Reloading: External Ballistics

We’ve accelerated the projectile, made it spin and waved goodbye at the muzzle. From here to the target things get complicated. The path of any free moving, un-powered projectile is its trajectory. In a gravity-free vacuum it would simply continue to travel in a straight line trajectory at undiminished speed. However, gravity, air density, wind speed and direction conspire to screw things up. The rule is simple: bullets cannot travel in straight lines due to external forces such as drag and gravity trying to pull it from the sky, resulting in a variable rate curve.

If the bore of the rifle was actually aligned with the target, the bullet would miss. Given that the sights or scope are generally above the line of the bore it means that the bullet will cross the line of these sights twice. The first generally happens within 20 to 30 metres and is called the initial point, the second as the trajectory descends and is known as the zero point.

GRAVITY SUCKS

With all other things being equal, the greatest drop in trajectory to a target is when the muzzle and target are at the same height above the centre of the earth. If the target is at the same distance but either higher or lower than the muzzle (the slant range) then the trajectory will have to be reduced by lowering the muzzle. Why? Because gravity exerts its greatest percentage influence on a projectile that is travelling perpendicular (right angles) to its direction of influence… the greatest amount of drop for a given distance. Just to qualify that, imagine firing from a first floor window to a target in the garden… a component of the drop has already been introduced by the direction in which you are aiming. Gravity has not changed, but over the bullets flight its impact is reduced. The drop correction is marginally less when shooting downhill than the equivalent uphill angle as gravity is a slightly larger component in the maths. However, if a target is raised or lowered vertically from a known linear distance from the muzzle then the distance from the muzzle to the target will increase… the simple hypotenuse theory applies. This is known as the slant distance. In extremis this additional component will counteract the first factor.

The shape of the bullet trajectory reflects the deterioration of velocity due to drag and the continual acceleration caused by gravity at 32.174 ft/sec² (9.807 m/sec/ sec) towards the centre of the earth. As drag slows the projectile so the acceleration due to gravity becomes a larger component of the trajectory. If the bullet was fired at a sufficient altitude then the bullet trajectory would finally decay to only a vertical component – drop due to gravity. Put another way, on level ground, if you drop a bullet from the height of the muzzle and FIRED a bullet at exactly the same time, they would both strike the ground at the same time.

A BIT OF A DRAG

Drag is the resistance created by the atmosphere; if you’ve put your hand out of the car window at speed then you’ll know what I mean. At 1000mph the air is like jam…. at the 18,000 mph re-entry speed of a space capsule it’s nearer to the perceived consistency of water. Drag increases at the square of speed. Other factors that will determine external ballistic performance include altitude (air density), air temperature, humidity and wind. All other things being equal, a rifle bullet will impact between ½ and 1 moa higher for every 5,000 ft increase in elevation. The colder the air, the more dense, the more drag!

To cheat this resistance we modify the shape of the projectile. The relative efficiency of these shapes is distilled into a numeric value, its’ Ballistic Coefficient or BC. It is a function of mass, diameter and drag coefficient (cd) and quantifies the ability of the bullet to resist drag. Simplified, BC is the result of dividing the Sectional Density by the Form Factor. Sectional density is derived by dividing the mass of the bullet in pounds or kilograms by the square of the calibre in inches or metres. The result will be in either lb/in² or kg/m². The Form Factor is a standard model number that is the result of dividing the bullet drag coefficient (cd) by a reference G1 bullet model. For a flat-based, blunt-nose bullet this number is 1. We can rework the equation to an easier format by dividing the Mass by the result of the bullet Diameter squared multiplied by the Form Factor. Unlike cd, the bigger the number the better!

BY THE BOOK

Some manuals, mostly those produced by bullet manufacturers, quote a value of BC for each design, weight and calibre. Of more importance, they then interpolate this figure into their ballistic tables giving time of flight, drop and other data against distance. Two variants of BC exist, whilst only differing by 2%, mixing them will skew your long range calculations. The simple solution is to use the actual maker’s data for your bullet rather than using the quoted coefficient as a generic in any tables. Getting down to the finer details, there are at least half a dozen bullet drag curve models, each based around a particular style or design of bullet.

Other forces apply to; precession, the Coriolis effect, the Magnus effect and transonic shock. All but the last are generally of little significance and anyway, are an unavoidable part of bullet flight dynamics. However, transonic shock is something to be avoided wherever possible. As its name implies, it is the transition of the shock wave created by supersonic travel through the atmosphere from the tip of the bullet to the base as its speed degenerates to subsonic. As the shock wave moves along the bullet it can induce sufficient yaw to destabilise it to the point where it may even tumble. This effect is particularly common at extended range as spin induced gyroscopic stability has kept the bullet pointing (more or less) along the bore line, an increasing angular deviation from its actual direction of flight (trajectory). It also affirms the fact that the rifling induced spin has been substantially retained for the duration of flight.

USING THE NUMBERS

Ballistic coefficients range from 0.050 to 0.560. We use it in conjunction with muzzle velocity to calculate the trajectory and from that, the maximum practical range for accurate use. I’ve quoted this before but it’s worth repeating. Let us compare a couple of 165-grain Speer .308 Spitzer bullets. The square base has a BC of 0.433 and from an MV of 2700 has a residual velocity of 1747 fps at 500 yards. The sleeker Boat Tail version has a BC of 0.477 and from the same MV is still travelling at 1834 fps at 500 yards. That’s a 4% improvement. As the zero range extends, so the benefits of the higher coefficient will increase. We can tweak exterior ballistic performance by careful bullet choice and optimum speed… and an awareness of the factors that will influence our winning shot.