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#### kg062007

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So according to a kinetic energy and momentum calculator, my 50lb recurve and 540-grain arrows are capable of 0.42 slugs of momentum. There are lots of charts that show the kinetic energy for different types of games, but I can't find one for momentum. Also, looking at the 2 equations, the kinetic energy equation seems to really value speed, whereas the momentum equation weighs them equally. So why is kinetic energy favored when talking about penetration?

#### Bowmania

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For me it's pretty simple. What the heck is a slug compared to a foot/pound of energy. Add in the fact that slugs aren't a whole number - 0.42.

That said, your are correct, momentum is a more accurate measurement of what we are interested. I just had my shoulder replaced and when I get back to shooting, I'm going to figure out what slugs I've been shooting. With my bow not my shotgun.

Bowmania

#### GameWarden

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So according to a kinetic energy and momentum calculator, my 50lb recurve and 540-grain arrows are capable of 0.42 slugs of momentum. There are lots of charts that show the kinetic energy for different types of games, but I can't find one for momentum. Also, looking at the 2 equations, the kinetic energy equation seems to really value speed, whereas the momentum equation weighs them equally. So why is kinetic energy favored when talking about penetration?
I think KE is easier for some to comprehend and it favours speed which sells bows. My 610 grain arrows are .47 slugs and 41 ft/lbs.

#### Carboniac

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KE is what is used by the broad head to cut a hole as well as anything else the arrow accomplishes. .
Momentum is what keeps the arrow moving. When all of the momentum is transferred to the the target the arrow stops going forward.

#### Glynn

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I'm curious how much KE it takes to drive a razor sharp sword 8 or 10 inches into flesh..and say.. rib bones?

Arrows and broadheads don't have to displace or disrupt anything like bullets do. They just have to cut soft stuff that bleeds a lot.

#### UglyJoe

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I'm curious how much KE it takes to drive a razor sharp sword 8 or 10 inches into flesh..and say.. rib bones?

Arrows and broadheads don't have to displace or disrupt anything like bullets do. They just have to cut soft stuff that bleeds a lot.
... Which still requires energy...

To the OP,

TL; DR: KE is used because that's what we actually independently control in an archery system; momentum doesn't quantify anything related to penetration or lethality; the charts are essentially useless because they don't account for HOW your arrow uses its KE.

This is a very slippery topic, and one almost entirely misunderstood by the archery community. Even the more educated with engineering backgrounds and the like usually get this wrong.

First, DO NOT think of KE as "one-half mass times velocity squared". That is NOT what energy is. Energy is the capacity of a system to influence (do "work" on) other systems. Likewise DO NOT think of momentum as "mass times velocity". Each are there own physical quantity and these equations are simply ways of calculating these quantities.

For example, if your floor is a rectangle its area can be obtained by the equation Area = base × height. There is no sense in which the area physically is base times height—a circular floor has an area but has no base and no height. Area is a completely different physical quantity than length. The equation is just a relation between area and lengths that happens to be true for a rectangle (and not always; the equation is only strictly true in Euclidean spaces). Same for KE. It can be calculated with the equation KE = 0.5 × mass × v^2, but it isn't in any sense "mass times velocity squared".

There are 4 quantities that are meaningful when it comes to archery, particularly for hunting (let's ignore the vector nature of velocity and momentum, as this is a finer detail that is unimportant in an elementary analysis for archery).

Velocity: how "fast" a system moves from here to there within a chosen reference frame

Energy: the capacity of a system to do work on external systems. In the case of an arrow the only important energy is kinetic, as its internal and potential energies change very little during a shot/animal penetration.

Mass: a measure of the inertia of a system. Inertia is the tendency of a system to resist acceleration, so the more massive a system, the greater the forces needed both to get the mass moving as well as to stop the mass once it is moving.

