Stored Energy 
    4H Archery

All archery bows need an archer to draw them. When drawn, the 
bow stored energy. This is called potential energy because it has the potential to do work. 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
                              1            2
	E(stored) = ---- * M * v
           2                 2
where v = v * v.

How much energy is stored in the bow ?

When not drawn, the string is a distance equal to the brace (string) height from the front of the bow, typically 8 inch. Let x(brace) = 8 inch. When fully drawn, x(drawn), the bow exerts maximum force which we measure with a weight scale. Weight (W) is measured in kilograms (kg) in Europe where they use SI or Scientific International units. Weight is the force of gravity on an object which is its mass (M) times the acceleration of gravity (g), or W = M*g. In SI units g = 9.8 m/s/s, meters per second per second. 

	Force(drawn) = mass * gravitational acceleration
	         =  (20#) 9.091 kg * 9.8 m/s/s
	         =  89.1 Newton.		
 	Distance = ( x(drawn) - x(brace) )/12 inch/ft 
		  = (24 inch - 8 inch )/39.37 inch/m
		  = 0.4064 m

During drawing the bow, the force increases from initially zero to the maximum, almost linearly for a recurve bow. So the average force
is 	Force ave = (Force(initial) + Force(drawn) )/2
		   = (        0          + Force(drawn) )/2 
		   =  Force(drawn)/2.

The energy stored in the drawn bow is then

	Energy(drawn) = Force(drawn) * Distance / 2
		          = 89.1 Newton * 0.4064 m
		          = 18.1 Joule.

The Joule has the units of Newton*meter, and the abbreviation J.

Examples:
1) A 20# bow with a brace (string) height of 8 inch drawn to 24 inch has a stored energy of 
    0.5*20#*(24-8)/12ft = 13.33 ft*lb = 18.1 Joule, where 1 ft*lb = 1.3558 J.

2) By comparison, a 5 cm rubber band drawn to twice its original length has a stored energy of 
    0.5#*0.5# * 5 cm /2.54 cm/inch /12 inch/ft = 0.04101 ft*lb = 0.0556 J.
(# = lb)

(c) 2009 Lee F. Donaghey 91115