Advanced equations

Before the advent of computers, mathematicians developed shortcuts and approximations for calculating trajectories. These shortcuts were often based on a principle called the “flat-fire assumption”. This assumption held that the geometry contained no extreme angles and that the horizontal downrange component of motion was the only important dimension of the problem; it assumed that the projectile was in level flight. This assumption leads to errors when the line-of-sight is angled upward or downward. Modern handheld devices have the compute power to do much better than this, but unfortunately many apps still rely on this “flat-fire” shortcut. 

In the early days, marksmen often used the “Rifleman’s Rule”. In this approximation, only the horizontal distance to the target was used in computing vertical drop. Multiplying the cosine of the “Look Angle” times the true distance to the target gave the effective horizontal distance.  This effective distance was used in formulating a drop correction. Marksmen treated uphill and downhill scenarios as equivalent. Modern day numerical physics shows that this shortcut quickly loses accuracy at extreme uphill or downhill angles, especially at distances beyond 1,000 yards. 

There are sophisticated techniques for solving differential equations using a computer that most computer programmers would not know. Only physicists and engineers with experience in scientific applications tend to know these techniques. They are difficult to master but they make good use of the powerful floating-point processors in today’s mobile devices, and they are able to pinpoint the point of impact that other products can only estimate.