Momentum (symbol p)... This is the trickiest one, and by far the most misunderstood, because it is an abstract quantity that doesn't seem to mesh with everyday experience. It is NOT "mass in motion". That's something physicists say when they are being lazy and don't want to try and explain what this nebulous quantity actually is. In fact, a beam of light has momentum, but no mass—clearly momentum can't then be "mass in motion". It doesn't quantify anything. You will hear people say it is conserved, but this is only true for an isolated system coordinatized in an inertial reference frame. The truth is that momentum is a point (EDIT: "point" is wrong; the right term is "coordinate") in phase space used to define the state of a system (or the states of the constituent particles of that system). It is also the quantity whose rate of change is equal to the force acting on the system. While unbelievably useful in the theoretical structure of classical and modern physics, the one thing it simply does NOT do is quantify the amount of penetration your arrow may obtain in an animal. Contrary to what you have probably read countless times on archery forums, you should simply ignore the momentum of your setup. It quantifies nothing. When archers say "you need a healthy dose of momentum for good penetration", they almost always mean "you need a heavy arrow for good penetration". Heavy, of course, being a relative term.

So four quantities that characterize our arrow in flight. Additionally, there are two independent constraint equations, which may be written as:

KE − 1/2 × m × v^2 = 0
p − m × v = 0

I write them in these unfamiliar forms to emphasize that each of these quantities are unique and not somehow defined in terms of the other quantities. The truth is we can rearrange these equations however we wish to make them the most useful with respect to our individual problem/system.

Because there are two constraint equations, we can choose 2 quantities to be independent variables and choose the others to be dependent variables. The question is, which two do we choose for archery? This is one of those examples where nature and theory reflect each other precisely. We have two choices in theory and two choices in archery. We get to choose our arrow and we get to choose our bow. Clearly our arrow choice determines mass, so we should take mass as one of our independent variables. The other is our choice of bow. What quantity does the bow correspond to? It corresponds to energy. The bow is an energy transfer device. It converts the chemical energy stored in your body into the potential energy stored in the bow at full draw and finally to the kinetic energy of the arrow after the shot. While (assuming that draw length is consistent shot to shot) the actual independent variable is the potential energy stored in the bow, to first approximation the kinetic energy produced by the bow is independent of arrow choice. (This is essentially true for compounds; it is less so for traditional bows. Nevertheless, to first approximation, thinking of a bow as a constant KE device is reasonable for initial considerations.) We now rearrange the equations above to:

v = SQRT(2 × KE / m)

p = SQRT(2 × KE × m)

Velocity and momentum are dependent variables. We can't independently choose either; they depend on the combination of bow and arrow we choose; they depend on the combination of energy and mass we chose. Mathematically, we would say that momentum and velocity are functions of KE and mass.

This is why it is essentially wrong to consider a chart of "required momentum" to kill an animal of a particular species. We don't have explicit control over momentum in an archery setup. We do however have explicit control over energy and mass. You can build an arrow to whatever mass you wish, independent of your bow choice. You can build a bow to convert whatever energy from chemical to kinetic we wish, independent of our choice of arrow. But we cannot control momentum directly; we have to choose an arrow and a bow (mass and energy) and momentum gets determined for us once we make those choices.

This is also why a "KE chart" is a useless piece of fiction. Penetration/lethality, like momentum, is a dependent variable that depends on both the arrow choice and the bow choice. The KE charts make more sense when accompanied by an indication of arrow mass. If you are talking about shooting a 300 grain arrow, then you need a lot of KE because that amount of mass will not resist deceleration in animal tissue and needs more KE to get it to through the animal. If you're shooting a 650 grain arrow you need a lot less KE because that arrow will resist deceleration much more than the light arrow. The heavy arrow is more efficient with its energy than the light arrow. This also, of course, depends on many parameters such as broadhead choice, blade sharpness, arrow diameter, etc. In short, it's hard to boil down arrow lethality to just two independent variables (arrow mass and bow energy) because the parameters of the arrow are so important (2 blade head, 3 blade head, mechanical, fixed, COC or no, etc.), and impossible to boil it down to one (energy only). The charts aren't helpful.

The truth is, for a soft tissue shot with a fixed, sharp, COC, efficient broadhead, it takes almost no energy for an arrow to penetrate an animal, assuming the arrow is heavy enough. And heavy enough doesn't mean 800 grains. A 500 grain arrow (or even less) with 30 ft·lbf KE in the right spot with a good broadhead will kill most animals on the planet. In the wrong spot it won't go in two inches. The KE charts you usually see should probably be called "overkill charts". For example, a quick Google search finds bowhunting.com suggesting 65 ft·lbf for a grizzly bear. That corresponds to a 700 grain arrow going 205 fps. That will kill just about anything on the planet, with the possible exception of some African dangerous game. That also corresponds to a 250 grain arrow pushing 342 fps, which I wouldn't take to hunt turkeys. You simply can't quantify penetration with one number only; you need to specify both independent variables, and charts that only suggest one (such as KE) are essentially useless.

