Arrow Velocity

When the bow is drawn and the arrow released, the bow propels the arrow forward.

Before the draw the string is stretched taught, but the string is a distance equal to the brace height from the front of the bow, and there is now force applied to the arrow. When drawn, the string is an arrow-length away from the front of the bow, and the force is equal to the bow weight. When released, the force decreases to zero as the string moves back to the brace height.

At full draw, the stored energy in the bow is equal to the Bow-weight x (Arrow-length - Brace-height). This is called potential energy because it has the potential to do work. After release, the string returns to the brace-height, and the force is zero. So, for the release, the average energy is E = [Bow-weight x (Arrow-length - Brace-height) + 0 ] / 2 = [Bow-weight x (Arrow-length - Braced-height)] / 2.

The energy is converted into kinetic energy by launching the arrow which then flies with a velocity v and mass M.
A small amount of energy is lost in heating the bow, so the energy balance is
	E(stored) = E(kinetic) + E(heat).
Because the heat energy is small, we can approximate this as
	E = M * v * v / 2

So, the arrow velocity is 

           v = sqrt (  2 x E / M )

For example, for a 20# bow with an arrow length of 24 inch and a brace height of 8 inches, the stored energy is 90 N.

For an arrow mass of 20 gm = 0.02 kg, the arrow velocity is

           v = sqrt (  2 x 90 N / 0.02 kg ) = 95 m/s, where "sqrt" means the quare root.

Or, in terms of the bow properties, 

	v = sqrt ([Bow-weight x (Arrow-length - Brace-height)] / M  )

How fast do your arrows fly ? To determine this, you will need to measure your arrow-length, your bow-weight,  your brace-height, and your arrow-mass. 

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(c) 2010 Lee F. Donaghey