<- continued
We have discovered the first control parameter for our archery system—the kinetic energy that the system produces. We control this by our bow choice; what type of bow are we using (trad or compound, etc.), what is the bow's draw weight, how efficient is the bows cam system, etc. Once we have settled on a bow, the kinetic energy we can expect to obtain out of the system is effectively fixed. If you are fiddling with arrow weight in an attempt to "maximize" your kinetic energy, you are doing it wrong. So why then do so many archers feel that kinetic energy is "unimportant" when considering penetration issues? The answer is simple. Any bow that is legal for hunting purposes will produce enough kinetic energy to achieve enough penetration to kill an animal for which that bow is a legal method of taking. That’s why the bow is legal! If the bow did not produce enough KE to kill efficiently and quickly, then the bow would not be legal as a method of take. Furthermore, the amount of KE required to deeply penetrate a large game animal is actually quite small. Dr. Ashby showed that with a low poundage trad bow, producing (if I remember correctly) ≈22 ft·lb of kinetic energy (see the 2008 update, part 1), he could successfully breach the rib cage of a Cape buffalo bull 100% of the time and the arrow still had enough energy to penetrate more than half of the thoracic cavity after smashing through the heavy ribs of the bull. My SR6, pulling 53 lbs, produces well over twice this much kinetic energy. Every bow suitable for hunting produces more than enough kinetic energy to fully penetrate essentially any game animal in North America.
So why is it that we have all seen examples where a hunter drawing 70+ lbs with a 340+ IBO bow producing >80 ft·lb of kinetic energy fails to get more than 4 or 5 inches of penetration on a whitetail doe shot in the rib-cage? The answer lies again in the definition of energy. Energy is a measure of the total amount of influence a system may have on its surroundings; how that energy is used to influence or alter the system's surroundings depends on the details of the system. A gallon of gasoline contains a certain amount of chemical energy. If we simply light the gasoline with a match that energy will be used to heat up its surroundings. If we burn that gasoline in a combustion engine, we will heat its surroundings, but we may also use that energy to accomplish useful work, perhaps transporting a shipment of medicine from one town to the next. The same energy reserve is used for two entirely different purposes.
Our choice of bow determines how much energy we have available to use. Our choice of arrow determines how we budget that energy resource. How we build our arrows determines where our kinetic energy is spent. If we top our arrows with a mechanical broad head, it should be unsurprising that we have chosen to use a significant amount of our energy resources—energy that could be used for arrow penetration—to do the work of opening our mechanical. Additionally, mechanicals generally have very wide cutting diameters; this increased cutting diameter requires more work—energy—to penetrate the animal. This is one reason why use of wide-blade mechanicals is strongly discouraged for those shooting low-energy bows—the energy reserves to open the mechanical and still allow good penetration simply aren't there.
So how do we build an arrow that maximizes our energy use for penetration (I hope we can agree that maximizing penetration is the best way to ensure arrow lethality; if you think other factors are more important than penetration you will chose to build your arrow to maximize energy use with those factors in mind)? Many would argue to turn immediately to Ashby's 12 factors; they may increase FOC, carefully select a particular broadhead, etc. This, however, is the incorrect approach.
There is a well-defined property of a system, called its inertia, that describes the system's resistance to a change in motion. A moving system, such as our arrow, that is very difficult to stop has a large inertia. The inertia of a system is quantified by one measurable parameter and one measurable parameter only: the system's mass. This is why we often refer to mass as inertial mass. When you measure the mass of an arrow, what you are actually measuring is how difficult it is to cause that arrow to start moving, or, conversely, how difficult it is to cause that arrow to slow down. Arrow penetration, which is a direct measure of how difficult it is for the animal's internals to stop the arrow's motion, is a function of the arrow's mass.
This is why it is wrong to state that arrow penetration depends on on arrow momentum. It simply does not. I've seen multiple threads on these forums where one poster shows a system built with a low mass, high velocity arrow and another with a high mass low velocity arrow; both arrows have the same momentum. The poster will inevitably conclude that both have the same penetration potential. A second poster will then—usually following Ashby logic—claim that "not all momentums with the same value are equal" or some such. That somehow momentum "built" from large mass and small velocity is different than momentum "built" from small mass and large velocity, and therefore the penetration potential of the two systems is different. This is nonsense—momentum is momentum, and quantities that truly depend on momentum do not care if the momentum describes a heavy system in slow motion or a light system in fast motion; once the momentum of a system is parameterized it "forgets" what factors went into defining it. The truth is that the heavier arrow penetrates better because it has greater inertia—greater mass—not because it has "mass-derived momentum". I will repeat myself for clarity's sake. Increasing momentum DOES NOT result in an increase in penetration.
