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slow_fast_arrowThe instant that an arrow leaves the string, it is at its maximum velocity and will immediately start to slow down.  There is much talk about how much kinetic energy and momentum that an arrow has, but most of it centers around the initial velocity produced by the bow.  However, what is much more important is how much speed, kinetic energy and momentum the arrow has at the point of impact, especially in bowhunting conditions where the arrow must effectively penetrate an animal.  This article will be broken into three sections:

  • Part I: a discussion of how arrow weight affects downrange velocity, kinetic energy and momentum
  • Part II: mathematical look at the above discussion to better understand what is really happening
  • Part III: real world results from experimental shooting

Most readers will benefit from Part I as I discuss in a verbal fashion what happens after the arrow leaves the string.  Many people will not appreciate Part II as I delve into the math and physics, though many will enjoy it, and it won’t hurt my feeling if you skip that part!  I’m a engineering and physics geek so bear with me.  Part III will probably be the most interesting section as actual experimental numbers will be used to test out the theories and descriptions of the previous parts.

Part I:  Heavy vs. Light Arrows, a Discussion of Arrow Deceleration

Arrows achieve velocity as the bow is drawn (energy added by the archer) then released (energy transferred from the bow limbs through the string and to the arrow.) At the instant an arrow leaves the string,  there is no longer a force pushing and accelerating the arrow.  Once the arrow is in flight, the only outside force acting on it that affects its speed and power is the air resistance slowing it down called the drag.  The total amount of drag on an arrow is a factor of the shape of the arrow, the speed of the arrow and the density of air.  Because there is no thrust or anything else to power the arrow’s flight after leaving the string, the drag alone will determine the arrow’s deceleration over the remainder of the flight.

Consider two identical arrows on the outside, same shaft diameter, fletchings, point and nock.  One arrow is very light, and one much heavier.  For the sake of keeping the argument simple, we’ll consider both arrows to have the same spine as well.  When shooting both arrows out of the same bow, the lighter arrow will of course be faster at point blank range.  However, the heavier arrow will leave the bow with more kinetic energy and momentum due to the bow being more efficient at delivering energy into the arrow (for more discussion on this, see the Arrow Kinetic Energy and Momentum article.)

To determine how much drag is experienced by an arrow in flight, the drag coefficient must be known.  The drag coefficient is a dimensionless number (a number having no units such as inches, pounds, etc. associated with it) that describes how aerodynamic an object is.  This number is determined by the shape of the body and can be found either experimentally or by looking it up in tables found in fluid mechanics texts or other places.  In the case of our light and heavy arrows, this number is the same because on the outside they are dimensionally equivalent.

Once the drag coefficient  and speed of the arrow are known, the drag force on the arrow in flight can be determined.  As the speed of the arrow increases, so does the drag force on the arrow.  The faster the arrow, the higher the force trying to slow it down.  Thus the faster arrow is going to have more force slowing it down initially.

For this section, one equation (I promise, only one!) needs to be understood; Newton’s second law of motion: F=m*a (force = mass times acceleration.)  This simple law states that the higher the force placed on an object, the faster it will accelerate.  Also, the higher the mass, the harder it is to accelerate.  In the case of the flying arrow, the force is acting against the forward motion of the arrow and the arrow will experience deceleration.   The force on the lighter, faster arrow will thus slow the arrow down at a faster rate than the heavier, slower arrow.

At typical archery distances, the lighter arrow will almost always maintain a higher speed than the heavier arrow.  Even though the lighter arrow is slowing down faster, it started out much faster and the heavier arrow is also slowing down.  Because the heavier arrow is decelerating at a slower rate, it will maintain a higher percentage of it’s original speed than the faster arrow.  Also remember that the heavier arrow has more kinetic energy and momentum than the lighter arrow at launch already.  This gap only grows larger as the arrows progress downrange.

This entire discussion would tend to favor the slower, heavier arrows for having more power at impact for multiple reasons.  However, there is a drawback.  The lighter arrow is going to drop less over the full distance to the target and is thus less dependent on accurate distance judging (notice I said less dependent; accurate distance judging is still extremely critical!)  For more details on speed and drop over distance, head on over to the One Pin to Forty Yards article.

Hopefully this all makes some basic sense.  Continue reading for the mathematical proof and a better understanding behind the physics of what is happening.