#### Carboniac

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You took a lot of time to give functionally the same answer I did.

#### UglyJoe

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You took a lot of time to give functionally the same answer I did.
Momentum doesn't "keep the arrow moving". That would be mass, if anything.

But your point is well-taken. I've written long papers, put together presentations, considered making videos and doing podcasts, etc. on this subject... what is the point? The "real" answers are more technical than the general archery public cares to engage, and almost never worth the effort. The truth is so simple that it is only people who read bad science in bowhunting articles who get confused.

Shoot a heavier bow, get more lethality. Shoot a heavier arrow, get more lethality. How heavy of a bow and how heavy of an arrow do you need? Blurred lines. Boring enough that I can only be bothered to generate exposition on it once, maybe twice a year.

#### hybridbowhunter

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Thank you UJoe for that great post.

#### Styksnstryngs

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... Which still requires energy...

To the OP,

TL; DR: KE is used because that's what we actually independently control in an archery system; momentum doesn't quantify anything related to penetration or lethality; the charts are essentially useless because they don't account for HOW your arrow uses its KE.

This is a very slippery topic, and one almost entirely misunderstood by the archery community. Even the more educated with engineering backgrounds and the like usually get this wrong.

First, DO NOT think of KE as "one-half mass times velocity squared". That is NOT what energy is. Energy is the capacity of a system to influence (do "work" on) other systems. Likewise DO NOT think of momentum as "mass times velocity". Each are there own physical quantity and these equations are simply ways of calculating these quantities.

For example, if your floor is a rectangle its area can be obtained by the equation Area = base × height. There is no sense in which the area physically is base times height—a circular floor has an area but has no base and no height. Area is a completely different physical quantity than length. The equation is just a relation between area and lengths that happens to be true for a rectangle (and not always; the equation is only strictly true in Euclidean spaces). Same for KE. It can be calculated with the equation KE = 0.5 × mass × v^2, but it isn't in any sense "mass times velocity squared".

There are 4 quantities that are meaningful when it comes to archery, particularly for hunting (let's ignore the vector nature of velocity and momentum, as this is a finer detail that is unimportant in an elementary analysis for archery).

Velocity: how "fast" a system moves from here to there within a chosen reference frame

Energy: the capacity of a system to do work on external systems. In the case of an arrow the only important energy is kinetic, as its internal and potential energies change very little during a shot/animal penetration.

Mass: a measure of the inertia of a system. Inertia is the tendency of a system to resist acceleration, so the more massive a system, the greater the forces needed both to get the mass moving as well as to stop the mass once it is moving.

Momentum (symbol p)... This is the trickiest one, and by far the most misunderstood, because it is an abstract quantity that doesn't seem to mesh with everyday experience. It is NOT "mass in motion". That's something physicists say when they are being lazy and don't want to try and explain what this nebulous quantity actually is. In fact, a beam of light has momentum, but no mass—clearly momentum can't then be "mass in motion". It doesn't quantify anything. You will hear people say it is conserved, but this is only true for an isolated system coordinatized in an inertial reference frame. The truth is that momentum is a point in phase space used to define the state of a system (or the states of the constituent particles of that system). It is also the quantity whose rate of change is equal to the force acting on the system. While unbelievably useful in the theoretical structure of classical and modern physics, the one thing it simply does NOT do is quantify the amount of penetration your arrow may obtain in an animal. Contrary to what you have probably read countless times on archery forums, you should simply ignore the momentum of your setup. It quantifies nothing. When archers say "you need a healthy dose of momentum for good penetration", they almost always mean "you need a heavy arrow for good penetration". Heavy, of course, being a relative term.

So four quantities that characterize our arrow in flight. Additionally, there are two independent constraint equations, which may be written as:

KE − 1/2 × m × v^2 = 0
p − m × v = 0

I write them in these unfamiliar forms to emphasize that each of these quantities are unique and not somehow defined in terms of the other quantities. The truth is we can rearrange these equations however we wish to make them the most useful with respect to our individual problem/system.

