This article is being superseded by two new articles:
Kinetic Energy, Momentum and Arrows: a Simplified Approach
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.
Kinetic Energy Quick Reference Chart
Momentum Quick Reference Chart
There is also a page with javascript calculators where the speed and weight can be input and the momentum and kinetic energy calculated:
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.
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{ 15 comments… read them below or add one }
You are not quite correct on your discussion of KE and Momentum of the arrow after the shot. I’d just like to point out that two arrows of identical external dimensions with different velocities have the same drag coefficient (Cd), but the faster arrow will have a higher air resistance (Drag Force) since the drag force equation has the velocity squared in it. The equation is F=0.5*Cd*p*V*V*A (where p is air density, A is cross sectional area, and V is velocity). Just put your hand out the car window while driving to experience this. Your hand dimensions haven’t changed, but the force on your hand increases with the speed of the car. Therefore, F=ma does in fact explain why the heavier arrow retains more energy downrange, but it is important to note that the force in this case is not the same for the two arrows. If the drag force were the same, then the distance required to stop the two arrow’s flight would be exactly the same for both arrows (even though the lighter arrow decelerates at a higher rate under this scenario, it’s mass is also proportionately lower in order for a contant force to be exerted on it). Thanks for all the other great info you’ve provided, keep up the good work.
Chris,
You can also read a more in-depth discussion of the subject here: http://archeryreport.com/2011/01/heavy-vs-light-arrows-speed-power/
You may have to enlighten me more because I fail to see where I have contradicted anything you said; in fact you restated exactly what I said! Perhaps I need to work on my wording
Michael,
I love this site. I think a more accurate calculation for most of us flatlanders is to use the acceleration due to gravity = 32.16 ft/sec/sec. This yields a denominator of 450240 for the KE calculation. I have seen others use this figure and I believe it applies to locations 1000 ft above sea level.
Heya Redneck, Thanks for reading!
It would be more accurate to use a gravitational constant of 32.174 instead of rounding up to 32.2 which would result in a denominator of 450436. For illustration purposes and making it easier for people to remember, I usually use the simpler 32.2 and 450800 which is only a difference of about .08%. But since I’m an ultra-anal engineer that has spreadsheets and MathCad workbooks to do my calculations, I actually have a box to compensate for additional, yet extremely minor, losses of gravity due to altitude. In my case I live at about 5000 feet, so the correct value here would actually be 450653. Hardly worth sneezing about, but when you have the computational ability of a computer, why not!
All the math and physics is great up to a point, but how do all the numbers translate to what bowhunters really want to know? If I place my arrow in the kill zone does it have what it takes to effectively and humanely bring down big game? I am shooting a 388 gr arrow 300 fps. This gives me a ke of 77.46 and a mo of .5164. My bow is maxed out (60 lbs) so while I can increase arrow weight speed will decrease accordingly. If I increase arrow weight 100 gr to 488 my speed reduces to about 270 fps. My ke increases to 81 and mo increases to .5845. Does 3.5 lbs of ke and the increase in mo make any real killing difference on deer or elk out to 50 yards at these levels?
Rick, you’ve hit the dilemma right on the head. Each archer/hunter needs to decide for themselves where they want to be. Lighter arrows will certainly kill game when placed in the right spot and give a flatter trajectory to help minimize errors in yardage judgment. Heavier arrows will do a better job of penetrating through bone and flesh with the shot isn’t perfectly broadside as well as drift less and maintain more energy downrange. I hunt elk in the west where the animals are huge and tough and the distances long so I opt for a 470 grain arrow and practice my yardage judging every chance possible. It’s not my job to judge anyone’s setup, but rather to just lay the facts out and let everyone decide!
Thanks Michael for your effort in all of this. I killed a bull elk with the set up I mentioned above last season. I missed all bone and had most of the arrow penetrate thru the off side. 38 yard broadside shot. I hit him a little far back but was blessed and took out the main artery. He went 50 yards and tipped over. I am going to try adding some weight tubes to my current arrow setup to see how I like them. At 3 gr/inch they will put me right up near your set up @478 gr.
I have found several charts stating what is the sufficient KE for killing different size big game, but is there a chart out there that has the recommended MO (momentum) for killing different sized animals (ie. elk, deer, sheep, moose, ect.)?
