## External Ballistics Part III – Understanding Trajectory

Thank you for returning to our third and final discussion on External Ballistics. In this article, I will discuss the elements that affect projectile trajectory, such as sonic vs. sub-sonic velocities, ballistic curve and “zeroing” the firearm. This last article on external ballistics ties other articles in this series together in order to provide you with an understanding on how to leverage trajectory and consistently place your shots where you want them. This also establishes a solid baseline for the following articles on terminal ballistics, which describe projectile effects on impact.

### Trajectory

Trajectory is simply defined as the curved path a projectile travels from point of initial velocity to impact. As we discussed in previous articles, upon exiting the firearm bore, the projectile’s initial velocity, momentum, and direction are affected by the external forces of gravity, air resistance, yaw, precession, and nutation. These external forces combine in a unique way to send the projectile along a relatively predictable path. Understanding the effects of these forces provide the shooter with the ability to combine accuracy with mechanical precision.

This may be a good time to dispel a common myth. Many have described projectile trajectory as “rising” during its flight to target. Although the projectile travels in a modified parabolic arc, the myth of projectile “rise” is not true in relation to the bore axis (line from the bore to the target). The projectile initially exits the bore in a straight line. The moment gravity and air resistance act on the projectile, it is “pulled toward the earth” and the projectile follows a downward arc. Yaw, nutation, and precession guide the tip of the projectile in the direction of this downward arc which keep it oriented toward the target. Therefore the projectile never “rises” above the line of the bore axis. The line of sight and the bore axis are NOT parallel. The bore axis and the line of sight are offset to compensate for projectile trajectory. Although subtle, the line of sight is a straight line between the shooter’s dominant eye and the target… but the bore axis is tilted slightly upward to ensure the projectile trajectory terminates on the intended target. Herein lies the source of the misperception: the projectile never “rises” above the bore axis… but it most certainly rises above the line of sight.

Also, note that I’ve described projectile trajectory as a modified parabolic arc instead of a true parabolic arc. If gravity was the only external force acting on a projectile, then it would travel in a true parabolic arc. However, air resistance causes drag on the projectile and “slows it down” throughout its travel to the target. The further the projectile travels, the longer air resistance affects the projectile and thus decays the parabolic path to target and increases the bullet drop.

### Sonic vs Sub-Sonic

“Sonic speed,” or the speed of sound in the atmosphere is of particular interest to shooters. The speed of sound is the rate at which small pressure disturbances will be propagated through the air and this propagation speed is solely a function of air pressure. The speed of sound is directly related to temperature and air density at altitude. In general terms, sonic flight is 1,125 feet per second (f/s) at 68 degrees F at sea level (767mph, or 1 mile in 5 seconds). The speed of sound increases as the temperature increases, and decreases as temperature decreases. In most cases, we can use 1,120 f/s as sonic flight for a projectile.

So why is this important to shooters? We’ve discussed how air resistance creates drag, slows projectile velocity, and decays the parabolic arc. At velocities below the speed of sound, air is considered incompressible. Therefore, air resistance simultaneously builds at the tip and along the entire surface of the projectile (laminar flow). However, at supersonic speeds (above the speed of sound) there is no build-up of air resistance at the tip of the bullet and the projectile travels with greater efficiency. Projectiles traveling above 1,120 f/s maintain greater aerodynamic efficiency. When drag eventually reduces this velocity below 1,120 f/s, aerodynamic efficiency is lost, parabolic arc decays, and trajectory drop increases.

### Ballistic Curve

Now that we have solidified the fact that the projectile does not “rise” in relation to the bore axis, but it does, in fact, follow an arc from the point of origin to the striking point in relation to the line of sight, we can now confidently describe this as the ballistic curve. From the shooter’s point of view with the sights aligned to the target, the upward orientation of the bore in relation to the sights or scope sends the projectile in an arc that “rises” along an ascending branch, reaches its maximum ordinate (max ord) and then follows a downward path along the descending branch until it reaches the point of impact on the target. The constant effect of gravity combined with the varying effect of air resistance (in relation to projectile velocity) alter the path of the descending branch in relation to the ascending branch… i.e. the drop on the descending branch is more dramatic than the rise on the ascending branch. Note that high velocity rifle projectiles fired at long range can experience trans-sonic ascending branch decay as they transition from efficient supersonic aerodynamic flight to less efficient subsonic flight. Altogether, this describes the ballistic curve.