Because there are two constraint equations, we can choose 2 quantities to be independent variables and choose the others to be dependent variables. The question is, which two do we choose for archery? This is one of those examples where nature and theory reflect each other precisely. We have two choices in theory and two choices in archery. We get to choose our arrow and we get to choose our bow. Clearly our arrow choice determines mass, so we should take mass as one of our independent variables. The other is our choice of bow. What quantity does the bow correspond to? It corresponds to energy. The bow is an energy transfer device. It converts the chemical energy stored in your body into the potential energy stored in the bow at full draw and finally to the kinetic energy of the arrow after the shot. While (assuming that draw length is consistent shot to shot) the actual independent variable is the potential energy stored in the bow, to first approximation the kinetic energy produced by the bow is independent of arrow choice. (This is essentially true for compounds; it is less so for traditional bows. Nevertheless, to first approximation, thinking of a bow as a constant KE device is reasonable for initial considerations.) We now rearrange the equations above to:

v = SQRT(2 × KE / m)

p = SQRT(2 × KE × m)

Velocity and momentum are dependent variables. We can't independently choose either; they depend on the combination of bow and arrow we choose; they depend on the combination of energy and mass we chose. Mathematically, we would say that momentum and velocity are functions of KE and mass.

This is why it is essentially wrong to consider a chart of "required momentum" to kill an animal of a particular species. We don't have explicit control over momentum in an archery setup. We do however have explicit control over energy and mass. You can build an arrow to whatever mass you wish, independent of your bow choice. You can build a bow to convert whatever energy from chemical to kinetic we wish, independent of our choice of arrow. But we cannot control momentum directly; we have to choose an arrow and a bow (mass and energy) and momentum gets determined for us once we make those choices.

This is also why a "KE chart" is a useless piece of fiction. Penetration/lethality, like momentum, is a dependent variable that depends on both the arrow choice and the bow choice. The KE charts make more sense when accompanied by an indication of arrow mass. If you are talking about shooting a 300 grain arrow, then you need a lot of KE because that amount of mass will not resist deceleration in animal tissue and needs more KE to get it to through the animal. If you're shooting a 650 grain arrow you need a lot less KE because that arrow will resist deceleration much more than the light arrow. The heavy arrow is more efficient with its energy than the light arrow. This also, of course, depends on many parameters such as broadhead choice, blade sharpness, arrow diameter, etc. In short, it's hard to boil down arrow lethality to just two independent variables (arrow mass and bow energy) because the parameters of the arrow are so important (2 blade head, 3 blade head, mechanical, fixed, COC or no, etc.), and impossible to boil it down to one (energy only). The charts aren't helpful.

The truth is, for a soft tissue shot with a fixed, sharp, COC, efficient broadhead, it takes almost no energy for an arrow to penetrate an animal, assuming the arrow is heavy enough. And heavy enough doesn't mean 800 grains. A 500 grain arrow (or even less) with 30 ft·lbf KE in the right spot with a good broadhead will kill most animals on the planet. In the wrong spot it won't go in two inches. The KE charts you usually see should probably be called "overkill charts". For example, a quick Google search finds bowhunting.com suggesting 65 ft·lbf for a grizzly bear. That corresponds to a 700 grain arrow going 205 fps. That will kill just about anything on the planet, with the possible exception of some African dangerous game. That also corresponds to a 250 grain arrow pushing 342 fps, which I wouldn't take to hunt turkeys. You simply can't quantify penetration with one number only; you need to specify both independent variables, and charts that only suggest one (such as KE) are essentially useless.
Well I thought this was a good read. Learning isn't always a bad thing.

kg062007

#### Carboniac

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Momentum doesn't "keep the arrow moving". That would be mass, if anything.

But your point is well-taken. I've written long papers, put together presentations, considered making videos and doing podcasts, etc. on this subject... what is the point? The "real" answers are more technical than the general archery public cares to engage, and almost never worth the effort. The truth is so simple that it is only people who read bad science in bowhunting articles who get confused.

Shoot a heavier bow, get more lethality. Shoot a heavier arrow, get more lethality. How heavy of a bow and how heavy of an arrow do you need? Blurred lines. Boring enough that I can only be bothered to generate exposition on it once, maybe twice a year.
Well momentum not mass is the vector derivative of KE so I think its more representative of the tendency to keep moving on one direction which is what penetration is kind of about.

