/s) shows that in experienced hands a muzzleloader should have good shot placement at 200 yards.
Actual calculations attached below confirm this prediction.
If you want to calculate the trajectory on any range just enter the range in cell B11 and click the green button to recalculate.
This will calculate the correct projection angle for that range.
You can also change the short- and long-range estimates (cells I18 and I19) to see the effects of an incorrect estimation of the target range.
Finally, you can play with the hunter’s height above/below the target to convince yourself that only the horizontal distance to the target has a significant impact on shot placement.
In all cases, if the projectile hits the target in less than 0.3 seconds, the predator is relatively immune to target range estimates.
Simple effective range estimation (multiply launch speed by 0.3 seconds)
|Weapon||Launch Speed (ft/s)||Effective Range (Years)|
Have fun playing with trajectory calculations.