## Kinetic Energy, Momentum and Arrows: a Simplified Approach

It is often debated in archery and bowhunting circles which is more important: kinetic energy or momentum.  Nearly as often the terms are not fully understood!   In this article we tackle the basic definitions of both KE and momentum, how to calculate them and why it’s important.

### Kinetic Energy of Bows and Arrows

The kinetic energy (KE) of an object is the energy of the object due to its speed and mass.  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 friction in the bow parts, noise, vibration and other inefficiencies experienced by the bow. Energy that is absorbed by the arrow is converted into multiple forms, the majority resulting in the forward velocity. The kinetic energy of the arrow that archers care about and calculate is the energy due to its forward motion. As the arrow travels downrange, the total energy diminishes mostly due to air.

The standard formula for kinetic energy is:

where m is the mass of the moving object and v is the velocity.

For an arrow, the kinetic energy is calculated by taking the weight in grains, multiplying it by the square of the speed in feet per second, and dividing by the constant 450800.

### 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. There is a difference in terminology between the words speed and  velocity.  Speed is just a number, velocity is speed with a direction.  For KE, either can be used and the resulting number, the energy, has no direction.  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 its direction, times its mass.

Momentum is calculated using the following formula:

And in the case of arrows:

### 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:

ArcheryCalculator.com

### Who Cares About Kinetic Energy and Momentum?

Archers and bowhunters care of course! There is always a lot of debate in the archery world 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 Bowtech Destroyer 350 set at 30? and 70 lbs. as it is shot with various arrow weights:

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.

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 steadily but is beginning to level off slightly. Of all the testing done to date, I have not found any cases where the kinetic energy and momentum 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 1500 grains for bows I have tested.

Why is this? Why doesn’t an increase in arrow weight reduce the velocity exactly in proportion to the kinetic energy and momentum? The answer lies in the efficiency of the bow. As 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 and 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 determining 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 slows down. 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 profile, the air resistance will be similar and 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 slow down at a lower 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. See “Heavy vs. Light Arrows:  Downrange Speed and Power” for more details on this subject.

So why not shoot solid steel 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 tradeoff 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.

When 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, and resistance to bodily fluids that is much higher than when flying through air. In the case of mechanical broadheads, energy is required to open the blades as well.

Many people will argue that either KE or momentum is the better indicator of penetration potential. I differ from their opinion in that I believe both are good indicators of penetration potential and prefer to use both numbers together. Energy is used and transferred from the arrow as the blades cut, slice, break and otherwise move through an animal. Momentum gives a good indication of resistance of the arrow to slowing down. My suggestion is to learn to understand and use both numbers rather than relying on only one or the other.