Ballistic Coefficient Effects on Bullet Trajectories
For a given bullet fired at a known muzzle velocity, the ballistic coefficient of the bullet determines its trajectory. This is because drag is the strongest force acting on the bullet, and the ballistic coefficient governs the amount of drag. Strictly speaking, this observation is only approximately correct, but it is almost true for bullets which are well stabilized in flight.
The two trajectory parameters in which shooters are most interested usually are drop and remaining velocity as functions of range from the firing point. A question which comes to mind is, how strongly dependent are drop and remaining velocity on the ballistic coefficient? This question is very important when a shooter is trying to select a bullet for a specific hunting or target shooting purpose. As well see, the best bullet is not always the one with the highest ballistic coefficient.
Another question closely linked to the one above is, what effects on the trajectory are caused by small changes in ballistic coefficient? This question is important because there are experimental errors in measuring the ballistic coefficient of a bullet, and also because (as we will describe later) a bullets ballistic coefficient changes with its velocity. So, we need to know how serious a small error or a small change can be.
This section will answer the questions posed above, and it will also address one other question which often arises. If two bullets of different designs, calibers, and weights have the same ballistic coefficient, how do the ballistics of these bullets compare?
Effect of Ballistic Coefficient on Bullet Drop and Remaining Velocity
Ballistics tables for sporting arms are always computed for the case of flat fire, that is, with the gun barrel level. The results are good for small elevation angles, up to perhaps 10 degrees unless the range is very long. When a bullet starts out level, its path curves downward continuously under the action of gravity. Drop is always measured from a line going out the bore, and at any range the drop is the vertical distance between this line and the bullet path.
Drag always acts to slow the bullet down. It can be thought of as a vector tangent to the bullet path but pointed backwards along it. As the bullet path curves downward as the range increases, the drag vector tips upward and a small component of drag then begins to act along the vertical direction, parallel with but opposed to gravity. The main part of drag always acts in the level direction.
For modern bullets the trajectory curvature is never very large. Target bullet trajectories have a slope of the order of 1 to 2 percent at 1000 yards, when the trajectory starts out level and the muzzle and it gets rapidly smaller at shorter ranges. So, the conclusion is that not a very large part of drag ever acts in a vertical direction, if the trajectory starts out level. Pretty much all of the drop is caused by gravity, and what little drag does act vertically tends to oppose gravity, making the drop a little less than if it were caused by gravity alone.
The effect of ballistic coefficient enters mainly through the time of flight the drop at any range is nearly proportional to the square of the time of flight (assuming drag has a small effect). It is clear then that a bullet with a shorter time of flight to some range will have less drop than one with a longer time of flight. Time of flight in turn is affected by drag, since drag slows a bullet down. Since drag slowing gets less as
gets larger, larger
tend to make for less drop. However, time of flight also depends on muzzle velocity, so it is not possible to say that if you want small drop you simply pick a bullet with a large
. A large, heavy bullet typically has a pretty large ballistic coefficient, but it also has a fairly low muzzle velocity limit, so it can have more drop than some other small caliber, low
bullet which can be fired much faster.
To illustrate, refer to
Table 4.2-1(rifle) Table 4.2-1(handgun)
which shows remaining velocity and drop data for three hypothetical bullets. The first is a heavy bullet with very good ballistic efficiency so that it has a ballistic coefficient
equal to 0.6. The second is a bullet of medium weight and moderate ballistic efficiency with
equal to 0.4. The third is a light bullet with low ballistic efficiency and
equal to 0.2. Remaining velocity and drop are shown for three muzzle velocities, 3500, 3000, and 2500 fps.
