Sir Isaac Newton’s first law of motion states that the velocity of a body remains constant unless the body is acted upon by an external force. The bow string force accelerates the arrow from the bow until the arrow reaches the launch velocity, drag forces slow the arrow as it flies through the air, and gravitational force eventually brings the arrow to the ground. In archery, these forces (F) are directly related to the acceleration (a) and the mass (m) of the arrow.
F = ma (1)
a = F/m (2)
Large forces result in large acceleration, while large masses are very hard to accelerate or decelerate (think about pushing a car versus a bicycle on a level street). This is why a lighter arrow leaves the bow at faster speeds, but loses velocity faster during flight.
To calculate the trajectory of an arrow we only need to define the three forces that accelerate or decelerate the arrow: 1) force of acceleration from the bow toward the target, 2) force of acceleration toward the earth due to gravity, and 3) force of deceleration due to aerodynamic drag on the arrow. The acceleration forces of the bow and gravity are well understood and covered in full detail in past posts. This post addresses the deceleration forces on the arrow due to aerodynamic drag. Computing the drag is not simple, but it is critical in defining the trajectory of an arrow (and all ballistic projectiles).
The Drag Equations
Ballistic engineers have been working on models for aerodynamic drag for over 250 years to solve the trajectories of bullets, cannon balls, or arrows. All arrows moving through the air experience a deceleration drag due to friction of air on the arrow tip, shaft, and feathers. A typical arrow flies at over 150 mph so think about putting your hand out the window of a performance race car and you get a sense of the drag force on the arrow. For reference, the drag force is about 2 to 3 times the gravitational force. The drag force is related to the shape of the object, viscosity of the air (think about putting your hand in the water from a speeding boat), and the cross sectional area of the object as defined by the drag equation.
Fd is the drag force (units of Newton or lbf)is the density of the fluid (kg/m3 or lbm/ft3)
v is the velocity of the object (m/s or ft/s)
Cd is the drag coefficient (dimensionless)
A is the cross sectional area of the projectile (m2 or ft2)
Since F=ma we can divide the drag force by the arrow mass (kg or lbm) to get the drag deceleration of the arrow (d)
where d is used instead of a to differentiate between acceleration from the bow and deceleration due to drag (d has units of m/s2 or ft/s2).
The bottom line is that deceleration of the arrow due to aerodynamic drag increases with density, velocity and area and decreases with mass. Heavy and slow arrows have less drag deceleration. Large diameter arrows have more drag deceleration. The effect of arrow or bullet shape (feathers, nock, point angle) on the drag are incorporated in the drag coefficient. Projectiles with lower drag coefficients are more aerodynamic. Values of Cd for a range of objects are listed in the table below and make intuitive sense. It is always better to throw a football compared to a brick!
Figure 1. (http://en.wikipedia.org/wiki/Drag_coefficient)
Computing Aerodynamic Drag for Real Objects
If we know the drag coefficient it should be possible to calculate arrow trajectories to better than a few centimeters at shooting distances of many meters.
The problem is that Cd is not a true constant but varies as a function of speed, flow direction, object position, object size, fluid density and fluid viscosity. Bummer! The figure below shows the measured (blue circles) and calculated (red line) value of d for the G1 reference bullet. The calculated values use Cd = 0.5191 at all velcocities.
Figure 2. Measured and Calculated vales of drag deceleration (d) as a function of G1 projectile velocity. Notice the wiggle around the speed of sound, 330 m/s.
The G1 projectile is a 1 lb, 1 inch diameter bullet diameter bullet with a flat base, a length of 3 inches, and a 2 inch radius tangential curve for the point. The G1 standard projectile originates from the "C" standard reference projectile defined by the German steel, ammunition and armaments manufacturer Krupp in 1881 (Wikipedia).
The important thing to notice is that theory using a constant Cd does a good job of estimating d at many velocities, but does a pretty bad job before and after the speed of sound (330 m/s). The wiggles in the actual data make it nearly impossible to write a continuous function that defines the drag deceleration. Fortunately, the wiggles are fairly constant from arrow to arrow or bullet to bullet. Therefore, one practical solution is to define drag coefficient, Cd, in terms of the measured drag coefficient of the G1 reference projectile (Cg) and scale the coefficient up or down by the mass, size, and shape of each projectile. This was the big idea behind the development of the Ballistic Coefficient, BC. BC is a dimensionless scaling factor that defines the performance of a projectile relative to a G1 or other ballistic standard.
where SD is the sectional density
and i is the form factor of the projectile
m is the mass of projectile (kg or lbm)
d is the projectile diameter of projectile (m2 or ft2) - different d from the deceleration constant.