### Firearm Zero

A rifle or pistol is “zeroed” when the point of impact matches the point of aim. Most pistols are zeroed at 25 yards. Rifles can be zeroed at any distance in relation to the intended target. ** Applicability**: Once a pistol or rifle is zeroed at a distance, engaging a target closer or farther than the zero can alter the point of impact in relation to the point of aim. Above, we discussed that the projectile follows a ballistic curve in relation to the line of sight. If a pistol is zeroed at 25 yards, which means that the point of impact matches the point of aim at this distance, and a target is engaged at 10 yards, the point of impact can be “higher” than the point of aim due to impact at the ascending branch of the projectile’s flight. If the same pistol engages a target at 50 yards, the point of impact can be “lower” than the point of aim due to impact at the descending branch of the projectile’s flight. Similar principles are true for rifle shooters and the difference between point of impact and point of aim are more pronounced due to the greater distance traveled, higher max ord, and greater effect of arc decay on the descending branch. In practice, shooters should know their firearm’s zero and make either sight/scope adjustments or apply the proper sighting “offset” to ensure the projectile hits the intended point of impact.

Many online resources provide ballistic calculator software. Some of them are free and some incur a charge. In many cases, you get what you pay for. Hornady’s web site provides a free ballistic calculator which calculates basic data points that many shooters could find useful. Just as a test, I entered my AR-15 data into the calculator. Using a 69gr hollow point boat tail .223 projectile with an initial velocity of 2,700 f/s zeroed at 100yards, the Hornady ballistic calculator computed the following.

Range (yards) |
Velocity (fps) |
Energy (ft.-lb.) |
Trajectory (in) |
Come UP in MOA |
Come UP in Mils |
Wind Drift (in) |
Wind Drift in MOA |
Wind Drift in Mils |

Muzzle | 2700 | 1117 | -1.5 | 0 | 0 | 0 | 0 | 0 |

100 | 2409 | 889 | 0 | 0 | 0 | 0 | 0 | 0 |

200 | 2136 | 699 | -4.5 | 2.2 | 0.6 | 0 | 0 | 0 |

300 | 1882 | 542 | -16.7 | 5.3 | 1.5 | 0 | 0 | 0 |

400 | 1649 | 416 | -38.8 | 9.3 | 2.7 | 0 | 0 | 0 |

500 | 1441 | 318 | -73.8 | 14.1 | 4.1 | 0 | 0 | 0 |

What this means is that at 100 yards, the projectile would impact the target at the point of aim at a velocity of 2,409 f/s. At 200 yards, it would impact 4.5 inches lower than the point of aim at a velocity of 2,136 f/s and would require a sight/scope adjustment of 2.2 minutes of angle elevation. At 500 yards, the projectile would impact the target 73.8 inches lower than the point of aim at a velocity of 1,441 (approaching trans-sonic) and would require a sight/scope adjustment of 14.1 minutes of angle elevation. NOTE: these are simply computer calculations and should be tested to verify and add to a comprehensive data book.

Using the same calculator, I entered my 1911 data using a 230gr lead round nose .45ACP projectile with a ballistic coefficient of .207 and initial velocity of 772f/s. The Hornady calculator computed the following:

Range (yards) |
Velocity (fps) |
Energy (ft.-lb.) |
Trajectory (in) |
Come UP in MOA |
Come UP in Mils |
Wind Drift (in) |
Wind Drift in MOA |
Wind Drift in Mils |

Muzzle | 772 | 304 | -0.5 | 0 | 0 | 0 | 0 | 0 |

25 | 760 | 295 | 0 | 0 | 0 | 0 | 0 | 0 |

50 | 748 | 286 | -3.3 | 6.2 | 1.8 | 0 | 0 | 0 |

75 | 737 | 277 | -10.4 | 13.2 | 3.8 | 0 | 0 | 0 |

100 | 726 | 269 | -21.6 | 20.6 | 6 | 0 | 0 | 0 |

Although the table doesn’t calculate any distances closer than 25 yards, you can surmise that the point of impact would be higher than the point of aim at distances closer than 25 yards, match the point of aim at 25 yards, and fall below the point of aim by 3.3 inches at 50 yards, 10 inches at 75 yards, and 21.6 inches at 100 yards. Since this pistol has an adjustable rear sight, I can either add elevation at the rear sight or leave it zeroed at 25 yards and elevate my point of aim corresponding to the degree of total elevation required to hit the target. Both require practice!

An internet search for the term “ballistics calculator” or “ballistics software” will produce a rather extensive list of resources. Hell, there are even smartphone apps for “handy” ballistic calculations. For the example used above, Hornady’s ballistic calculator can be found at the following link: http://www.hornady.com/ballistics-resource/ballistics-calculator

### Conclusion

At this point, we’ve debunked the myth of the “rising” projectile and honed our understanding of trajectory. Come back next month when we put it all together and discuss range estimation, wind deflection, ballistic slope error (shooting uphill or downhill), and defining minutes of angle. In the mean time, stay safe and shoot straight!

~ Howard Hall