First, let us compare the ballistic performance of these three bullets when all three are fired at the same muzzle velocity, say 3000 fps. Looking at the numbers in
Table 4.2-1(rifle) Table 4.2-1(handgun)
for the three bullets at the muzzle velocity of 3000 fps, we see that at both 500 and 1000 yd. ranges the bullet with the highest ballistic coefficient (
= 0.6) retains more velocity and shoots flatter (less drop) than the other two bullets. Similarly, the bullet with
= 0.4 retains more velocity and shoots flatter than the bullet with
These comparisons can be generalized. If we compare any bullets
fired at the same muzzle velocity,
the bullet with the highest ballistic coefficient will retain more velocity and shoot flatter than the others at any range. But we know that this type of comparison is not exactly fair. Thats because heavy bullets cannot be loaded to the same muzzle velocities as light bullets. So, we need to consider the muzzle velocity limits of different bullets when we compare their ballistics.
To further illustrate these points, lets take a specific example, the .308 Winshooting, and so a hunter or target shooter has a wide choice.
Tables 4.2-2 (a)
list trajectory parameters for six hunting bullets and four target bullets from Sierras line of .30 caliber bullets, which we have chosen only to illustrate our points with this particular cartridge.
Consider the hunting situation first. The bullets in
Table 4.2-2 (a)
range from the 110 grain Hollow Point with a ballistic coefficient of 0.188 at 2500 fps, to the 200 grain Spitzer Boat Tail with a ballistic coefficient of 0.552 at 2500 fps. The muzzle velocity for each bullet is a safe load a little below the maximum recommended in this Manual for the weight of the bullet. The table then shows the remaining velocity, energy, drop, and crosswind drift for a 10 mph crosswind, at ranges of 250 and 500 yards. Remaining velocity and drop shown in
Table 4.2-2 (a)
follow the same description given for
Table 4.2-1(rifle) Table 4.2-1(handgun)
. We have added two other important parameters, bullet energy, which depends on remaining velocity, and crosswind drift, which depends on time of flight, muzzle velocity, and wind velocity.
The first objective for a hunting cartridge is to deliver adequate energy downrange, and adequate energy of course depends on the size of the game. It is interesting to note that the muzzle energies of all the bullets in the table are not much different, but the downrange energies differ quote a lot. If a hunter is after varmints, the 110 grain Hollow Point bullet in
Table 4.2-2 (a)
delivers adequate energy out to 500 yards. For medium game, such as white tail or mule deer, the 110 and 125 grain bullets lose energy rapidly beyond 250 yards, and a heavier bullet should be chosen. For large game, like elk or bear, the 200 grain Spitzer Boat Tail delivers the most energy downrange, and it would be the obvious choice. At 500 yards this bullet retains more than half its muzzle energy because it has a high ballistic coefficient.
For smaller game, though, other objectives can be considered. It is very important to have a trajectory as flat as possible, because it is very difficult to judge distances in the field. It also is highly desirable to have minimum crosswind sensitivity. For varmints at ranges less than 250 yards, the 125 grain Spitzer is a better choice than the 110 grain Hollow Point because it is much less sensitive to crosswinds, although the two bullets shoot about equally flat out to 250 yards.
For medium game the choice is between the 150 grain, 165 grain, and 180 grain Spitzer Boat Tail bullets. The 150 and 165 grain bullets shoot about equally flat and have about equal crosswind sensitivities at ranges out to 500 yards. The 180 grain Spitzer Boat Tail has considerably more drop at longer ranges than either the 150 or 165 grain bullet, because it can be started out only at 2500 fps rather than 2700 or 2800 fps, and its sensitivity to crosswinds is only a little less than the other two bullets. The choice for medium game leans toward the 165 grain Spitzer Boat Tail because it retains between 5 and 10 percent more energy downrange than the 150 grain bullet, and it does not suffer as much drop as the 180 grain bullet.
The objectives for choosing a bullet for target shooting are very much different than for hunting. Bullet energy downrange is a consideration in target shooting only for the case of silhouette matches, in which the steel targets must be tumbled, not just struck, in order to score. In all target matches the ranges from firing point to the targets are fixed. Trajectory drop, therefore, is not a strong consideration because the rifle sights can be adjusted for the known ranges. The most important consideration for shooting in outdoor target matches is crosswind sensitivity, and minimizing crosswind drift is a major objective in choosing a bullet for target matches.
