The Physics of Archery

By Kate Bassett, Naomi Burnell,
Jeanne Case, and Josh Walker
The 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.
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)
the distance of the offset from the initial position of the string
• The Draw Weight (W) is
proportional to the
Draw Length (Δx), thus
this follows Hooke’s
• Hooke's Law Deformation of an
object (the limbs of the
bow curving back) is
proportional to the
force (W) causing it.
• 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.
• 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)
• 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.
The 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 at
If 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:
What We Learned
• Uses many of the concepts • Hooke’s Law
we learned in class.
• Having 2 limbs effects the
– Kinetic Energy
calculations for energy and
– Potential Energy
draw weight
– Conservation of Energy
• How the conservation of
– Calculations of initial velocity,
energy is bad for a bow
the distance an arrow travels
when dry fired
(x and y), weight, Force,
kinetic and potential energy. • The spine or stiffness of an
arrow creates the Archer’s
Your Challenge Should You Chose
to Accept it:
• Go to:
• Use your Physics knowledge to master the art
of shooting a bow and arrow with a mouse!
Works Cited

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