Pellets In Drag

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Many keen airgunners enquire as to the ballistic coefficient (BC) values of certain pellets, without any explanation as to where the BC value came from, or how it was obtained. And, often, the quoted BC values for the same pellet are different! So what exactly is a BC, how can we get a correct figure and, perhaps more importantly, is it of any real use to airgunners?

Simply put, a BC is merely a ratio comparing the ballistic retardation of a pellet with a standardised projectile – most commonly the G1, a bullet shaped form that measures up at one-inch in diameter with a weight of one imperial pound (figure 1).

FIG1_G1

Figure 1: G1 standard projectile

As you can see, the G1 standard projectile is shaped nothing like a conventional airgun pellet – and the problem in quoting a single value of BC for a particular airgun pellet is that we are assuming the drag of the pellet reacts in the same way as that of the G1 projectile. However, this isn’t actually the case.

Let’s take a hypothetical example of drag law – figure 2 – which shows the drag coefficients for the G1 and our airgun pellet (at 600fps). A drag law is merely a table or a graph which shows how the drag coefficient (expressed as Cd) varies with speed. The G1 drag law is shown by the red line, with the drag of a typical roundhead pellet shown as the green line.

DRAG LAW

Figure 2: The Drag Law shows the relationship between the G1 Projectile (Standard and Modified) and a standard airgun pellet

As you can see, they are very dissimilar – but one of the functions of the BC is to effectively move the G1’s drag law up the scale to match that of the pellet’s drag law. If we take the G1 law and move it to match our pellet’s value, then we get the blue drag curve – though it’s still not much like our pellet’s drag curve (in green).

This is because a BC value is only valid at one speed or velocity, but you never see the velocity at which the quoted BC is supposed to be valid. Some sources seem to use a fixed value of the drag coefficient (Cd) instead of the G1 drag law, possibly in an effort to simplify things. (A ‘constant’ Cd value of 0.319 is popular for some reason.) The problem doesn’t get resolved, though, because a constant Cd is still not a good match to an airgun pellet’s Cd as the velocity varies. Hence a BC will still only be valid at one, specific velocity.

One way to calculate the BC value of a particular pellet is to measure its speed at two different points, a known distance apart. The appropriate calculations can then be applied to get an estimate of the BC value needed to match those two measured velocities – but just how accurate is the BC calculated in such a way? Well, it will all depend on the speed of the pellet and the distance between the two measuring points! Table 1 shows the speed of a .177 calibre, 8.7-grain pellet leaving the muzzle at 788fps. Its BC values are shown for both the G1 drag laws (at this velocity) and the ‘constant’ 0.319.

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As you can see from the table, the value of this pellet’s BC varies, depending on the range, the associated speed used to calculate it, and the basis of the reference Cd used for its calculation (either G1 or 0.319). So, in order to use the BC to tell us anything about the external ballistics of our airgun pellet, we’ll need some kind of trajectory software – and luckily there’s plenty to be found on the internet, most of which will give you an estimate of BC if you input the range and pellet’s velocity. The calculated BC can then be used to compute the residual velocities at range, helping us to calculate drop-over-distance and, accordingly, where best to set our sights.

However, as Table 1 proves, we need to choose our range carefully – using the wrong BC will introduce errors. Take the figures in Table 1. Using the BC computed from measuring the pellet’s velocity at the muzzle and 10 yards, the calculation to determine the pellet’s drop at 50 yards equates to 9.8 inches – whereas the ‘true’ answer is actually 9.65 inches. Okay, not much difference, but it will get bigger as we increase the range. The calculated pellet speed at 50 yards using the 10 yard BC value is 495fps, but the ‘real value’ is actually 517fps; in energy terms, that’s 4.7ft/lb versus a true 5.2ft/lb.

And, as muzzle velocities increase, the errors will be compounded due to the curvature of the Cd/velocity curve shown in figure 2. If the same calculations are used for a lightweight, high-drag pellet with a flat front, rather than a heavy roundhead pellet, the results are different. This time the true drop at 50 yards is 10.7 inches and the speed is 412fps – but the calculated values using a 10-yard BC calculation are 11.5 inches and 368fps. Quite a difference – but even this inaccuracy is nothing compared to the error which will be induced by basing our BC on the wrong reference. As you can see in Table 1, there can be a vast difference in the BC value depending on the reference drag coefficient used.

So the moral of the story is don’t mix up your software sources; always use the BC associated with the software you’re using. Yet even though a fixed BC can (by way of software) give us some idea about our sights and how much energy there may be remaining in the pellet at a given range, it can’t give a full indication of the wind effect on the pellets. Yes, a BC may indicate a large or small horizontal crosswind effect, but it cannot tell us how much the pellet will fly high or low in a crosswind. Neither does it really tell us if the pellet is a low-drag or high-drag design. The weight of the pellet will have a first order effect on the BC, as will the pellet’s drag coefficient – hence comparing drag coefficients between two pellets using BC is only valid if they have the same weight.

So, I have to conclude that the BC of an airgun pellet is of fairly limited value, particularly when it is quoted without giving any detail of the basis for the value. As I’ve demonstrated in Table 1, a BC based on a fixed drag coefficient of 0.319 will look much better than a value based on the G1 drag law for most pellet speeds. Thus we can have two completely different BC values for the same pellet… in addition to the differences due to range and speed!

That is why in military ballistics, we don’t use BCs – and haven’t done for the past 50 years, since the use of computers became common. We now exclusively use the Cd value, because the results are much more robust.

– Miles Morris

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