Part II:  Mathematics and Physics of Heavy vs. Light Arrow Flight

In this section we are going to take a deeper look into the physics and math behind what is happening after an arrow leaves the bow. We will start with the very basic equation that was briefly mentioned above: F=m*a.  When the bowstring is first released, the energy stored in the limbs (Potential Energy, PE) accelerates the bowstring and thus the arrow forward.  There is a positive force F in the right hand direction and a positive acceleration a in the same direction.  The arrow picks up speed until it leaves the string, at which point there is a net force of zero for an infinitesimal moment.

The instant after the arrow has left the string, the only force acting on the arrow to slow it down is F(drag), which is the drag caused by the air on the arrow (we are neglecting the force of gravity pulling the arrow down for now.)  This force is acting to the left.  Now that the force has reversed directions, so does the acceleration and at this point the arrow is no longer accelerating, but rather decelerating in the negative (left hand) direction.  Throughout all of this the mass, m, has remained unchanged.

Now let’s take a look at the case of two arrows of absolute identical external characteristics (shaft diameter, vanes, point, etc.), identical spines but having different weights.  For the sake of initial simplicity, we will assume that they both leave the bow at exactly the same speed.  This of course means that more energy was put into pulling the bow back in order to transfer more energy into the heavier arrow.  Once both arrows have left the bow, they both experience exactly the same drag force as each other.  This is because the drag force is a function of the external physical characteristics of the arrow.  If both bows have the same F(drag), but different masses, then for F=m*a to hold true, the heavier arrow must have a lower deceleration!  If both arrows leave the bow at the same velocity, the lighter arrow with it’s greater deceleration will lose velocity faster and arrive at the target a slight amount later and a slower speed.

Unfortunately most cases are not this simple.  Generally speaking the archer/bowhunter is concerned about how different arrows from the same bow, shooting the same draw weight, will perform.  In this case, assuming both arrows are properly spined, the heavier arrow will leave the bow with a lower velocity.  However, the heavier arrow will also leave the bow with more momentum and kinetic energy because bows are better able to convert potential to kinetic energy with heavier arrows (see the article on momentum and kinetic energy for more details.)  Since the formulas for momentum and kinetic energy both involve the velocity of the arrow, and because the heavier arrow will retain more of it’s initial velocity downrange, it will maintain a higher percent of it’s original kinetic energy and momentum!

Now let’s get even more complicated!  Recall in the first case that because both arrows had the same velocity, they also experience the same drag force.  In the second example, the arrows had different velocities because the draw weight of the bow was held constant.  In this case, the drag forces are not the same.  This is due to difference in the coefficients of drag.  This can be calculated with the following formula:

Here F(drag) is the drag force, p (rho) is the density of air, V is the velocity of the air and A is the cross-sectional area of the arrow when looking at it head-on.  The coefficient of drag is a constant that is fully dependent on the geometry of an object.  Because we are shooting arrows of identical external characteristics, the coefficient of drag for both arrows is equal. In our case the area is the same, the density of air is the same and of course the coefficient of drag is the same.  Thus the only two variables are the drag force and the velocity.  Looking carefully at the formula one can see that as the velocity increases, the drag force must also increase in order to keep the coefficient of drag the same.  In fact, the drag force increases in proportion to the square of the velocity.  This means that as velocity increases, the drag force increases a great deal!

Back to our example:  if the lighter arrow is traveling faster than the heavier arrow it will be experience a higher drag force than the heavier arrow due to its higher velocity.  Remembering what was discussed about F=m*a and now knowing that the force acting against the lighter arrow is higher than the heavier arrow, it’s easy to see that the lighter arrow will experience an even higher deceleration rate than if it were traveling the same speed as the heavier arrow.

To sum it up:  lighter arrows shot from the same bow as heavier arrows have two things that cause them to decelerate faster than heavier arrows: first the mass is lower thus the drag force has a larger effect and second, the higher velocity will cause the drag force to be larger.

An extreme example would be to take an arrow made completely of foam (but that was somehow stiff enough to be tunable) and a regular arrow.  Assuming both arrows have the same external characteristics, and that they are shot from the same bow, the foam arrow would leave the bow and a much higher initial speed but intuition says that it would slow down very quickly.  The regular, heavier arrow would leave the bow much slower but maintain a much higher percent of its velocity downrange.  In fact, the regular arrow would even pass the foam arrow downrange even though it started slower!  Generally speaking, with most arrows within the norm of what archers shoot, and where weight differences are not so extreme, the heavier arrow would never pass the lighter arrow, but it would maintain more kinetic energy and momentum.  Because the heavier arrow starts with more kinetic energy and momentum, this can have a significant impact on the difference of energy and momentum that the two arrows have at the impact point.

