Understanding Ballistics of Long-Range Shooting

Long-range shooting of 1,000 yards to ranges of well over a mile is all the rage today, and manufacturers are putting huge resources into marketing and development of everything long range.

But technology is only as good as the shooter’s knowledge of the ballistics impacting every pull of the trigger. We have all heard the shooting and trajectory considerations that have to be made for long-range shooting: muzzle jump, spin drift and Coriolis Effect (what I like to call earth-based effects). But how many of us know what the magnitude of these effects are and physically why they occur?

Let’s explore each one of these effects, why they occur and just how much impact they have. Most of the calculations of effects were done with the Hornady 4DOF ballistics solver with a Sierra 168-grain MatchKing projectile.

The Right Hand Rule

There is a reality of physics that has to be understood to visualize many of the physical things that happen to a projectile in long-range flight. A spin-­stabilized projectile is a rapidly spinning object that for all practical purposes is a gyroscope and will behave like a gyroscope. This gyroscopic effect is what gives an otherwise aerodynamically unstable projectile stability, preventing it from tumbling in flight. Gyroscopic stability is fairly simple. It is the resistance of a rotating body to a change in its orientation.

This gyroscopic stability is a wonderful thing physics has provided us shooters, but as in most things in life, it comes with side effects. The principle side effect that shooters are concerned with is that gyroscopically stabilized objects have some nonintuitive responses to outside forces.

In particular, any gyroscopically stabilized object will respond to an external force 90 degrees to the direction of the input force. This is easy to visualize by using your right hand to show what is called the right-hand rule (see Figure 1).

Put your right hand in the configuration shown. Your index finger represents the projectile and points in the direction of travel of the projectile. Your middle finger represents the direction of a force input to the projectile. Your thumb then shows the direction of the response of the projectile to the force input. Practice this for a few minutes because we will be using the right-hand rule to illustrate several of the long-range shooting effects. You can rotate your hand to get different directions of force input and reaction.

Muzzle/Aerodynamic Jump

Muzzle jump is a phenomenon that occurs when the projectile experiences a sudden crosswind. This most commonly happens when the bullet leaves the muzzle and is immediately hit by a crosswind. The greater the crosswind, the greater the amount of the muzzle jump. Go back to the right-hand rule illustration. If the crosswind is from the right, the net response of the projectile will be to move up.

If the crosswind is from the left, the response of the projectile will be to move down. The effect is an angular change of the path of the projectile from the point it is impacted by the force. So how much effect does muzzle jump have on a projectile? Table 1 shows that the muzzle jump for the Sierra 168-grain MatchKing is strictly a function of the value of the crosswind.

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This response of the projectile can also be a factor when shooting in the mountains while shooting off a ledge or cliff. If you fire your rifle and the projectile passes over a cliff or ledge into free air but experiences an updraft of wind at the edge, according to the right-hand rule, the projectile will jump to the right. If there is a down draft, the projectile will jump to the left.

Spin Drift

Spin drift is an effect that is another by-product of a spinning, gyroscopically stabilized projectile. Spin drift obeys the right-hand rule. Before we get into the specifics of spin drift, we have to have a short discussion on projectile stability.

As a projectile travels downrange, its stability or resistance to changing its orientation steadily increases. There is very little that causes the projectile’s spin rate to decrease. As the bullet flies downrange and relatively rapidly loses its velocity, the spin rate hardly changes. The projectile becomes steadily more and more stable. This increase in stability will play a prominent role in spin drift at long ranges.

There are several causes of spin drift. The first is a direct result of a spin-­stabilized projectile. Any spin-­stabilized projectile fired from a right-hand twist flies with the nose slightly above and to the right of its line of flight. This is called the yaw of repose. Figure 2 shows this in a greatly exaggerated depiction.

As I said above, as the bullet flies downrange, the stability steadily increases and so does the yaw of repose. There is a small but steadily increasing force from the airflow acting on the bottom of the bullet because the yaw of repose is steadily exposing more and more of the bottom of the projectile to the oncoming airflow. This acts just like a crosswind coming from below the projectile. Figure 3 shows actual values of the yaw of repose of the Sierra 168-grain MK as it flies downrange and becomes more and more stable.

The yaw of repose is very small but increasing. The projectile is actually not flying exactly point first. As this yaw of repose increases so does the force on the bottom of the bullet. If you use the right-hand rule, you see that a force acting on the bottom of the spinning projectile will cause it to move to the right.

