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) and 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).
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.
where:
is the density of the fluid (kg/m^{3}
or lb_{m}/ft^{3})
Since F=ma we can divide the drag force by the
arrow mass (kg or lb where d is used instead of a to differentiate
between acceleration from the bow and deceleration due to drag (d has units of
m/s 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 C Figure 1. (http://en.wikipedia.org/wiki/Drag_coefficient)
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 C 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 C
where SD is the sectional density
and i is the form factor of the projectile m is the mass of projectile (kg or lb d is the projectile diameter of projectile (m C C 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
As a practical matter, we want C 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/m
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, C
Figure 3. Calculated values for C
_{d} 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 C 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. References: 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 |