The Long and Short of Hitting the Mark

    In the last post I detailed the physics of arrow flight including the real effects of aerodynamic drag on the trajectory of an arrow.   These calculations will not make you a better archer, but they do provide significant insight into the techniques and equipment that can improve archery success.  While turkey hunting the other day with Mike Guarino of Maine Wilderness Tours we got into a discussion on the appropriate correction for shooting at a target above or below the archer.   For the bow hunter, I argue that estimating range is the single biggest factor in hitting your target and that shooting elevation has almost no effect on shot placement.  This blog details the calculations used to support this claim.   

    Consider the cartoon below.  How much error will result from errors in range estimation to each target?   How much should an archer compensate for shooting up hill or down from a tree stand?  These questions are easily evaluated by calculating arrow trajectory as a function of bow elevation.

    The exact solution to these questions is possible using Newton's equations of motion in a frictionless world.  In the real world, however, aerodynamic drag is significant and I use the numerical techniques detailed in the last post to do the calculations.   Briefly, the calculations start with launch angle and arrow velocity.  From the velocity I calculate the arrow deceleration due to drag, and from the launch angle I calculate the arrow deceleration due to gravity.  These "accelerations" change the X and Y (horizontal and vertical) velocity of the arrow.  These new velocities change the arrow flight path.  After milliseconds of flight time the calculations are repeated and a new position is obtained.   This process is easily repeated hundreds of times in an Excel spreadsheet (attached below).   A plot of the X and Y position of the arrow for each small increment of time yields the complete flight path.  The figure below shows the trajectory of an arrow shot at a target 30 meters away.  The initial velocity of the arrow is 260 ft/s and a drag coefficient of 0.0054 1/m.

Figure 1.  Arrow trajectory for a 30 meter target using an arrow launch velocity of 260 ft/s.

    Inspection of the trajectory tells us many useful things.   One, the arrow starts about 3 inches below the plane of the target.   This is because you sight the bow using your eye, and your eye is above the arrow on the bow.  A typical eye to arrow distance is 9 cm or 3.6 inches.   In the first 15 meters of flight the arrow rises past the plane of the target, crosses the archers sight line (you can always see the arrow rise during a shot), reaches a maximum height about half the distance to the target, and hopefully hits the target at an elevation of zero - the bullseye.  The red and blue lines on the plot show the effect of drag on the arrow.   Aerodynamic drag significantly slows the velocity of the arrow.   The slower the arrow the less kinetic energy the arrow carries to the target.   The purple line is the percent kinetic energy of the arrow as a function of distance.   At 30 meters the arrow has 73% of the initial energy provided by the bow.   In future posts I will describe how to experimentally measure the bow speed and aerodynamic drag for individual bows and arrows.   For now, using an estimate of your bow speed and my estimate of aerodynamic drag (0.0054 1/m) will allow any archer to use the attached Excel spread sheet to calculate arrow trajectories and kinetic energy.  The next figure is the same calculation for a bow shooting arrows at 320 feet/s.  During my turkey hunt, Mike as shooting some fancy arrows that were more aerodynamic than my standard carbon arrows with conventional fletching. We haven't tested Mikes arrows on the range, but I give him a 5% decrease in drag because the arrows "looked cool" and will use a drag coefficient of 0.0051 1/M in the calculations.

Figure 2.  Arrow trajectory for a 30 meter target using an arrow launch velocity of 320 ft/s.

    The results of the calculations are interesting in two respects.  One, shooting a bow with 25% more speed results in a arrow trajectory that is twice as flat.  Bow speed makes a big difference in arrow trajectory!   Two, a five percent change in the drag coefficient is inconsequential in arrow performance as shown the nearly identical relative (percent) kinetic energy plots in both simulations.  If I am looking for improved archery performance I would put my money into bow speed, but read on to see what really matters to most hunters - hitting a target under hunting conditions.