For practical purposes momentum keeps the arrow moving(even if you want to get technical/philosophical and say it doesnt exist because only mass and energy exist) . When all the momentum has transferred into the target the arrow stops.

but anyway
We can think just in terms of KE if you like for now.

The moving arrow has KE. That energy is leaves the arrow as it
-makes the hole. Both by breaking and cutting the structure it encounters and by displacing the media outwards for a channel
-deforms the arrow itself( if that happens)
-pushes the target forward as its transferred from the arrow by friction
-some of it goes into heat friction(and this heavily overlaps with creating the channel if you have an elastic medium.)
The friction vs making a hole thing gets messy but ultimately it pushes the target forward, it breaks the structure of the target and it self, and it creates heat.

So it ultimately comes down to a heavy arrow for a given KE is more efficient at travelling though a medium. Because less energy wasted on friction(which both pushing the target forward and creating heat) and less energy is expended in creating the channel.(at much higher velocities the KE might clear a path or change the mechanical properties of media but that isnt much relevant to bows and arrows.)

So I agree with you that more KE is better than less KE and more mass is better than less mass.

But you are unreasonably discounting momentums significance. It may seem like Im being pedantic but the velocity portion of momentum actually matters as it is conserved( and only lost in pushing forward) just as much as the mass portion is.

And it maybe that momentum is better than the other options in predicting penetration in situation where drag matters more than mechanical deformation. -Im not sure.

#### UglyJoe

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Well momentum not mass is the vector derivative of KE so I think its more representative of the tendency to keep moving on one direction which is what penetration is kind of about.

For practical purposes momentum keeps the arrow moving(even if you want to get technical/philosophical and say it doesnt exist because only mass and energy exist) . When all the momentum has transferred into the target the arrow stops.
Momentum does happen to be the derivative of KE with respect to velocity, but that is an extremely technical detail that doesn't mean much for this discussion. In Lagrangian mechanics we actually take this to be the definition of canonical momentum (the derivative of the Lagrangian function with respect to a generalized velocity coordinate). In most cases this turns out to be somewhat more complicated than the simple p = mv relation of a free, massive particle. All this to say that, while what you say regarding momentum being the derivative of KE is true, it's not really a factor in this particular discussion.

What is NOT true is that momentum is representative of the "tendency to keep moving in one direction..." That's inertia, plain and simple. The inertia of a system, quantified by its mass, is the property you are describing. Momentum is just a description of the system's state within a given reference frame. How big or small (how many "slugs", although that is not a unit of momentum) the momentum is does not play into determining how much penetration the arrow can do, at least not directly.

It's not a semantics game. If I tell you that an arrow is carrying 30 ft·lbf of KE, then the maximum amount of work that that arrow can do is 30 ft·lbf. End of story. If I tell you an arrow has 0.40 slug·ft/s, what does that tell you? Not much. If we assume the arrow comes to a stop (not a requirement, by the way, as anyone who has seen an arrow deflect or bounce off an animal can attest), all this tells you is that the impulse applied to the arrow was 0.40 slug·ft/s, but the momentum carried by the arrow does not determine why the impulse was what it was. Impulse (generally a function of velocity in collision/penetration mechanics) determines how the momentum of the arrow changes, not the other way around. The momentum of the arrow doesn't determine or quantify the impulse of penetration.

but anyway
We can think just in terms of KE if you like for now.

The moving arrow has KE. That energy is leaves the arrow as it
-makes the hole. Both by breaking and cutting the structure it encounters and by displacing the media outwards for a channel
-deforms the arrow itself( if that happens)
-pushes the target forward as its transferred from the arrow by friction
-some of it goes into heat friction(and this heavily overlaps with creating the channel if you have an elastic medium.)
The friction vs making a hole thing gets messy but ultimately it pushes the target forward, it breaks the structure of the target and it self, and it creates heat.
Agreed. Also, let's throw in sound generation, which is another vector for energy loss. If shooting a mech then we have opening the broadhead, which can actually cost significant energy at high velocity because of additional losses.

So it ultimately comes down to a heavy arrow for a given KE is more efficient at travelling though a medium. Because less energy wasted on friction(which both pushing the target forward and creating heat) and less energy is expended in creating the channel.(at much higher velocities the KE might clear a path or change the mechanical properties of media but that isnt much relevant to bows and arrows.)