Table 4.2-2 (b)
has been prepared for the rifle silhouette game with the .308 Winchester cartridge, and four of Sierras Hollow Point Boat Tail MatchKing bullets are considered in the table. Ballistic coefficients are showded in this Manual) is listed for each bullet. The energies of all the bullets are sufficient to tumble, drop, and drift for a 10 mph crosswind are shown at 200 m (for the chicken) and at 500 m (for the ram the least crosswind drift, even though it starts out with the lowest muzzle velocity. For this specialh also is the heaviest bullet), and we should try to launch it with the highest muzzle velocity that we can load safely. This will assure that the crosswind sensitivity is minimized.
Note that crosswind drift scales directly with crosswind velocity. Thus, a 5 mph crosswind would cause half the drift shown in
Tables 4.2-2 (a)
, and a 20 mph crosswind would cause twice the drift shown. We will explain this in detail in a later section of this article.
All these considerations illustrate the point that our bullet choice depends on our objectives, and our objectives certainly are different for hunting or target shooting. Our objectives also may be different for certain kinds of hunting or certain kinds of target matches. It is not always best to choose the bullet with the highest ballistic coefficient to accomplish the purpose we have in mind.
Effect of a Small Change in Ballistic Coefficient on a Bullets Ballistics
A question often arises about the effect of a small change in ballistic coefficient on a bullet trajectory. The question sometimes comes up because there may be a small uncertainty in the value of the ballistic coefficient of a bullet, and it is natural to ask how that uncertainty affects the bullet trajectory. A change in altitude or atmospheric conditions can cause such an uncertainty.
To gain some insight refer again to
Table 4.2-1(rifle) Table 4.2-1 (handgun)
. For the low, medium, and high value of
(.200, .400, and .600), a 1% and a 10% change have been added to each value, and trajectories have been computed for all these values with three different muzzle velocities. The table shows the values of remaining velocity and drop at the two ranges, 500 and 1000 yds, for each case.
It is generally surprising that a change in
, even a relatively large change like 10 percent, causes such relatively small changes in remaining velocity and drop. The changes in these parameters are not in the same proportion as the change in
A couple of general observations can be made from the table. First, it can be seen that a change in
has a larger effect on remaining velocity at higher muzzle velocities than at lower ones. For example, when
equals .600, a 10 percent change to .660 causes the remaining velocity at 500 yds to increase by 69 fps when the muzzle velocity is 3500 fps, but only 55 fps when the muzzle velocity is 2500 fps. This is a general trend in the data.
The second observation is that a change in
has a larger effect on drop at lower muzzle velocities than at higher ones. For example, when
is .600, a 10 percent change to .660 causes the drop at 500 yds to decrease by .84 inch when the muzzle velocity is 3500 fps, but by 1.79 inches when it is 2500 fps. This is also a general trend.
Relationships between Different Bullets with the Same Ballistic Coefficient
Suppose that two bullets of different weight or different caliber have the same ballistic coefficient; how do the ballistics of the two bullets compare?
To start with, assume that the two bullets are fired with the same muzzle velocity. This gives a common basis for a generalized comparison. Since the bullets have equal
, they have equal drag decelerations, and they slow down equally fast. Consequently, if the two bullets start with equal muzzle velocities, at any range they will have equal velocities, equal times of flight, and equal drops. Unless they have equal weights, they will have different energies, however. Since energy is proportional to weight, the heavier bullet will have proportionately more energy at any range.
Usually the lighter bullet can be fired at a higher muzzle velocity than the heavier. In the case where the
are equal, the lighter bullet will always have the edge over the heavier one in velocity, time of flight, and drop, since it can be started at a higher velocity. The energy comparison cannot be generalized, however. The lighter bullet can catch up energy-wise with the heavier one if it can be driven at a high enough velocity to make up for the weight difference.
It is very hard to make any generalized comparisons for bullets with unequal
. Specific comparisons can be made after trajectories have been computed, and that is one of the uses of ballistics tables.