For those that really want to get into the math, I’ve made a short example to show how to find the drag force and deceleration.  In this case, I calculated an the actual drag force on a real arrow by using CFD (computational fluid dynamics.)  This can also be found experimentally using an air tunnel and force gauges:

Knowing the actual coefficient of drag of this arrow geometry now makes it possible to calculate the deceleration of similar arrows of different weights and initial velocities.  It is very important to understand that these calculations are done with perfect tuning and perfect arrow flight as an assumption.  If the weight of an arrow is increased by adding more point weight, the spine of the arrow will be reduced and the arrow may not as efficiently absorb energy from the bow on the shot (due to excessive flexing) and will continue to lose energy down range at a higher rate.  In reverse, removing too much point weight can make an arrow too stiff and with too low of an FOC to allow for consistent, stable flight and thus will also experience higher energy loss.  These are just two examples of variables that can cause all sorts of variations in real world arrow flight.  That being said, the formulas and theory discussed here hold true and when understood correctly can become important factors in an archer and bowhunters selection of arrows.

Part III:  Experimental Data of Heavy vs. Light Arrow Flight

This part of the article is going to deal with the real world numbers when a light arrow is shot vs. a heavier arrow and the speeds measured at different distances. By measuring the speed of different arrows at various distances, it is easy to calculate which arrow maintains more of its speed and thus its kinetic energy and momentum farther downrange.

Introduction and archery equipment for testing

Test equipment for this round of shooting

Test equipment for this round of shooting

If you care about all of the math and the theory behind how arrows decelerate after leaving the bow, please make sure to read all of the above article first. For this testing, I’ll be using two arrows that are virtually identical to each other on the outside, but one will be weighted with an extra shaft and weight layered on the inside.

These first numbers come from using two Victory VForce HV arrows, one standard that weighs in at 326 grains, the other layered with a 1516 aluminum shaft on the inside that weighs 580 grains. The heavier arrow also has additional weight on the inside near the point in order to keep the FOC of both of the arrows nearly identical. Both arrow use FOBs for the fletchings.

All arrows were shot from my Elite Envy set at 29″ and 60.2 lbs. and were chronographed with an Easton Pro Chronograph.

For this testing, I shot seven shots with each arrow at point blank range, 10 yards and 20 yards; then I threw out the highest and the lowest speeds, leaving five speeds to be averaged for the results. All of the speeds in each set were within +/- 0.4 fps.

Results of arrow speed testing

Speed (fps) KE Momentum Change from 0 Yards
326 Grains 0 Yards 316.4 72.3 0.457 KE Momentum
10 Yards 308.6 68.8 0.446 4.85% 2.45%
20 Yards 299.2 64.7 0.432 10.55% 5.42%
580 Grains 0 Yards 243.1 76.0 0.626 KE Momentum
10 Yards 239.6 73.9 0.617 2.81% 1.42%
20 Yards 235.3 71.3 0.606 6.27% 3.18%

The actual testing results follow what would be expected from the theory and math discussed earlier. I would like to test the speeds out to further distances and will do so in the future to get a better picture of behavior downrange.

It’s interesting to see just how much more the lighter arrow slows down and sheds its kinetic energy and momentum. The lighter arrow is losing speed at a rate 40-45% faster than the heavier arrow. At point blank range, the heavier arrow starts with 3.7 ft-lbs. of KE more than the light, and by only 20 yards it had 6.6 ft-lbs. more KE. I would say that is a significant difference!

Of course the extra KE and momentum come at a cost, trajectory. The heavier arrow is going to drop significantly more at every distance and yardage estimation becomes much more critical. A three to five yard mis-judgement in yardage with the lighter arrow could still result in a clean kill shot on an animal, while with the heavier arrow it most likely would result in a much poorer shot or even perhaps a complete miss.

That being said, having a fast arrow is no substitute for practice in yardage estimation or using a range finder when possible. Each archer needs to know their own equipment and make their own decisions on arrow weight depending on the game being hunted and where they are hunting. Know your equipment and practice with it constantly!

Future testing

In the future I plan on doing testing out to forty or more yards, and also with some lighter and heavier arrows. I have some 262 grain Speed Pro Max arrows donated to the cause that should be fun to play with (yes, I will be shooting them at 60 lbs, not recommended!) I will also use some different types of fletchings (4″ feathers, 1.6″ vanes) and varying helical/offsets as well to see how much they affect the speed.

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