Understanding Arrow Trajectory
The slower the speed of a projectile, the more it will drop over the course of its flight. To compensate for this drop, a projectile must be fired in an arched path to successfully strike the intended target. The arrow starts below the archer's line of sight, rises above it, and then drops back to his line of sight at the target. This curve in an arrow's path between bow and target is known as trajectory.
Trajectory is a crucial concern in bowhunting because a bowhunter simply can't escape its effects. People who contend that modern bows are so fast they're almost like rifles simply don't understand trajectory. Trajectory is directly related to the speed of a projectile, and the very fastest bows shoot about 300 feet per second (fps) compared to the fast rifle at 3,000 fps. An arrow fired from such a bow has a greater trajectory over 30 yards than does a rifle bullet over 300 yards.
Instinctive shooters who must visualize the path of their arrows to the target know the importance of trajectory. But sight shooters with an understanding of trajectory find it easier to make good sight settings and are better able to avoid branches and other obstacles. And to accurately shoot uphill and downhill, an understanding of trajectory is essential.
Hitting obstacles ranks as one of the major reasons for missed shots, especially for hunters pursuing whitetail deer and other forest animals. A clear understanding of trajectory can help you prevent many of these misses.
First, remember that an arrow starts below your line of sight-as much as 6 inches or more if you're a release-aid shooter with a low anchor point. When your line of sight is well above an obstacle, there is a tendency to assume the arrow will clear with no problem. But because the arrow remains below your line of sight until it is 4 to 5 yards away from the bow, it can slam into a nearby obstacle if you don't elevate high enough. Many hunters have blown easy shots by hitting stumps, rocks and other objects directly in front of them.
Learn to visualize arrow trajectory so you can cleanly shoot under or over limbs and other obstacles. It's possible, of course, to compute trajectory mathematically, but it's not necessarily practical and may not help you in the field. Some simple steps on the target range will help you actually see the trajectory of your arrows.
Make sure your bow is sighted in at regular intervals, such as 20, 30, 40 and 50 yards. To determine the trajectory of your arrows at the halfway mark, you'll be shooting target groups for each sight pin from a distance equal to half their sighted distance. First, stand 10 yards from the target and aim at the bullseye with your 20-yard pin. Shoot several groups, measuring the distance of each arrow above the center of the target; then average these numbers. This average is your midrange trajectory at 20 yards.
Shoot additional groups at 15 yards with your 30-yard pin; 20 yards with your 40-yard pin; 25 yards with your 50-yard pin. With increasing distance, you'll find that your arrows hit progressively higher, giving you a clear picture of the trajectory at each distance. For a bow shooting about 220 feet per second (fps), the midrange trajectory at 20 yards will be roughly 3 inches; at 30 yards, 6 inches; at 40 yards, 12 inches; and at 50 yards, 20 inches. At higher arrow speeds, the trajectory will be less; at slower speeds, greater. This method doesn't give peak trajectories, which are slightly closer to the target, but it does give you a practical picture of midrange trajectory.
How do you put this knowledge to practical use? Let's assume you've drawn a bead on a buck standing 30 yards away, and there is a tree limb hanging in your path 15 yards out, directly in the sight path between your bowsight and the deer's chest. An archer with no knowledge of trajectory might believe he has to maneuver to shoot either above and below the branch, but from your range test experience, you know that your arrow will be 6 inches above your line of sight at the midway point in its flight. Fire away; your arrow will clear the branch cleanly. Similarly, if the offending branch is 6 inches or so above the sight line, many hunters would shoot, unaware that the arrow is likely to strike the branch. You, however, will seek a new shooting position, looking for a sight line which ensures that the arrow's midpoint trajectory is a safe distance away from the tree branch.
Remember also that when a sight pin rests on an object at the same distance as the sighted distance of the pin, the arrow will likely hit the object. Imagine, for example, that you've put a pin on a deer 30 yards away using your 30-yard pin. But your 20-yard pin falls on a limb about 20 yards away. Your shot will miss the deer, and the arrow will hit the limb.
At any given distance, you have some leeway when estimating range, and this leeway is called a sight window. Understanding the sight windows for your bow can help with shot selection. Again, the best way to learn about sight windows is by thorough practice-range work.
To gauge the window for your 20-yard sight pin, for example, shoot at an 8-inch target, which is roughly the size of a large deer's kill zone when it is standing broadside. Start at 20 yards, and move closer, 1 yard at a time, always aiming at the center of the target with your 20-yard pin. The arrows will gradually move up until they begin hitting above the 8-inch target. That's the minimum end of your sight window. Now start again at 20 yards, and move back, 1 yard at a time, until your arrows hit below the circle. That's the maximum end of your sight window.
Repeat this test for each sight pin; for the 30-yard pin, start at 30 yards; for the 40-yard pin, start at 40 yards; and so forth. Move from the starting point toward the minimum end of each sight window; then move back from the starting point to determine the maximum end. The sight windows will become proportionately shorter as you move to the pins sighted for longer distances, due to the increasing arc of the trajectory.
Shooting on an Angle
Bows are generally sighted in for horizontal shots, where arrow trajectories are predictable. However, when you shoot from a high tree stand or on an uphill or downhill slope, the rules for trajectory change.
