Shoot the Horizontal

    In my past posts I described trajectory calculations that show that only the horizontal distance matters when aiming a bow or riffle.  The horizontal distance is referred to as the ballistic distance by some range finder manufacturers. You should aim your weapon based on the horizontal distance even though the true distance to the target is longer. The calculations are confirmed by our experience shooting from a tree stand - aim based on the horizontal distance.  However our calculations and experience don't answer the question - Why does this work?   

Figure 1.  Shooting from elevation.
    Consider the figure to the right.  When you shoot down from a tree stand (or up from a gully) the total distance of the shot is longer.   This is because you are shooting the hypotenuse of two legs of a triangle and this can add as much as 40% to the shot distance for some extreme shots.  Unless you believe in some alternate dimension in the space-time continuum, the arrow must fly this extra distance and the extra distance means that the arrow takes up to 40% more time reach the target.  This also means that the arrow will fall further due to gravity.  Contrary to the hopes of some researchers, we don't know how to beat gravity and all objects fall the same distance in a given time.   A golf ball released from your hand will drop the same amount as an arrow or bullet shot at any velocity if all three "fly" for the same time.  Gravity is a constant for all projectiles and the longer a bullet or arrow is in the air the more it will drop.  It doesn't matter why the arrow the arrow takes longer to reach the target.   Arrows shot from below or above the target both experience more drop due to gravity compared to a horizontal shot.

    If the arrow drops more when shooting from above or below the target why dies it still hit the target?  Remember that the sight pins on a bow are set to provide the necessary upward arc to the arrow to compensate for arrow drop due to gravity.   The 30 yard pin on my bow provides a 1.5 degree upward arc to my shot to compensate for 7 inches of vertical drop that occurs during the 0.39 seconds of flight time.   No matter how high or low my vertical position in a tree, the accurate shot to a target 30 yards away in the horizontal uses the 30 yard pin.   Take out your bow and aim at targets above and below your shooting position.   Notice that the sight pins don't angle your bow upward relative to the Earth (the horizontal reference point) but relative to the line from your eye to the target (sight line).  The changing angle of the sight line is the reason that shots taken based only the horizontal shot distance hit the target.

    Remember from your high school physics that the total velocity of a projectile launched at any angle can be partitioned into vertical and horizontal velocities based on the launch angle.

Vx = Vtotal cos(launch angle)        Equation 1

Vy = Vtotal sin(launch angle)         Equation 2

When I take a 30 yard shot at a target on the level my 30 yard pin tips the bow upwards by 1.5 degrees.  The cosine of 1.5 degrees is 0.9996 (almost 1) so the horizontal velocity of my arrow is 239.9 ft/second, or 0.9996 times the launch velocity of 244 ft/second.  In this case almost all of the total arrow velocity is directed toward the target and only a tiny part of the velocity is directed upward producing a 7 inch high vertical arc along the path to the target.  This arc compensates for the drop in the arrow due to gravity.

    Now if take a 30 yard shot from 30 yards up in a ridiculously high tree the angle of the sight line is -45 degrees and the 30 yard pin will raise the bow by 1.5 degrees for a final angle of -43.5 degrees.   The cosine of -43.5 is 0.725 so the horizontal velocity is much less (177 ft/second).   The vertical velocity is almost as big as the horizontal velocity at -168 ft/second (headed toward the ground).   The reduced horizontal velocity is why it takes the arrow 27.5% longer to travel the 30 meter horizontal distance to the target.   The increased vertical velocity is the reason the arrow moves from high in the tree downward to the target on the ground.   So what about the drop???   The effect of gravity should be larger if the arrow spends 27.5% more time in the air.   The drop is larger, almost 14 inches due to the increased flight time.  The reason that the arrow still hits the target is that the arrows arc above the sight line from the archer to the target is also bigger, twice as big in this example.  When you shoot from an elevation above or below your target the 1.5 degree up angle from the 30 yard pin results in a larger upward arc relative to the target sight line.  The larger arc exactly compensates for the increased arrow drop due to increased flight time.

    I have spent many hours sitting in a tree stand thinking about this observation.   How can the arrow arc be larger if it is always referenced to the sight line between the archer and the target?   If I shoot 1.5 degrees above the sight line shouldn't the arc above the sight line be the same at any launch height?   NO.   The arc of the arrow is caused by vector sum of the horizontal and vertical velocities.  The arrow has an arc because the arrow is moving both up and away from the archer at launch.   As the arrow travels toward the target gravity slows the upward velocity of the arrow decreasing the angle of arrow flight from 1.5 degrees to zero degrees at half way to the target.  This is the top of the arc.  Past the half way point the velocity of the arrow is increasingly downward resulting in final angle of -1.5 degrees at the target.    The flight trajectory is not perfectly symmetrical due to aerodynamic drag, but it is remarkably close.  

    When shooting from above or below the target I still launch the arrow at 1.5 degrees above the sight line.  However, remember that the horizontal velocity is slower when the sight line is above or below a line parallel to the Earth.  It turns out that the decrease in horizontal velocity is more than the decrease in vertical velocity (both change as the sight line angle changes).  This means that the vector sum of the horizontal and vertical results in a longer, higher arc.  The arc in my shot from a 30 meter high tree is twice as high and 27.5% longer in length.   This is exactly what is required to hit the target.

What is going on here???  The simple answer is that trigonometric functions are not linear, and sin and cosine functions change by different amounts as the shot angle changes from 1.5 degrees to a new angle from elevation.  An interested student should plot the sin and cosine as a function of angle to observe this relationship for themselves.   

    The bottom line for the hunter is to shoot the horizontal distance (ballistic distance) irrespective of vertical elevation.  However, remember that the arrow or bullet will take longer to reach the target and the projectile will travel in a larger arc above the sight line from the hunter to target.  If you are planning to make a tough shot around branches or other obstacles the differences in projectile arc may become important.    For most mortal archers shooting from a 4 yard high tree stand, shoot the horizontal distance and focus on the other aspects of good marksmanship to produce an accurate, ethical shot.