The second cause of spin drift is caused by the rapidly increasing stability of the bullet and the trajectory of the projectile at long ranges. As a projectile is shot to longer and longer ranges, the maximum height of the trajectory becomes higher and higher, and the fall of the projectile gets steeper and steeper near the end of the trajectory. At the same time, the projectile is much more stable.

Remember that gyroscopic stability and its value is the measure of the resistance of the projectile to any change in its orientation. What’s happening is that at long range when a bullet reaches the maximum height in its trajectory and begins to rapidly fall, it is rather sluggish in wanting to nose over and follow the line of flight. This results in rapidly increasing side force on the bullet from below and corresponding increased drift to the right. An exaggerated depiction of this is shown is shown in Figure 4.

This effect was a significant problem with long-range artillery in World War I because they did not understand this effect at the time. Some artillery pieces had twist rates that were too fast, so projectiles were not impacting nose first on the fuse and were not exploding. This accounted for a lot of dud projectiles. Fast twist rates have been shown to give better projectile performance at long ranges. A faster twist also increases spin drift.

How great an effect is spin drift? Table 2 shows the spin drift of the Sierra MatchKing from the muzzle to 1,750 yards. The table shows two different twist rates to illustrate that as the bullet spins faster, the yaw of repose and stability is greater. This also means spin drift is greater.

Earth-Based Effects

“Coriolis effects” is used as a catch-all for all of the seemingly strange effects on a projectile because of the spherical spinning earth. I like to use the term “Earth-Based Effects” because there are several different things going on here. There are two effects of the earth that are hard to separate unless you only shoot either north/south or east/west. Usually you are shooting somewhere in between.

Let’s look at shooting east or west first. Shooting east or west causes the projectile to impact high or low at extreme ranges. The earth rotates to the east at approximately 1,000 mph at the equator. If you fire to the east, the velocity of your projectile is working with that rotational velocity. For the short time of projectile’s flight, it wants to get farther away from the earth because of the greater centrifugal force. Your projectile will hit a little high.

Conversely, if you fire west, your projectile is now working against the earth’s rotational velocity, feels less centrifugal force and wants to go closer to the earth and impact a little low.

This effect is real. We had to prove it to ourselves during my tenure at Hornady. We set up a test where we shot at 1,500 yards directly east and then fired directly west carefully monitoring the wind and its effects. With a 7mm Remington Magnum firing a 175-grain bullet, we measured a 7.5-inch difference in point of impact between firing east and west.

Shooting north, the projectile is departing from a larger circumference, faster rotating point on the earth than the target. The bullet keeps that higher velocity to the right, or east, as it flies because there is nothing to change this velocity. As your bullet is flying, the target is not rotating quite as fast as the projectile’s initial velocity to the right, or east, and as you see it, the projectile will impact to the right of where it was aimed.

Shooting south, the opposite occurs. The projectile has a higher velocity to the left, or east, than the target, which is at a smaller circumference on the earth. As you observe the flight of the bullet, it is moving to the left, or east, more rapidly than the target and will impact left of where it was aimed. This can all be a little hard to get your head around, but remember the earth’s rotational velocity is the highest at the equator and gets slower and slower as you go towards the poles. Figure 5 shows an exaggerated view of what is happening.

Much has been said about how significant Coriolis Effects are and that they have to be accounted for. The truth is, with small arms, the distance over the surface of the earth is so small, relatively speaking, that these effects are largely insignificant. We’re not talking 16-inch naval guns firing 25 miles with minutes of flight time. For small arms, they can virtually be ignored and are covered up by many other much larger effects.

The most significant earth-based effect is the change in elevation firing to the east or west. Table 3 shows trajectory differences for the Sierra load fired in the cardinal compass directions at a totally incredulous range of 2,800 yards so the differences would be big enough to show. The drift column is the combination of spin drift and “Coriolis Effects.” As you can see, any differences in trajectory are very small other than the difference between firing east or west.

Hopefully after reading this you will have a better understanding of the unique effects associated with long-range shooting. Most of these effects are not large enough to consider at ranges under 600 yards. Muzzle jump and spin drift are rather straightforward to understand once the physics of what’s happening are understood. Earth-based effects can be difficult to understand. Get a globe and try to picture what is happening and you will understand what is occurring.

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