I LIKE THE CHARTS AND ALL THE MATH, SEEMS TO BE CLOSE. REAL WORLD TEST, I TOOK
ONE ARROW 400 GRAINS AND ANOTHER 313 GRAIN, SHOT BOTH OUT OF SAME BOW. 313 GRAIN ARROW SHOT 330 FPS, 400 GRAIN ARROW SHOT 291 FPS, SHOT BOTH ARROWS INTO SAME TARGET, A 3 FOOT HIGH BEAR TARGET, FIRST SHOT BOTH OF THEM AT 20 YARDS, THE 313 GRAIN ARROW SUNK 11 INCHES INTO TARGET, 4 SEPERATE SHOTS OF SAME ARROW. THE 400 GRAIN ARROW SUNK 10.5 INCHES IN TARGET SAME 4 SHOTS, WENT TO 33 YARDS AND 313 GRAIN 9.25 INCHES 400 GRAIN ARROW 8.5 INCHES. MATH DOESNT SEEM TO ADD UP.
Scott, a couple of different things in your case: first, were each of the arrows tuned properly and did you check that they were? If so, how? Changing from arrow to arrow can drastically change the tune of the bow/arrow combination and result in a loss of efficiency. Also, foam acts differently in stopping arrows than most other media. I have seen faster arrows with less KE and momentum penetrate better than heavier/slower arrows with more KE and momentum. My working theory is that the foam closes up around an arrow at a certain velocity and the faster arrows are able to penetrate more than slower arrows before the friction of the closing foam stops the arrow. See this article for more: http://archeryreport.com/2010/01/friction-penetration-archery-target-animal-penetration/
BTW, please refrain from typing in all caps. It’s more difficult to read and there is no need to shout
Oh sorry bout the caps i wasnt shouting, work computer is set up that way and i never changed it. I will check out that article.
Well lol I think u are all wrong I shoot a high country prox10 at 390 fps with a 249 grain arrow and have complete pass thru on a whitetail at 83 yards thru both front shoulders and my pro x11 high country I shot one with a 330 grain arrow at 411 fps and at 50 yards thru both front shoulders had a complete pass thru but my arrow also drove in ground 10 and half inches so kentiec energy ain’t [edited: no profanity on this site please] its how much flex your arrow has when entering that object my arrows have no flex what so ever leaving the bow are in slow motion where we slowed down 6000 times per frame when hitting target never flexed like all the others my arrows are from high country made out of military carbon 190 a dozen and only animal I had no pass thru was a elk I shot at 98 yards it came out 6 inches on opposite side of shoulder but that’s still great it ran 10 steps. And Fell
Kody,
Firstly, using some proper punctuation and grammar would make your post and point much easier to understand. Also, profanity is not allowed on this site.
Secondly, you can argue all you want for lighter arrows, but the simple truth is that they do not have as much KE, momentum or penetration potential as their heavier counterparts. There are multiple articles and tests on this site, and done by others, that prove the same. This is not to say that a light arrow won’t get the job done, especially on thin-skinned whitetails. Many people have done perfectly well with light arrows on many animals.
And I also am a bow tech and only have a 261/2 drawand I’m the best at what I do and high country knows it contact them and ask to speak to Donna she will tell u kody is the best and not two many people have a president of the company call and talk for a hour on phone a lot get a bow that’s quick forgiveing and regardless always make the best shot and have fun god bless butbut air flow over the arrow and wind drift is not a factor with my bows I’ve shot a lot and practice to the point I know when I release on a animal at 80 yards its mine regardless and I shoot thru a graph 20 feet in front of a bow
Well sorry for punctuation, but still KE is not a factor with the arrows that you can get this day and time,yes we have been thru the test also arrow dia. Plays a factor in this length of arrow broad head length also forgive me for spelling on android and sometimes I cannot catch all the words and I apologize for language it auto spells on me but I will say I have seen a bow shoot with 110 fp of KE and did not pass thru a elk I was shooting my hoyt supertec at 360 fps and a 367 grain arrow but your right you can kill a deer with as little as 30 pounds draw weight and a 200 grain arrow but I will say high country archery is a step up in the industry as of this month there new bows will exceed the 400 + mark I’ve seen them but you do have some good points but as you know we all do are own test and I’m glad I’ve talk to you and when I finish our testing we would send you video of it if you like it will be on highcountry web site buy first of year thank you and once again I apologize for the language god bless