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Bow Mechanics


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. 

A diagram showing the calculation for projectile energy.


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. 

A diagram showing how the release angle affects where the arrow travels.


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.

Fspring = -kx

A diagram illustrating Hooke's law


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.

work = PE

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. 

A diagram illustrating elastic potential energy


elastic potential energy = 12kx2


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.

n = KEarrowPEbow= 12 * mass *velocity212 * k * x2


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

A diagram showing the different modes of motion of an arrow.