So I agree with you that more KE is better than less KE and more mass is better than less mass.

But you are unreasonably discounting momentums significance. It may seem like Im being pedantic but the velocity portion of momentum actually matters as it is conserved( and only lost in pushing forward) just as much as the mass portion is.
Actually, in this analysis, momentum and velocity are not conserved. They are only conserved for isolated systems in inertial reference frames. Our system here is not isolated (forces act between the archer and the earth through their feet and between the animal and the earth through their feet; significant forces act on the arrow through interaction with all of the air molecules generating drag). Unless you intend to keep up with the trajectories of every particle in the entire closed system, i.e. Planet Earth, then momentum is not conserved in this problem, and becomes even less useful as a predictor of any kind.

And it maybe that momentum is better than the other options in predicting penetration in situation where drag matters more than mechanical deformation. -Im not sure.
It turns out that when looking at drag forces in a viscous fluid under the appropriate condition such that the drag is proportional to velocity (and not velocity squared, which is the usual case for ballistics), the particular combination of mass and energy that determines position as a function of time (i.e. "penetration") is exactly the momentum of the system. So in this very particular case (which is not the case for animal penetration or for drag forces acting on the arrow as it moves through air) the total momentum does predict the penetration depth of the system. This is more of an accident of nature than anything else, and is certainly not indicative of any general principle. For systems where the only forces are sliding or kinetic friction which is independent of velocity, it turns out that mass drops out of the solutions and the position function only depends on energy (which is closely related to how archery targets are designed to work). Again, this is not a general case. You can always find weird exceptions when looking at isolated problems. The most general case is that trajectory are functions of projectile mass and energy—momentum only plays into the story tangentially (that's a maths joke).

Let me emphasize again my main point; momentum is not a control variable in archery, so this immediately disqualifies it as something you should manipulate to determine arrow lethality. The control variables are mass of the arrow and energy of the bow. The momentum is whatever it happens to be, but we determine penetration potential and arrow lethality by manipulation of our independent control variables, as well as by adjusting the parameters of the arrow (broadhead shape, FOC, shaft diameter, etc.).

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#### UglyJoe

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This is an excellent description of momentum and kinetic energy. I found most the key points there. Momentum vs Kinetic Energy: The Key Differences (& How They Are Related!) – Profound Physics

I was a quantum mechanics guy so I had to deal with quantum equations when looking at energy and momentum.
Hank, I just wrote a response touching on some of this. Apparently it's being held up for moderation for some reason. I'm reading over the link you posted now.

What did you do in your past life? I'm still a quantum guy. Usually elementary quantum mechanics is treated with the Hamiltonian approach, where the energy is a function of momentum... essentially the exact opposite of what we have in archery, where the momentum is a function of energy.

EDIT: Just finished reading through the article. It's mostly pretty good, though the author makes a few common mistakes. First, there is no notion of a "negative" vector, so momentum, and its magnitude, are only positive. Components of the momentum may be negative in a particular choice of reference frame, which is what the author was getting at, but since the article intends to be technically accurate, this is an important point. I'm not a huge fan of thinking of KE or momentum as "relying on" velocity. As I pointed out in my first post in this thread, KE, p, and v are each unique physical properties that are related by constraint equations. They all depend on each other but none are in any real sense defined by each other.

The only place the author really went wrong was in understanding what quantum mechanical operators and wave functions are. Operators represent physical properties in QM. These properties are the only "real" thing in QM. The wave function itself is simply a mathematical construct representing the state of the system and has no physical manifestation. The author seems to imply the opposite conclusion (that KE and momentum aren't "real" in QM but the wave function is).

#### Carboniac

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1 Inertia is the tendency to remain in your current state
Momentum is the tendency of that mass to move in particular direction. Its mass x velocity
This isnt complicated. You are trying to make it seem complicated then wave your hands and say it doesnt fit for nebulous reasons. And then you do switch to a semantics game of saying momentum is a result rather than a cause - but that irrelevant as we are only determining if momentum is predictive.

2 Actually momentum is conserved - you just may not be be able to track it because there are so many elements.

If I shoot a ball in the air the total momentum of the ball and arrow remains the same. Except whats transferred to the air behind the ball. .