Why then the confusion? Take a look at the derivation presented below. Most are familiar with the definition of kinetic energy in terms of mass and velocity used earlier. Another definition may be derived in terms of momentum (see Eqn. 3). This definition is actually the better definition and the most commonly used equation for kinetic energy in non-relativistic classical (and even elementary quantum) physics. KE is the momentum squared divided by twice the mass. Rearrangement of this equation gives the magnitude of the momentum (||p||) as a function of kinetic energy and mass. (Momentum is a vector quantity; when we assign a value of "momentum" for an arrow we are actually talking about the magnitude of the momentum vector, properly indicated by the double bars on either side of the vector p.) Momentum is a function of the product of kinetic energy and mass. We have already shown that any legal bow has enough kinetic energy to adequately penetrate a North American game animal. If we increase arrow mass, we will, by the definition given in Eqn. 4, also increase the arrow momentum, since increasing arrow mass has little effect on arrow KE. Increasing arrow momentum does not effect an increase in arrow penetration; increasing arrow mass increases arrow penetration while simultaneously increasing arrow momentum. Momentum increase and arrow penetration increase may be correlated; one, however, does not cause the other!
So now we have established our two control parameters; kinetic energy, as determined by our choice of bow, and arrow mass. These are the parameters we are free to change and vary as we wish. Notice that arrow velocity (similar to momentum) is determined by our choice of these two control parameters. Because arrow trajectory is largely determined by arrow velocity, we must carefully choose our bows (KE) and our arrow mass to achieve our minimum acceptable arrow trajectory. Once we have nailed down the values of the two controls that we find acceptable for our personal shooting pleasure, we can begin to talk about the other "factors" of arrow building that come into play: arrow components, broad head selection, front of center, etc. This is the most logical progression for a "archery build" that one can follow when developing a new hunting rig.
We have discovered the first control parameter for our archery system—the kinetic energy that the system produces. We control this by our bow choice; what type of bow are we using (trad or compound, etc.), what is the bow's draw weight, how efficient is the bows cam system, etc. Once we have settled on a bow, the kinetic energy we can expect to obtain out of the system is effectively fixed. If you are fiddling with arrow weight in an attempt to "maximize" your kinetic energy, you are doing it wrong. So why then do so many archers feel that kinetic energy is "unimportant" when considering penetration issues? The answer is simple. Any bow that is legal for hunting purposes will produce enough kinetic energy to achieve enough penetration to kill an animal for which that bow is a legal method of taking. That’s why the bow is legal! If the bow did not produce enough KE to kill efficiently and quickly, then the bow would not be legal as a method of take. Furthermore, the amount of KE required to deeply penetrate a large game animal is actually quite small. Dr. Ashby showed that with a low poundage trad bow, producing (if I remember correctly) ≈22 ft·lb of kinetic energy (see the 2008 update, part 1), he could successfully breach the rib cage of a Cape buffalo bull 100% of the time and the arrow still had enough energy to penetrate more than half of the thoracic cavity after smashing through the heavy ribs of the bull. My SR6, pulling 53 lbs, produces well over twice this much kinetic energy. Every bow suitable for hunting produces more than enough kinetic energy to fully penetrate essentially any game animal in North America.
So why is it that we have all seen examples where a hunter drawing 70+ lbs with a 340+ IBO bow producing >80 ft·lb of kinetic energy fails to get more than 4 or 5 inches of penetration on a whitetail doe shot in the rib-cage? The answer lies again in the definition of energy. Energy is a measure of the total amount of influence a system may have on its surroundings; how that energy is used to influence or alter the system's surroundings depends on the details of the system. A gallon of gasoline contains a certain amount of chemical energy. If we simply light the gasoline with a match that energy will be used to heat up its surroundings. If we burn that gasoline in a combustion engine, we will heat its surroundings, but we may also use that energy to accomplish useful work, perhaps transporting a shipment of medicine from one town to the next. The same energy reserve is used for two entirely different purposes.
