Welcome back to our study of ballistics. Now that we’ve covered the many aspects of internal ballistics and how ammunition interacts with the inner workings of firearm components to produce mechanical accuracy, we will discuss external ballistics. Since there are so many forces that affect a projectile’s flight from the muzzle to the target, I’ve broken the subject material down into a two part series. In this article, I will introduce some basic physics principles, discuss projectile characteristics and focus on transitional ballistics.
In order to fully explain some concepts related to external ballistics, I need to briefly review the principles on which all ballistics rest: Newtonian Physics. While many of us learned our first physics lessons from Wylie E. Coyote, nearly everything depicted was incorrect. Even though we cannot observe the forces acting on a projectile the moment it leaves the muzzle, we know those forces significantly affect its actual path. In order to set the record straight, I will cover Newton’s three laws to help us better understand ballistics. Now, on to the physics.
Newton’s First Law: “A body at rest tends to remain at rest, and a body in motion tends to remain in motion at the same speed and in the same direction unless acted upon by a force.” Applicability: If it wasn’t for the force of gravity or air resistance, a projectile would travel endlessly in a straight line from the barrel.
Newton’s Second Law: “When a body is acted upon by a constant force, the resulting acceleration is inversely proportional to the mass of the body and is directly proportional to the applied force.” Applicability: Force = Mass X Acceleration. For this discussion of ballistics, the projectile’s mass is constant, so force and acceleration are directly proportional. Therefore, a change in acceleration results in an equal change in force.
Newton’s Third Law: “Whenever one body exerts a force on another, the second body always exerts on the first a force which is equal in measure, but opposite in direction.” Applicability: For every action there is an equal and opposite reaction as can be seen in felt recoil, pressure build-up from air resistance and terminal impact.
Why is this important to ballistics? All three laws act together to describe the interaction of forces that affect a projectile’s flight to target. At the moment the primer ignites the propellant and rapid gas expansion thrusts the projectile forward, the projectile experiences dramatic acceleration. The third law describes how the force required to propel the mass results in an equal and opposite force we know as felt recoil. The very moment the projectile leaves the barrel, friction from air resistance combines with gravity to simultaneously decelerate the projectile and alter its path in a modified parabolic arc (trajectory) toward the earth. As the projectile impacts the target, it decelerates from its residual terminal velocity to zero with an energy transfer that is equal to one-half of the projectile mass multiplied by velocity squared to produce a ballistic effect (third law). But there is more…
Projectile design significantly affects performance from ignition, travel through the bore, trajectory, and target impact. While there are many variables, I will briefly describe the characteristics common to most.
Ogive: This is the gradual radial reduction from the shoulder to the meplat or tip.
Shoulder: This is the transition point from the bearing surface to the Ogive.
Bearing Surface: This is the somewhat longer surface area along the length of the projectile that presses against the inside of the bore. The outside diameter of the bearing surface equals the caliber of the projectile.
Cannelure: As mentioned in Internal Ballistics Part III, the projectile is secured within the case by friction as the result of a process called crimping. Some rifle and pistol projectiles have a cannelure somewhere along the bearing surface. The cannelure is a set of tooling marks or series of indentations designed to better secure the crimp and prevent projectile set-back.
Heel/Base: Ranging from flat base through boattail (tapered), the shape greatly affects the amount of drag exerted on the projectile during its flight. Base drag is caused by the partial vacuum that occurs behind the projectile during its flight. A tapered, or boattail base reduces this drag. However, studies indicate that the benefits of a boattail are negligible under 200 yards. Therefore, handgun cartridges and short-range rifle cartridges do not benefit from a boattail projectile.
OK, now it is time for the heavy physics stuff. If you recall from our discussion on mechanical precision, barrel rifling will rotate a projectile along its center of mass and the barrel crown exerts the very last influence on the projectile as it exits the bore. In a perfect world, which doesn’t exist, the a perfectly concentric projectile with its uniform density dispersed evenly along its longitudinal center of mass travels down a perfectly aligned bore. It is provided the last bit of thrust with a perfectly machined crown as it enters a vacuum (no air resistance) and travels along a mathematically perfect parabolic arc. This process occurs so rapidly that most shooters would believe that a projectile exits the bore in a straight line toward the target. Again, Wylie E. Coyote’s version of physics does not apply. There is a brief period of instability as the projectile exits the barrel before all of physics principles align and stabilize the projectile. This period is called Transitional Ballistics.
Transitional Ballistics, Yaw, Precession and Nutation: Simply defined, Yaw is a deviation of a forward moving aerodynamic object from its longitudinal axis. Precession is an angular force applied to a rotating object caused by its torque. Nutation, literally “nodding,” is the opposing force in a rotating object that gradually “normalizes” along the longitudinal axis during projectile flight.
Wow! So what does that all mean? During the projectile’s travel down the bore, the rotational acceleration is controlled solely by friction applied by the rifling. As the projectile exits the muzzle, its rotational momentum will cause it to continue to rotate in the same direction as it moves under forward momentum. Keep in mind, however, that the bore axis of a firearm is tilted slightly upward in relation to the line of sight. Therefore, although the firearm appears level, the projectile exits the bore at a slight upward angle. The very moment the projectile is free of the bore, atmospheric resistance exerts pressure simultaneously against the nose and under the ogive of the projectile at the same time. These pressures combined with any imperfections in the barrel crown cause the projectile to yaw away from its center axis. Concurrently, air resistance across the full surface area of the rotating projectile exerts an angular force at 90 degrees in relation to the direction of rotation and the orientation of the longitudinal axis, which is precession. At the same time, the projectile’s release from controlled rotation in the rifling to “free flight” outside of the bore requires a short time for the rotational inertia to stabilize along the longitudinal axis. In this process, the nose of the projectile varies in a helical motion until it is dampened to stabilized ballistic flight, which is nutation.
OK… let’s try it again in plain English: There is a brief moment when a projectile leaves the barrel in which it is somewhat unstable. This instability is caused by a combination of the projectile’s rapid transition from controlled rotation to free rotation and the introduction of air resistance. During this period, the projectile “wobbles” in a helical pattern as it moves forward before it fully stabilizes. You can also see this in slow-motion replays of long football passes. When the football leaves the quarterback’s hand, it is rotating along its longitudinal axis, but is wobbling for a short distance before it stabilizes into and travels to the receiver (or not). The same is true for a projectile!
How does this affect the average shooter? Pistol shooters should know that these principles exist, but have little to be concerned with. The short length of pistol projectiles allow them to stabilize quickly and within just a few feet. Rifle shooters, on the other hand, fire longer projectiles that require up to 48 feet to stabilize. Matching the projectile shape and weight to the propellant charge and barrel rifling will minimize (but not eliminate) the intrinsic instability of the projectile as it leaves the muzzle. This will allow it to stabilize more quickly and efficiently allowing it to travel with greater precision and energy conservation to the target.
The bottom line is there are a lot of forces that simultaneously act on a projectile the moment it leaves the barrel and it takes a short time for the projectile to stabilize into predictable ballistic flight. Now that we’ve covered these details, we are set up to discuss how gravity, ballistic coefficient, wind and other atmospheric effects alter a projectile’s flight to target. Check back with us to continue the discussion with External Ballistics Part II.
Until then, stay safe and shoot straight!
– Howard Hall