At a given distance, gravity has the greatest effect on trajectory for an arrow flying horizontally, and has gradually less effect as the shooting angle is raised or lowered. The reason for this is that the trajectory arc is determined by the horizontal distance the arrow travels, not the overall distance. When shooting at steep angles, either up or down, aiming for the actual distance to the target is a mistake, since this measurement may be considerably larger than the horizontal distance that governs the arrow's trajectory. For example, if you shoot at a target 40 yards away down a steep hill, using your 40-yard sight pin, your shot will probably go high of the mark. At steep angles, you must learn to compensate by aiming low.
Contrary to common opinion, this principle applies equally to uphill and downhill shots. It's true that uphill trajectory begins to differ from downhill trajectory at some point, because gravity slows an arrow flying up faster than one flying down. But within standard hunting distances-50 yards and less-the difference is not significant. For practical purposes, then, slant range can be computed identically, whether the slope is uphill or down.
The trick, of course, is determining how low to aim at different distances and slope angles. You can calculate this information geometrically, but to do so requires a calculator, electronic rangefinder, clinometer for measuring the angle of slopes and a good head for mathematics.
If you choose the mathematical method, you can simplify this work by using a chart like the example shown here. A complete chart for all practical shooting distances will help you make clean kills at steep angles. To put such a chart to practical use, however, you must know the distance to the target and angle of slope. Steep slopes make range estimation difficult at best, so a good rangefinder can help here. With practice you can learn to estimate slope visually, or you can buy a compass with a built-in clinometer to measure slope. Or, you can buy an expensive forester's clinometer, which measures angle of slope precisely.
To be honest, using mathematics to calculate shooting adjustments is not very practical for most hunters. In the field, game animals rarely give you enough time to make the measurements and computations, and few hunters are enthused by the prospect of carrying a rangefinder, calculator and printed charts into the field.
Most shooters find that simple practice gives them a feel for how upward and downward angles affect arrow trajectory. You can practice by shooting from a tree stand set to different heights, for example. At each height, shoot several groups at varying distances from your stand.
Flattening the trajectory of an arrow reduces the need for precise range estimation, streamlines the process for shooting over and under obstacles and simplifies adjustments needed when shooting at upward and downward angles. Several factors-fletching style, arrow weight and air density-can affect trajectory, but arrow speed has the single greatest influence. For this reason, manufacturers and archers continually seek ways to make their bows faster and faster.
You can increase arrow speed in several ways. One method is to reduce arrow weight. Each 5-grain reduction in arrow weight provides a speed increase of about 1 foot per second (fps). Thus, if you reduce arrow weight by 100 grains, you increase speed by roughly 20 fps.
Arrow speed can also be gained by increasing the draw weight of the bow. A 1-pound increase in draw weight yields an increase of about 2 fps in speed. Thus if you crank draw weight up by 10 pounds, you gain 20 fps in arrow speed.
Finally, shooting a bow with harder cams can increase speed. At a given draw weight, hard cams can provide 10 to 20 percent more speed than round wheels.
Up to a point, speed is good. But remember that the faster the arrow, the more "critically" it shoots. That is, faster arrows are more susceptible to uncontrolled oscillations and are more difficult to shoot with accuracy. At some point, the loss in accuracy and consistency that comes with ultrafast arrows may negate any gains you get from the flattened trajectory. Faster arrows also make it harder to tune the bow, especially when the arrows are equipped with broadheads.
Remember, accuracy is the number-one ingredient in clean kills. Lightning-fast arrows do you no good if they don't consistently hit the target.
Using Geometry to Calculate Arrow Trajectory
Here's how geometry can be used to calculate how arrow trajectory changes when shooting on an incline:
Visualize the hunting situation as a right-angle triangle, with two perpendicular legs; (A), the distance from the hunter to the ground; and (B), the distance from the animal to the ground directly below the hunter. The sloped hypotenuse, (C), is the distance from the hunter to the animal. In essence, the drop of an arrow fired along the inclined hypotenuse will equal the drop of an arrow traveling the horizontal distance (B).
For example, imagine you're on a cliff, where you spot a buck standing below at a 45-degree angle to your line of sight. Your rangefinder shows that the distance along the sloped line to the buck is 30 yards. Using a basic formula from geometry, called the Pythagorean Theorem (A2 + B2 = C2), you can calculate that side B, representing the horizontal distance to the deer, is roughly 21 yards. To hit the deer in the kill zone, then, you would use the 20-yard sight pin, not the 30-yard pin.
Trajectory refers to the curved path an arrow follows from bow to target.
Determine Sight windows for each of your sight pins. The archer shown here is finding the sight window for the 20-yard pin. In this case, the sight window was 8 yards-from 16 to 24 yards.
Sample Trajectory Adjustment Chart
Charts such as this one, taken from The Complete Book of Rifles and Shotguns by Jack O'Connor, can help you calculate trajectory changes when shooting on an incline
THE EFFECTS OF SLOPE on trajectory can be explained by the physics of gravity. Gravity exerts its influence only against the horizontal distance of an arrow's flight. When you shoot horizontally at a target 40 yards away (left), gravity exerts its influence for a full 40 yards, and the arrow's drop follows expectations. But when you shoot downhill (right) or uphill at a target 40 yards away, the horizontal distance can actually be much less, as shown above. In this situation, if you shoot using the 40-yard sight pin, your arrow will hit high on the target (inset), since it has not had time to complete its normal trajectory drop for 40 horizontal yards.