This article is being superseded by two new articles:
Kinetic Energy, Momentum and Arrows: an In-Depth Look at the Physics, Math and Reality (yet to be published)
Kinetic Energy of Bows and Arrows
The kinetic energy (KE) of an object is the energy of the object due to it’s speed. In order for the energy of an object to change, work must be done on the object. In the case of an arrow and archery, work is done by the archer’s muscles by pulling back the string and flexing the limbs. The energy is stored in the limbs in the form of potential energy; when the string is released the energy stored into the limbs is released, most of which is absorbed by the arrow. The energy not absorbed by the arrow becomes the sound, vibration and other movement that is experienced by the bow. Energy that is absorbed by the arrow is converted into multiple forms, the majority resulting in the forward velocity, other components are vibration/oscillation, sound, etc. The kinetic energy of the arrow that archers care about and calculate is the energy due to it’s forward motion; this is different than the total energy of the arrow which also includes kinetic energy due to vibration/oscillation of the shaft, potential energy stored in a flexed shaft, etc. As the arrow travels downrange, the total energy diminishes due to air resistance and friction in the arrow materials as the arrow flexes. Sound energy is created as the arrow moves through the air and creates sound waves due to the resistance of the air.
The formula for kinetic energy is:
For an arrow, the kinetic energy is calculated by taking the weight in grains, multiplying it by the square of the velocity in feet per second, and dividing by the constant 450800. The constant is derived from the “1/2” in the formula, converting between grains and pounds (7000 grains per pound weight) and the gravitational constant.
Calculating kinetic energy requires using the mass of the arrow, not the weight; thus the gravitational constant, 32.2 ft/s^2 is used to convert between mass and weight.
Momentum of an Arrow
The momentum of an object is the product of its mass times its velocity. Momentum is NOT a type of energy but it can be related to kinetic energy mathematically. Notice the difference in terminology between the words speed for kinetic energy and velocity for momentum. Velocity is used in the momentum calculation because momentum is a vector quantity; or rather momentum is a measure of the speed of the object along with it’s direction.
Momentum is calculated using the following formula:
And in the case of arrows:
The constant 225400 is calculated in the manner as for kinetic energy, except that momentum is not multiplied by ½.
To relate kinetic energy to momentum, the following can be used:
or for an arrow:
Arrow Kinetic Energy and Momentum Charts and Calculators
For convenience I have created quick reference charts that can be printed off and carried with you rather than having to use the formulas and a calculator/computer all of the time. All you need to know is your speed to the nearest 5 fps and arrow weight to the nearest 10 grains. These charts are also nice to make a quick comparison between two different setups.
Who Cares About Kinetic Energy and Momentum?
Archers care of course! By knowing how a bow performs with different arrows, and archer can improve accuracy at various ranges, and in the case of bowhunters, improve the penetration of an arrow on an animal. The remaining discussion will focus mainly on hunting, as that is where it is most applicable.
There is currently a lot of debate on various archery related message boards as to which is more important when trying to find the best hunting arrow, kinetic energy or momentum? I do not believe that either can be ignored but rather both should be considered.
The following graph shows the kinetic energy and momentum of a Hoyt Ultra-Elite XT2000 set at 29.5” and 60 lbs.:
And another chart for an Elite Envy set at 30″ and 70 lbs.:
For this testing, all arrows used are Easton X7 2412 aluminum arrows with different variations of other shafts inserted into them to get the various weights. This way all of the arrows shot have the exact same external dimensions and characteristics.
As can be seen, the kinetic energy and the momentum both rise as the arrow weight is increased. For the arrow weights tested, the kinetic energy tends to be leveling off but still gaining slowly, while the momentum is climbing almost steadily but is beginning to level slightly. Of all the testing done to date, I have not found any cases where the kinetic energy will decrease with increasing arrow weight. There is most likely a point where the arrow is so heavy that the bow cannot efficiently propel the arrow forward, but it is somewhere beyond 1450 grains for the bows tested.
So why is this? Why doesn’t an increase in arrow weight reduce the velocity exactly in proportion so as to have the same kinetic energy and momentum as a lighter arrow? The answer lies in the efficiency of the bow. As the arrow weight is increased, the bow is able to transfer a higher percentage of its stored energy (potential energy) into the arrow. Less of the bow’s energy is converted into wasted energy. A simple test is to take any bow and shoot two arrows of significantly different weights and the bow will be quieter and have less vibration with the heavier arrow. More of the energy goes into the arrow and thus less is converted into vibration that creates sound.
Kinetic Energy and Momentum of an Arrow After the Shot
Once an arrow leaves the string, the mass of the arrow continues to have a significant effect beyond the initial velocity. As good ol’ Sir Isaac Newton taught us, F=ma (Force=mass*acceleration). In archery terms, this simple equation states that the force slowing the arrow down (mainly air resistance) is proportional to the mass of the arrow and how quickly it decelerates. The greater the mass, the more force it takes to slow the arrow down. Considering two arrows of equal outside dimensions, including the point and vanes, but of different masses, the arrow with greater mass will take more force to slow it down. Because the two arrows have the same frontal profile, the air resistance will be the same and thus the lighter arrow will be subject to a greater deceleration. Of course the lighter arrow will begin at a higher velocity, but the heavier arrow will lose less of its initial energy downrange. Knowing that a heavier arrow will always have a higher kinetic energy and momentum to begin with, and knowing that it will also decelerate at a slower rate downrange, it becomes obvious that a heavier arrow will not only begin with more energy and momentum, but will retain a higher percentage of its energy and momentum downrange.
So why not shoot re-bar shafts that weigh in the pounds instead of grains? Such an arrow would have lots of energy to begin with, but very little velocity and would “drop like a rock” shortly after leaving the bow. It becomes a trade off between speed and how much an arrow will drop over distance, and how much energy/momentum the arrow will have when arriving at the target.
Once an arrow reaches an animal, energy and momentum are rapidly lost as the broadhead encounters resistance to cutting the skin, bones and organs, as well as friction from the same, and fluidic resistance that is much higher than when flying through air. In the case of mechanical broadheads, energy is required to open the blades as well. If the arrow shaft does not enter perfectly perpendicular to the animal body and deflects off of anything, energy and momentum will be transferred out of the direction of penetration and lost in side to side/up and down motion.
This article is a work in progress, and probably will progress for quite some time as more arrows are being shot, data is being taken and analyzed. Check back to see updates.