I always think it’s funny when different issues in human civilization are on fire and we make fun of people who actually care about details and mechanisms and first principles. Here's an excerpt that I think can close the loop for most people in about ten minutes of sitting still and reading:
*Not my work, don't take credit. Follow link for OP:
www.archerytalk.com
Let's begin by addressing some common misconceptions that we should eliminate immediately.
Misconception 1.) The one that is going to tick a lot of people of instantaneously—arrow momentum does not determine penetration. This notion is absolutely false. Linear momentum—
p, defined in Eqn. 2—is an incredibly useful tool for performing calculations. This stems from momentum being a conserved quantity. In more advanced formulations of classical mechanics momentum is a useful quantity to find the underlying symmetries of nature, so it is a very important quantity—nevertheless, the momentum of an arrow DOES NOT determine its penetration potential.
Now, let's be very precise. An increase in momentum may (and almost assuredly will)
correlate with an increase in arrow penetration, but we should be careful about the difference between
correlation, when two observations are usually found to occur in tandem, and cause and effect, or
causation, where one phenomenon causes another. It is
usually the case that an arrow with large momentum will penetrate deeply, but the large momentum does not
cause this deep penetration.
In addition, we should be careful about discussing momentum as something the archer actually has control of in their setup. Momentum in a hunting arrow is a
dependent parameter. There is no momentum control knob that one may turn on their bow to increase the momentum that their arrow has upon firing. Instead, we must change another
independent parameter—also called a
control parameter—which will in turn change the momentum of our arrow as it leaves the bow. We will discuss what the appropriate control parameters that we should consider are momentarily.
Misconception 2.) Archery calculators are useful over large ranges of their input parameters, such as arrow mass. To be perfectly honest, most archery calculators are poor, and they tend to give archers very bad information. The most common misconception that archery calculators reinforce is that there is a "sweet spot" of arrow mass that maximizes the kinetic potential obtained from firing the arrow from a particular bow. I have read what seems like countless posts (and shamefully have even posted the same misinformation occasionally myself before actually thinking about the way these calculators work) where the poster thinks that a calculated arrow mass will give the archer the most kinetic energy that they can get out of their bow. This is always incorrect. This will be covered more in misconception 3 and later.
As an example of how archery calculators can give truly horrible predictions, lets take
the backcountry bowhunting calculator as an example. I actually like this calculator when used appropriately, but let's see what happens when used incorrectly. Check the default settings on the calculator. The kinetic energy is given as 89.3 ft·lb. Now increase the arrow mass slider to 490 grains; the KE is 96 ft·lb. Above 510 grains the KE begins to decrease again. I assure you, this is NOT correct. This may be demonstrated by going further with the arrow mass slider. An arrow of 800 grains shows a kinetic energy of 74 ft·lb. Take a moment to actually think about this. All of the kinetic energy is transferred to the arrow from the bow, derived from the potential energy stored in the bow at full draw. The calculator is telling us that a 500-grain arrow will absorb 96 ft·lb of energy from the bow. This means that at full draw there must be at least 96 ft·lb of potential energy stored in the bow—in truth, no bow is 100% efficient, so there will actually be more than 96 ft·lb of potential energy stored in the bow. Now shoot an 800-grain arrow out of the same bow—the calculator is telling us that only 74 ft·lbs of kinetic energy will be accounted for in the arrow projection. Energy is a conserved quantity, so that means there is an additional >22 ft·lb of kinetic energy that the bow must be dissipating when it fires the arrow that is not accounted for in the arrow's KE. That much additional energy would result in a sound like a shotgun, and may severely damage the bow itself. We know this is not true in real life—the heavier arrow will result in a quieter bow with less hand shock and generally less energy left over after firing the bow that the bow must dissipate in other ways. The problem with the calculator can be shown in an even more extreme fashion by putting 1500 grains in as the arrow mass. The calculator now predicts the arrow to have 0.1 ft·lb of kinetic energy and to actually leave the bow with negative speed—it predicts that the 1500-grain arrow will be fired backwards. Hopefully this sufficiently demonstrates the pitfalls of using archery calculators to predict the kinetic energy, momentum, etc., of an archery system.
Misconception 3.) Choosing a particular arrow has any significant effect on the kinetic energy that the archery system will produce. If you are trying to achieve a kinetic energy goal by adjusting your arrow mass, you have already lost the battle. *This misconception has a slight caveat, which I will address shortly.*
With these misconceptions addressed, let's take a look at the real physics behind arrow penetration.
We begin with energy. Energy is the capacity of a system to do work. A system without any energy lacks any ability to change or influence other systems with which it interacts. Energy comes in two general forms: potential and kinetic. Arrows in a hunting situation lack any usable potential energy, so an arrow's entire ability to influence other systems—including the living system that is the game animal—depends entirely on its kinetic energy. As such, the first parameter in determining arrow penetration is—regardless of Dr. Ashby's insistence otherwise—the kinetic energy attained by the arrow in the archer's system. Eqn. 1 gives the usual definition of kinetic energy, 1/2·m·v[SUP]2[/SUP]. This definition has led to mass confusion in the archery community. Although the equation is, of course, correct, the general interpretation of the equation is incorrect in the case of an archery system. The mistake is in what is taken to be the
control parameter and what is the
dependent variable. Most on Archery Talk mistakenly take the kinetic energy of their bow to be determined by the mass and velocity of their arrow. In truth, the velocity of the arrow is determined by the arrow mass and the kinetic energy produced by the bow. Like momentum, velocity is a dependent variable and may not be controlled independently of other control parameters.
Let's readdress misconception 3 above. Many archers think (and have been misled by arrow manufacturers to believe) that one may choose their kinetic energy by choosing a particular arrow setup. This is essentially false. The kinetic energy obtained in an archery setup is almost entirely determined by your choice of bow. It is true that bow efficiency will increase when using heavier arrows, but the effect is minor and may essentially be ignored. A recent example that demonstrates this clearly is
this video by DIY Sportsman. The data given spans arrow masses from 379.4 grains to 1163.5 grains. That is an increase in arrow mass of 207%—a tripling of arrow mass. The kinetic energy of the arrow leaving the bow increased from 73.3-77.9 ft·lb., a mere 6% increase. A >200% increase in arrow mass yielded a <10% increase in kinetic energy. For all practical purposes, arrow mass does not affect arrow kinetic energy when the two arrows are fired from the same bow. For newer bows (5 years old or less) this effect is pretty general. Bow manufacturers have gotten so good at producing efficient cam systems with even low gpp arrows that there simply isn't much room for bow efficiency to increase with heavier arrows. As another example, an early review of the Realm SR6 showed a 1.7% increase in arrow kinetic energy for a 45% increase in arrow mass. Even with older bows, the above still holds true generally. The same may not be true of trad bows—finding data to check is more difficult, and I've seen reported that increasing arrow mass increases the efficiency of a trad bow more significantly, though I haven't seen data to support or refute that statement. However, even with a trad bow, it will still be true that it takes a huge increase in arrow mass—doubling or tripling—to see a relatively small 10-20% increase in the arrow kinetic energy produced by the bow.
Your choice of arrow has a minimal effect on the kinetic energy produced by your bow.
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