Our choice of bow determines how much energy we have available to use. Our choice of arrow determines how we budget that energy resource. How we build our arrows determines where our kinetic energy is spent. If we top our arrows with a mechanical broad head, it should be unsurprising that we have chosen to use a significant amount of our energy resources—energy that could be used for arrow penetration—to do the work of opening our mechanical. Additionally, mechanicals generally have very wide cutting diameters; this increased cutting diameter requires more work—energy—to penetrate the animal. This is one reason why use of wide-blade mechanicals is strongly discouraged for those shooting low-energy bows—the energy reserves to open the mechanical and still allow good penetration simply aren't there.
So how do we build an arrow that maximizes our energy use for penetration (I hope we can agree that maximizing penetration is the best way to ensure arrow lethality; if you think other factors are more important than penetration you will chose to build your arrow to maximize energy use with those factors in mind)? Many would argue to turn immediately to Ashby's 12 factors; they may increase FOC, carefully select a particular broadhead, etc. This, however, is the incorrect approach.
There is a well-defined property of a system, called its inertia, that describes the system's resistance to a change in motion. A moving system, such as our arrow, that is very difficult to stop has a large inertia. The inertia of a system is quantified by one measurable parameter and one measurable parameter only: the system's mass. This is why we often refer to mass as inertial mass. When you measure the mass of an arrow, what you are actually measuring is how difficult it is to cause that arrow to start moving, or, conversely, how difficult it is to cause that arrow to slow down. Arrow penetration, which is a direct measure of how difficult it is for the animal's internals to stop the arrow's motion, is a function of the arrow's mass.
This is why it is wrong to state that arrow penetration depends on on arrow momentum. It simply does not. I've seen multiple threads on these forums where one poster shows a system built with a low mass, high velocity arrow and another with a high mass low velocity arrow; both arrows have the same momentum. The poster will inevitably conclude that both have the same penetration potential. A second poster will then—usually following Ashby logic—claim that "not all momentums with the same value are equal" or some such. That somehow momentum "built" from large mass and small velocity is different than momentum "built" from small mass and large velocity, and therefore the penetration potential of the two systems is different. This is nonsense—momentum is momentum, and quantities that truly depend on momentum do not care if the momentum describes a heavy system in slow motion or a light system in fast motion; once the momentum of a system is parameterized it "forgets" what factors went into defining it. The truth is that the heavier arrow penetrates better because it has greater inertia—greater mass—not because it has "mass-derived momentum". I will repeat myself for clarity's sake. Increasing momentum DOES NOT result in an increase in penetration.
Why then the confusion? Take a look at the derivation presented below. Most are familiar with the definition of kinetic energy in terms of mass and velocity used earlier. Another definition may be derived in terms of momentum (see Eqn. 3). This definition is actually the better definition and the most commonly used equation for kinetic energy in non-relativistic classical (and even elementary quantum) physics. KE is the momentum squared divided by twice the mass. Rearrangement of this equation gives the magnitude of the momentum (||p||) as a function of kinetic energy and mass. (Momentum is a vector quantity; when we assign a value of "momentum" for an arrow we are actually talking about the magnitude of the momentum vector, properly indicated by the double bars on either side of the vector p.) Momentum is a function of the product of kinetic energy and mass. We have already shown that any legal bow has enough kinetic energy to adequately penetrate a North American game animal. If we increase arrow mass, we will, by the definition given in Eqn. 4, also increase the arrow momentum, since increasing arrow mass has little effect on arrow KE. Increasing arrow momentum does not effect an increase in arrow penetration; increasing arrow mass increases arrow penetration while simultaneously increasing arrow momentum. Momentum increase and arrow penetration increase may be correlated; one, however, does not cause the other!
So now we have established our two control parameters; kinetic energy, as determined by our choice of bow, and arrow mass. These are the parameters we are free to change and vary as we wish. Notice that arrow velocity (similar to momentum) is determined by our choice of these two control parameters. Because arrow trajectory is largely determined by arrow velocity, we must carefully choose our bows (KE) and our arrow mass to achieve our minimum acceptable arrow trajectory. Once we have nailed down the values of the two controls that we find acceptable for our personal shooting pleasure, we can begin to talk about the other "factors" of arrow building that come into play: arrow components, broad head selection, front of center, etc. This is the most logical progression for a "archery build" that one can follow when developing a new hunting rig.