Cd is the drag coefficient of the projectile (dimensionless)
Cg is the drag coefficient of the G1 reference projectile (dimensionless)
The sectional density “adjusts” a projectiles drag coefficient for cross sectional area and mass. The form factor “adjusts” the projectiles drag coefficient for shape. Very aerodynamic projectiles will have form factors less than 1 and a brick will have a form factor of about 1.6. Very aerodynamic objects with low mass will still experience significant drag deceleration while very heavy bricks will have little drag deceleration. The beauty of the Ballistic Coefficient is that it incorporates mass and shape into one number. Projectiles with larger BC values experience LESS drag deceleration.
As a practical matter, we want Cd for our actual projectile.
Tabulated BC values are defined in terms of the G1 projectile described in pounds mass and inches diameter. To work in SI units you must redefine the G1 projectile in kg/m2 by adding the conversion factor 0.0014223.
where mass is in kilograms and diameter is in meters.
The attached spreadsheet will help you implement these calculations. The first columns tabulate Cg for the G1 reference projectile. Don't mess with these unless you want to use a different ballistic reference. For each test projectile you will need to know the mass, diameter and the ballistic coefficient. The calculations are all in SI units so the spreadsheet starts by converting mass and diameter to these units. Next, Cd at any velocity is computed from Cg tabulated as a function of velocity. Finally, d is computed from Cd, velocity, test projectile mass and cross sectional area, and the density of air. The plots below compare values of Cd and d for several projectiles.
Figure 3. Calculated values for Cd for several test projectiles.
Values of Cd for bullets fall in range of 0.1 to 0.3 with a step upward at supersonic velocities (about 330 m/s). Bird shot has a Cd of around 0.5, typical of a sphere. Arrows have a much larger Cd due to the large surface area of the arrow shaft. This is reflected in the very low ballistic coefficient for an arrow compared to bullets. Like ballistic coefficients for bullets, BC values for arrows must be calculated from experimentally measured form factors (i). A value of 9 for i is a good starting point to explore the effects of other arrow properties (mass and diameter) on aerodynamic drag.
Figure 4. Calculated values of d for several test projectiles. Notice the log scale on the Y-axis. The dotted line is the acceleration of gravity for reference.
As expected, the drag deceleration, d, is much higher for arrows than bullets, but not as high as the drag deceleration on bird shot. Not surprisingly, arrows travel at least 4 to 5 times further than bird shot. As a reference, arrows with velocities over 200 ft/s experience more drag deceleration than gravitational acceleration. Of course, these forces are not oriented in the same direction. Drag slows the arrow along its flight path while gravity pulls the arrow toward the earth. Interestingly, a falling object reaches terminal velocity when the drag deceleration is equal and opposite the gravitational acceleration (9.8 m/s.s). This is the velocity of the crossing point of each test projectile drag curve in Figure 4 with the black dashed line. If you dropped all the projectiles from an airplane the G1 projectile would fall the fastest, almost 250 m/s. The G1 projectile is the most aerodynamic because of its large ballistic coefficient.
I fully expect some archers to object to my use of the G1 projectile reference for arrows. Arrows do not fly at supersonic speeds and it is not at all clear that a G1 projectile will have similar drag to an arrow at typical shooting velocities of less than 100 m/s. In my defense, the G1 reference is relatively flat at these low velocities and the errors associated in launching an arrow are probably greater than errors in the computed drag deceleration. It is also interesting to be able to compare arrows to other common projectiles. Comparing ballistic coefficients tells you a lot about relative aerodynamic performance (G1>Win 30> PowerBelt>Arrow>Bird Shot). Perhaps future researchers can develop an archery specific arrow reference based on the I.B.O standard arrow (30 inch, 350 grain arrow, 5/16” in diameter, with no fletching). This would produce a new set of BC values referenced to an I.B.O arrow. In the mean time, I am off to shoot a number of different arrows to experimenatlly determine the BC values for different fletch and shaft combinations. I have tabulated some predicted values of BC for different arrows below.
Table 1. Predicted BC values for different arrow types.
Notice that thin arrows are almost twice as aerodynamic than larger diameter arrows and thin, heavy arrows are best. This makes sense based on the increase in sectional density for thin, heavy arrows.
Future posts will detail how to measure the ballistic coefficients of arrows experimentally to confirm these predictions.
A good introduction to the Ballistic Coefficient: http://www.exteriorballistics.com/ebexplained/articles/the_ballistic_coefficient.pdf
Values of d for the G1 Ballistic Standard http://www.snipercountry.com/ballistics/
Values of Ballistic Coeffients for many projectiles: http://www.frfrogspad.com/bcdata.htm