In the world of archery energy is conserved, and the energy to launch the arrow comes from the archer.  As you draw the bow string you add potential energy to the bow.  No mater what type of bow we are shooting you must put energy in to get energy out.   It is not less work to draw a compound bow, it is simply easier to hold the bow at full draw because the cams reduce the draw weight at full draw.  A compound bow adds potential energy in the middle of the draw cycle while a conventional bow adds most of the potential energy at the end of the draw cycle.

We will begin our discussion of the bow draw curve by defining a few terms and then will look at the draw curves for a compound and conventional bow.

Draw Force - The "draw weight" on the string as you pull back the bow.   The draw force changes as the bow string is drawn and the shape of the draw force curve is dependent on the type of bow.

Draw Distance - The distance that the bow string is drawn from the rest position.  The total draw length of the bow is the draw distance plus the brace height.

Potential Energy - The energy stored in the bow at any draw distance.  Potential energy is a function of draw force and draw distance.

Kinetic Energy - The energy transferred from the bow to the arrow upon release.   For a modern bow this will almost equal the potential energy, with some (10%) of the potential energy lost to moving the bow and making noise.

The figure below shows the draw curve for a PSE Stinger compound bow and a conventional, long or recurve bow, both with a total draw weight of 60 pounds.   Notice that the X axis is nock travel distance from full draw so the zero position is at full draw and 19.5 inches is the point of arrow release.  The draw force of a conventional bow increases linearly to full draw, in this case 60 pounds.   In contrast, the compound bow rapidly reaches a draw force of 60 pounds in the first six inches of draw, remains above 50 pounds force for the next nine inches, and then lets off to about 10 pounds force at full draw.

The potential energy stored by each bow type is the area under each curve in units of ft-pounds or newton-meters.  To calculate the area the draw curve was divided up into 40 half inch intervals of draw distance and multiplied by the average draw force of each interval (simple numerical integration).  The sum of these 40 intervals is the total potential energy stored in the bow with units of force times distance (ft-pounds in the US and newton-meters for the rest of the world).  An example spreadsheet is attached to the end of this post.

The results for the two bow types are listed below.  Notice that the compound bow stores about 40% more potential energy than the conventional bow with the same draw weight.   Of course, this means that it takes 40% more energy to pull a compound bow, but because of the let off is much easier to hold the bow at full draw.

 Compound Conventional Units Potential Energy 73.0 50.0 ft-# Potential Energy 98.9 67.8 N-m Arrow Mass 436 436 grains Arrow Mass 0.0282 0.0282 kg Launch Velocity 274.5 227.3 ft/s Launch Velocity 83.8 69.3 m/s

In a perfect world, the potential energy stored in the bow is completely transferred to kinetic energy of the arrow.

potential energy = kinetic energy = 1/2(mass)(velocity)2

In the real world, modern bows are 85-92% efficient.  Assuming 100% efficiency for now, and knowing the mass of the arrow, it is easy to calculate the arrow velocity [KE in ft # = (mass in grains)(V in ft/s)2/450437]. The 60 pound compound bow will launch the arrow at 275 ft/s while the conventional bow of the same draw weight will launch the arrow at 228 ft/s.

A more subtile aspect of the draw curve for both bow types is the effect of draw curve shape on arrow acceleration.   The acceleration of the arrow may be calculated at any point during the arrow launch using the relationship between force and acceleration

Force = mass * acceleration

This means the acceleration curves look the same as the force draw curves after being scaled up or down for mass. A conventional bow has maximum acceleration at the point of arrow release.  The arrow must have sufficient spine for this acceleration..   In contrast, the compound bow accelerates the arrow more gradually and then reaches maximum acceleration for a longer distance.   Since the maximum acceleration is the same the arrow spine may be the same, but the longer maximum acceleration results in greater arrow velocity.

The next post will discuss the time and velocity profiles of an arrow released from compound and conventional bows.

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Whitney King,
Jun 17, 2011, 6:05 PM