Bow Mechanics

PROJECTILE MOTION

A form of motion where an object moves in a symmetrical, parabolic path. The path that the object follows is called its trajectory.

Figure: The trajectory of an arrow with the height, range, and launch angles indicated. 

ARROW TRAJECTORY

An arrow is a projectile. An arrow begins its flight at an upward angle, peaks at some point, then arches downwards until it hits the target. To hit the bullseye, the archer must consider the release angle ( θ ) and the range.

Figure: When an archer changes the release angle, it may cause the arrow to undershoot or overshoot the target. 

HOOKE'S LAW

A bow is an application of a two-armed spring. Hooke's Law states that the force (F) needed to stretch or compress a spring by a given distance (x) is proportional to the distance.

ELASTIC POTENTIAL ENERGY

Elastic potential energy is energy stored as a result of the deformation of an elastic object, such as the stretching of a spring or drawing a bow. Potential energy is equal to the work that must be done to move an object from one point to the other.

In terms of a bow, this can be visualized as the area under the draw curve:

Figure: Simplified draw curve to show that the area under the curve (which forms the shape of a triangle) is equal to one half of the spring constant (k) multiplied by the distance (x) squared. 

Thus,

BOW EFFICIENCY

When a bowstring is released, the potential energy from the bow is transformed and transferred into kinetic energy in the moving arrow. Although bows are elastic, they are not 100% efficient.

THE COMPLEXITY OF ARROW MOTION

Including the Archer's Paradox, there are at least 14 modes of movement associated with an arrow in motion.