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Details on arrow momentum and kinetic energy for hunting
Luis Pinel sheds more light on how to determine the momentum and kenetic energy of an arrow and how it effects penetration on an animal.
Since momentum = mv
and F = ma
For a short period of t (milliseconds)
F = m v/t
and Ft = mv
Thus, momentum = Ft
This implies that momentum can be understood as the magnitude of the force needed F acting over a period of time t which will bring a body of mass m and velocity v to a state of rest.
It appears that the force F can be smaller and act over a longer period t or it can be a bigger force acting over a shorter period.
As useful as this way of thinking about momentum is, it does not allow us to calculate the distance which the said body will travel before it comes to rest, unless we can accurately measure the time taken from arrow impact to coming to rest. This is not an easy undertaking.
Yet, it is necessary if we want to know, for example, how deep an arrow will penetrate into an animal.
But there is another way!
We know that to resist a force of magnitude F over a distance s requires an amount of work to be done equal to Fs. We also know that only the arrow's kinetic energy can be converted to work. Thus, arrow kinetic energy = Fs. Therefore, the penetration distance s = arrow kinetic energy/F.
Various attempts have been made at producing a mathematical model of the resistive force to arrow penetration in animals, with various degrees of success. One of the difficulties encountered is that animals are heterogeneous in terms of the types of substances which make up their bodies, ranging from solid bone to fluids like blood and other in-between matter like flesh and other tissue. The resistive force for solids like bone is not the same as that for fluids like blood, which is not the same as the resistive force for other body tissue.
Our only hope of coming to some meaningful conclusion regarding an arrow's ability to penetrate a hunted animal sufficiently so as to minimise the animal's suffering, is to take this problem and break it down into logical parts.
From the time an arrow leaves a bow it starts decelerating since it is subjected to a resistive force. It is true that the resistive force in air is proportional to the arrow velocity squared.
Thus, a relatively slow and heavy arrow will reduce the adverse effect of drag. However, we need to keep in mind that there is a small benefit to be gained by taking this option at this stage of the arrow's flight.
Typically, an arrow will lose about 1% of its velocity (and momentum) and 2% of its kinetic energy at 40 yards and 2% of its velocity (and momentum) and 4% of its kinetic energy at 80 yards. Of this, we are absolutely certain.
Having shed velocity and energy, the arrow now strikes the intended target. Dependent on the distance to the target, an adjustment can be made to the arrow's maximum velocity and energy if deemed necessary at this stage.
The second stage of arrow flight is target impact.
When hunting an animal, and to be on the safe side, we assume that the well-placed arrow strikes a rib which it has to break. The force which has to be overcome is constant for any specific animal, so the best one can do is to choose a suitable broadhead for this job in terms of its design and quality. In addition, the arrow design is also important in that it must be adequately fletched so that it strikes the target along the line of travel and it must have a suitable FOC so that arrow vibrations are minimised upon target impact. For shots taken oblique to the target surface, arrow momentum assists in ensuring that the arrow continues to travel along the line of flight.
Breeching a rib consumes a certain amount of arrow kinetic energy approximated by (according to SABA BProC on 15 November 2012):
KE rib (in foot-pounds) = 9,6 for an animal of W kilograms.
This value can range from about 20 foot-pounds for a 50 kg animal to
41 foot-pounds for a 2000 kg animal.
There is a third stage of arrow flight for those that are hunting animals.
Up to now the arrow has lost velocity and energy to get to the target and some more velocity and energy in order to breech a rib prior to penetration into the thoracic cavity.
The force which resists penetration into the thoracic cavity is affected by two types of factors: penetration inhibitors and penetration enhancers.
Both types of factors are within the control of the hunter. The factors which inhibit penetration are: broadhead resistance and arrow velocity. Broadhead resistance is in line with intuition, but arrow velocity warrants a closer look. We saw above that the penetration distance of a projectile like an arrow is equal to the kinetic energy divided by the resistive force. But the resistive force to penetrating pure animal tissue is proportional to arrow velocity ie more velocity equals more resistance. Then, the penetration distance must be proportional to arrow mass X arrow velocity X arrow velocity / arrow velocity ie it is proportional to arrow mass X arrow velocity.
This means that penetration into the thoracic cavity tissue, after breeching a rib, is proportional to arrow momentum.
However, an animal also has fluids in the thoracic cavity. And the resistive force in fluids is proportional to the arrow velocity squared. This means that penetration into the fluids in the thoracic cavity is proportional to arrow mass.
This does not mean that arrow momentum causes penetration in animal tissue or that arrow mass causes penetration in animal fluids.
Only arrow kinetic energy can be converted into work which results in penetration. However, the amount of penetration in this case is proportional to arrow momentum in pure animal tissue and it is proportional to arrow mass in pure animal fluids.
Since the thoracic cavity consists of neither pure tissue nor pure fluids, the amount of penetration which one will obtain will be proportional to neither pure arrow momentum nor pure arrow mass, but some hybrid of the two boundary conditions. The reason therefore is that the resistive force will oscillate from being proportional to arrow velocity, on the one extreme, to being proportional to arrow velocity squared, on the other, as the arrow traverses the different regions of the thoracic cavity.
It is not surprising to read articles by, amongst others, Dr Ed Ashby stating that his field experiments on arrow penetration into animals show that a heavier arrow with the same momentum as a lighter arrow penetrates deeper than the lighter arrow.
How much more kinetic energy do we need to traverse the full width of the animal's thorax after having broken a rib on entry?
This estimate is best given by:
KE needed = 15,72/m (rearranged from the equation in SABA BProC as on 15 November 2012).
Where: Ro = broadhead resistance in pounds
W = animal mass in kilograms
and m = arrow mass in grains
To maximise arrow penetration into any specific animal's thorax we need to minimise broadhead cutting resistance, and maximise arrow mass. These two variables are controllable by the hunter.
Thus, the penetration enhancers as mentioned above are: broadhead cutting resistance (low) and arrow mass (high).
In summary: we will not dwell on the arrow velocity and energy losses between bow and target for the purposes of hunting since they are relatively small. Penetration into the animal requires that we cater for breeching a rib on entry and this requires a specified amount of arrow kinetic energy as per the text above as well as a broadhead of adequate design and robust construction.
Further penetration into the animal's thorax requires a broadhead of adequate cutting resistance (so as not to rob the arrow of too much energy), this is particularly important for lower energy arrows relative to the animal being hunted. The second requirement for penetration into the animal's thorax is sufficient arrow kinetic energy with a high arrow mass in order to maximise the penetration depth.
What is a "heavy arrow"? I know that most traditional bow and modern compound bow hunters will probably give a different answer. But, I believe that the answer should be similar.
I, as a compound bow hunter, have come to value a heavy arrow, as used by traditional bow hunters for hunting purposes.
Traditional bow hunters have the benefit of many centuries of experience in extracting the maximum performance from their bows in general and for hunting purposes in particular. On the other hand, compound bow hunters are so blessed with the excessive arrow kinetic energies generated by modern bows for most hunting situations that arrow performance maximisation has not been necessary. But it sure becomes an issue when one is hunting big animals, even with modern compound bows.
I see an overwhelming majority of compound bow hunters using arrows under 450 grains at speeds between 275 and 300 plus feet per second. I believe that these hunters are missing a point.
If I am wrong, and these hunters are intentionally doing so, then I believe they are missing an opportunity to extract better performance from their equipment for the benefit of and mercy for their quarry.