Virtual Mass - The key to understanding arrow speed.

    In my last post I showed that accurate range estimation is the single biggest factor in hitting your target in a hunting situation.  However, shooting errors due to poor range estimates can be reduced by shooting faster arrows resulting in a flatter arrow trajectories and decreased down-range arrow drop.   Lots of archery products claim to increase arrow speed, but in reality only the potential energy of the bow (draw weight and draw length) and arrow mass will change arrow speed to any significant extent.   Purchasing a new, more powerful bow is a big investment.  This post explores how arrow speed is influenced by the mass and design of the arrow.

    Thomas Liston in his book The Physical Laws of Archery describes modifying his bow with an "over-draw" arrow rest so that he could shoot shorter, lighter, and therefore, faster arrows.  The gain in velocity was not what he had hoped because not all the potential energy stored in the bow is transferred to kinetic energy of the arrow.  This post describes how to calculate the velocities of arrows with different masses.  To understand arrow velocity we will begin with a simple energy balance for the bow and arrow. 

Potential Energy of the Bow  =  Bow Hysteresis + Kinetic Energy of the Bow + Kinetic Energy of the Arrow  (1) 

    The potential energy of the bow, PE, is the energy put into the bow by pulling the bow string.   The first post in this series provides details on measuring the potential energy of the bow.  When the archer releases the arrow some of the potential energy is lost to friction (heat) in the bow which we call Bow Hysteresis, Ehist.  The important aspects of Ehist is that it is a property of the bow, is independent of arrow speed, and can't be reduced without designing a better bow.  Ehist is about 10 to 20% of the potential energy put into the bow.  The rest of the potential energy goes into accelerating the arrow and accelerating the bow (kinetic energys).  Clearly the bow string and arrow both have to be moving the same speed during arrow release, and the kinetic energy of an object is related to both the objects mass and velocity.

Kinetic Energy (KE) = 1/2mV2    (2)

For the bow and arrow the complete conservation of energy is then,

PEbow = Ehist + KEbow + KEarrow =  Ehist + 1/2maV2 + 1/2mbV2  (3) 

    It is easy to calculate the KE of the arrow from the arrow mass and arrow velocity as the arrow leaves the bow.  However, calculating the KE of the bow is a bit more tricky since not all parts of the bow are moving.  Paul E Klopsteg in his 1943 paper on the Physics of Bows and Arrows proposed the concept of bow Virtual Mass to help solve this problem.  He assumed that any moving part of the bow moved at the same speed as the arrow.  Rearranging equation 3, we can calculate the kinetic energy of the bow, 

PEbow - Ehist - KEarrow = KEbow  = 1/2 mbV2  (4) 

where mb is the virtual mass of the bow.   Since we know the velocity of the bow is equal to the velocity of the arrow we can calculate the Virtual Mass of the bow that must be moving during arrow launch.  


equation 5 may also be rearranged to yield, 


providing the same result.   The concept of virtual mass is clearly an approximation since not all moving parts of the bow move at the speed of the arrow!   However, as we will see, the construct of virtual mass works remarkably well at predicting the kinetic energy of the bow.   The table below shows how I performed the calculations for two different arrows with three different tips producing arrow-tip combinations with six different masses.

    The total mass of the Beeman, ICS Bowhunter arrow with 100 grain tip was 432 grains (28.0 grams) as shown in the green column.  The potential energy of the bow was 105 joules (77 ft #).   Assuming the bow is 81% efficient, the bow hysteresis is 19.9 joules.   The actual speed of the arrow was 71.5 m/s (235 ft/s) giving a kinetic energy of 71.6 joules.  Plugging these numbers into equation 5 gives a virtual mass of 5.2 grams or 80 grains.   The virtual mass of the moving parts of the bow is about 1/5th the mass of the arrow.   This seems reasonable if the string is most of the moving mass.   

    Now the problem with this calculation is that the inital estimate for the bow hysteresis was a WAG (wild ass guess).  A solution to this problem is to make several arrow velocity measurements for different arrows with different masses.  I did this for six different arrow-tip combinations (blue columns) with masses from 374 to 475 grains.  This provides six equations and two unknowns (mb and Ehist).  Rearranging equation 5 it is possible to predict a new velocity for any arrow with mass, ma, given a constant bow virtual mass, mb


    Ehist is assumed to be a constant percentage of the bow's potential energy, PEbow.  I solved for Ehist by adjusting the bow efficiency until the predicted arrow velocity matched the observed velocity for all arrow masses (the error row is small).   The results are shown in the attached excel file and in the figure below.   Using a bow efficiency of 81% and a virtual mass of 5.2 grams I get excellent predictions (better than 1 ft/s) of arrow velocity for different arrow-tip combinations.   Inspection of the table and graph shows that it is possible to tune the arrow speed for my bow from 226 to 252 ft/s, about a 10% change in velocity.  If you want to do better than 10% then it is time to buy a new bow.

    Do these numbers make sense?  The bow efficiency seems a bit low compared to other published efficiency estimates, but this is probably due to some kinetic energy of slowly moving parts being assigned to Ehist.    If this bugs you, feel free to send me an email and I will explain why I think this OK.  Similarly, the virtual mass of the bow is fairly low, but this only includes the bow parts moving at the same speed of the arrow.  In the end, the ability of the model (red line in the figure above) to fit the experimental data (blue points) to better than 1 ft/s means the model works over a range of practical bow speeds and arrow masses.

    Using equation 7 is is also possible to calculate the expected arrow velocities for a similar bow at different draw weights and arrow masses.  As shown in the figure below, arrow speed can vary from 180 ft/s for a bow set to a 40 pound draw weight with a 500 grain arrow to 315 ft/s for an 80 pound bow shooting 300 grain arrows (an unsafe combination with commercially available arrows).  

    The table above also has a column for dry fire of the bow.  This is the expected string velocity when the bow is fired with no arrow (ma=0).  The dry fire velocity is almost 600 ft/s which is why it is a really bad practice to dry fire a bow!

    Bow manufacturers report the IBO speed on new bows to allow customers to have a common reference to compare bow performance.   IBO speed is measured with a bow set to 70 pound draw weight using a 30 inch, 350 grain arrow.   The figure above provides a qualitative measure of how arrow speed changes for conditions different than the IBO reference.  My bow has an IBO speed of 305 ft/s which is impossible to obtain because the draw weight is set to 62 pounds and my draw length is only 29 inches.  Even if the bow was set to 70 pounds draw weight the bow would only shoot at 280 ft/s.   

    This post may leave you thinking that shooting light arrows is a good option for improving bow performance due to increased speed.  However, as will be discussed in the next post, arrows must have sufficient spine or stiffness to launch reproducibly from the bow.   The mass of the arrow tip also determines the balance point of the arrow.   Often the best broadhead groups are obtained with an arrow balancing point about 12% forward of the arrow center.  Lighter is not always better.  But a slightly lighter arrow with a heavier tip might be a perfect combination.

Whitney King,
Jul 10, 2011, 6:12 PM