THE PHYSICS OF ARCHERY

By Kate Bassett, Naomi Burnell, Jeanne Case, and Josh Walker

The BowThe Bow is a basic machine called a “two-arm spring.” When an arrow is drawn back the bow stores mechanical potential energy in the part that bends.

Draw Weight

The force that the archer applies to the bow is called the Draw Weight. This force bends the limbs of the bow and adds elastic potential energy. Thus, the bow is like a spring that stores the energy to put into the arrow.

W = 2F (cosα) (cosβ)

Draw Length- the distance of the offset from the initial position of the string

• The Draw Weight (W) is directionally proportional to the Draw Length (Δx), thus this follows Hooke’s Law.

• Hooke's Law - Deformation of an object (the limbs of the bow curving back) is proportional to the force (W) causing it.

Potential Energy

• Usually you can use E = Fx / 2 to find the potential energy but because the bow is thicker in the middle than at the ends you need to use E = eFx / 2 where e is the efficiency. You divide by two because there a two limbs of a bow.

Transferring Energy• When the arrow is released if all of the potential

energy becomes kinetic energy. (ET = EK )• You can use ½mav2 = ½eFx (Conservation of

Energy)because the kinetic energy is what gives the arrow its velocity. (ma=mass of arrow)

WHAT IS DRY FIRING?• Dry firing is releasing the drawn bow

string without a knocked arrow.• If the bow is dry fired all of the kinetic

energy stays in the bow instead of transferring into the arrow.

• Because the bow was not made to handle this much energy it can crack, fracture and brake.

• Though the bow does use some of the kinetic energy when an arrow is fired (½ kmbv2, where mb is the mass of the arrow), this is not enough to damage it.

Archer’s ParadoxThe archer’s paradox is the flexing of the arrow as it flies toward its target. The arrow needs to have just the right amount of flexibility, also called spine, to flex and then flex back toward its target.

A video of this phenomenon can be found athttp://youtu.be/wGNslUNBrEMAnd http://youtu.be/WzWrcpzuAp8

SpineIf the arrow is too stiff, not enough spine, it will go off to the side the arrow head is pointing to. If it has too much spine, or flexibility, then the arrow will be deformed to much and go to the other side.

Hitting the Target

• Without air resistance the following formulas can be used:

• Range: dx= (vi2 sin2θ) / g

• Max Height: dy= (vi2 sinθ) / 2g• Where:

– vi is the initial velocity of the arrow calculated with (eFx/ma)½

– ma is the mass of the arrow– and g is gravity.

Why we Chose Archery:• Uses many of the concepts

we learned in class.– Kinetic Energy– Potential Energy– Conservation of Energy– Calculations of initial velocity,

the distance an arrow travels (x and y), weight, Force, kinetic and potential energy.

What We Learned• Hooke’s Law• Having 2 limbs effects the

calculations for energy and draw weight

• How the conservation of energy is bad for a bow when dry fired

• The spine or stiffness of an arrow creates the Archer’s Paradox

Your Challenge Should You Chose to Accept it:

• Go to: http://archeryphysics-mrsmith08.awardspace.com/bored.html

• Use your Physics knowledge to master the art of shooting a bow and arrow with a mouse!

Works Cited• http://www.archeryhuntingequipment.com/exclusivecontent/physic

s-of-archery• http://www.mrfizzix.com/archery/• http://archeryphysics-mrsmith08.awardspace.com/

• Videos:– http://youtu.be/wGNslUNBrEM– http://youtu.be/WzWrcpzuAp8

• Game: http://archeryphysics-mrsmith08.awardspace.com/bored.html