    The amazing thing about almost all bows on the market is just how good they are at launching arrows.   If you put a bow in a shooting machine and shoot ten arrows, each arrow will hit the same spot.  The physics and mechanics of bow action and arrow flight is very reproducible.  Most errors in arrow flight is caused by the archer, and the biggest error in shooting in real-world situations is judging target distance.  To get an idea of the errors in judging distance look at the arrow trajectory in figure two.   The trajectory plot shows the arrow height at any distance.  If the actual target distance is shorter than the 30 meter aiming distance the arrow will hit high. Conversely, an arrow shot at an actual target distance longer than 30 meters will hit low.  Figure three shows the high and low errors for a deer target at 25, 30, and 35 yards shot using the 30 yard pin.  (Notice I am mixing meters and yards!  The scientist in me prefers meters and most American archers are still using yards.   Good thing we are not designing a space telescope.)

Figure 3.  Arrow shot placement for a deer at 25, 30, and 35 yards shot using the 30 yard pin.

If the deer is at 25 yards, 5 yards closer than expected, the arrow will hit 4 inches high.   If the deer is at 35 yards the shot would be 6.3 inches low, a very significant miss.   In figure four I show the calculated the errors in shot placement for different shooting distances.  The gray band in the figure is the generally accepted error limit for hunting White Tailed Deer.   The diagram shows that archers that can't judge range to better than 5 yards should not attempt a shot longer than 30 yards.   It also shows why some archers use a single 20 yard pin when hunting deer.  This pin would provide a reasonable shot placement for target distances from 10 to 25 yards

Figure 4. Vertical shot errors for five different target distance errors computed for a 450 grain arrow launched at 260 ft/s.

    The diagram also demonstrates a very nonintuitive fact, shot error due to errors in target range estimation increase exponentially with distance.   A five yard error in estimating target range at 100 yards will result in a 30 inch vertical error in shot placement.  This is only a 5% error in estimating target distance!   In contrast, a 20% error in estimating distance at 20 yards will have still have every arrow hitting the kill zone of the deer.  The qualitative reason for the increase in errors with distance is the curvature of the arrow trajectory.   The greater the curvature of the trajectory the more variation in vertical distance with small changes in horizontal distance.   

    The down-range curvature of the arrow trajectory is less for faster bows.   Figure five is the same error plot calculated for a 320 ft/s bow.

Figure 5. Vertical shot errors for five different target distance errors computed for a 450 grain arrow launched at 320 ft/s.

Using the same vertical error band, the ethical shooting limit for a 320 ft/s bow is extended to 40 yards.  Compared to any other modification to your bow, increasing speed will have the most significant improvement in vertical accuracy assuming you can still pull and hold the more powerful bow.

    The final consideration in these calculations is the effect of shooting from below or above the target.   A shot from a tree stand is typically 2 to 4 meters above the target.   It is logical that changing the shooting height would change shot placement in much the same way that errors in target distance influence shot placement.   However, figure six shows that errors due to shooting height are almost insignificant.

Figure 6. Shot error due to shooting from above or below the target as a function of horizontal shot distance.

    Shooting above and below the target has almost no effect on arrow placement if the horizontal distance is used as the target distance.   If you consider that it is desirable to hit the target a bit high if shooting from above, the errors shown in the figure 6 become even smaller.   The answer to Mike's original question is that shooting height makes no real difference to the archer.   Aim your bow based on the horizontal distance as shown in the introductory cartoon and your shot will be true.  For best results,  purchase a laser range finder and measure the horizontal distance to the target to within a yard.  From a tree stand use several nearby trees to measure distance along your shooting lanes.   However, don't measure distance to the base of the tree, measure distance to a spot on the tree that is the same height as your stand.   It does pay to increase your bow speed if you plan on taking any shots past 30 yards.  Finally, use the attached program to test your own shooting conditions.  The results may surprise you.

A final thought - What happens if your misjudge the range when shooting from a tree stand five meters above your target?  Now things get a bit more complicated and this question will be a good topic for future posts.  For now, we will focus on getting the maximum possible speed from your existing equipment.
Whitney King,
Jun 17, 2011, 6:12 PM