If I shoot a ball on the ground that momentum is going to going into the earth.

Yes calculating it is impossibly complicated. But understanding it is simple. I can tell you know this. You are making a bad faith argument.

3 For our purposes the earth is an isolated system and since were focused on the momentum leaving the arrow and not concerned with where it ends up ultimately your point is irrelevant. Were talking about systems the initial target will have much much more mass than the arrow.

4 But we are dealing with those conditions. So even you concede after all, that momentum actually maybe useful as a general predictor for penetration when hunting.

5 Except in the real world its actually easy change your arrows momentum. You just make the arrow heavier. Its easy to figure the momentum you just multiply the new velocity by the new mass.

You are pretending this is more complicated than it is.
If you want to look at it as the better penetrating efficiency because the arrow is going slower with similar KE - fine(but its odd that you didnt explain that also simple concept in your first post). Frankly I think you just wanted to show that you know physics and didnt even bother to think about why more mass helps.
But momentum seems to actually be a useful predictor.

#### UglyJoe

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1 Inertia is the tendency to remain in your current state
Momentum is the tendency of that mass to move in particular direction. Its mass x velocity
This isnt complicated. You are trying to make it seem complicated then wave your hands and say it doesnt fit for nebulous reasons. And then you do switch to a semantics game of saying momentum is a result rather than a cause - but that irrelevant as we are only determining if momentum is predictive.
Mass doesn't have a "tendency" to move in any direction unless coupled to some sort of potential field, like a gravitational field. Momentum isn't a "tendency" to do anything; its a designation of the state of a system in a given reference frame.

2 Actually momentum is conserved - you just may not be be able to track it because there are so many elements.

If I shoot a ball in the air the total momentum of the ball and arrow remains the same. Except whats transferred to the air behind the ball. .

If I shoot a ball on the ground that momentum is going to going into the earth.

Yes calculating it is impossibly complicated. But understanding it is simple. I can tell you know this. You are making a bad faith argument.
You essentially just repeated what I've already stated. Yes, the total momentum of an isolated system is conserved. Our archery system is not isolated, so the momentum of the arrow is not a conserved quantity... clearly, it starts with no momentum, has a large momentum in flight, and ends with no momentum. There isn't anything deep in this, other than pointing out the fact that momentum is conserved for an isolated system doesn't really play into this discussion since the system isn't isolated and does in fact interact with its environment in such a way that we can't keep track of all of the exchanges of momenta in any useful way.

3 For our purposes the earth is an isolated system and since were focused on the momentum leaving the arrow and not concerned with where it ends up ultimately your point is irrelevant. Were talking about systems the initial target will have much much more mass than the arrow.
What point? Your making an argument that we know how much momentum the arrow has before it hits the animal and if we know that it has no momentum afterwards, then than momentum had to "go" somewhere. You've nowhere made a cogent argument indicating why this is an indicator of penetration potential/lethality.

4 But we are dealing with those conditions. So even you concede after all, that momentum actually maybe useful as a general predictor for penetration when hunting.
What conditions? The conditions of drag linear in velocity? I thought I made it clear that we most certainly are not dealing with those conditions in archery. Those conditions are for very tiny systems moving very slowly in a viscous fluid. An example is the Millikan oil drop experiment, where the drops experience essentially drag linear in velocity and the drops have a diameter of approximately one micrometer and a speed of about 10^(−5) m/s. That's clearly not what is going on in an archery shot, so what is your point?

5 Except in the real world its actually easy change your arrows momentum. You just make the arrow heavier. Its easy to figure the momentum you just multiply the new velocity by the new mass.
Hah. You just circled back to my main point. You change the arrow mass (independent variable) and this results in a change of the arrows momentum (dependent variable). It's not the momentum increase that causes the penetration potential to increase—it's the increase in the arrows inertia making it harder to stop that increases penetration potential.

You are pretending this is more complicated than it is.
If you want to look at it as the better penetrating efficiency because the arrow is going slower with similar KE - fine(but its odd that you didnt explain that also simple concept in your first post). Frankly I think you just wanted to show that you know physics and didnt even bother to think about why more mass helps.
But momentum seems to actually be a useful predictor.
I know why more mass helps; more mass means more inertia and so the arrow is harder to slow down... therefore more penetration. You say I make things more complicated, but the truth is introduction of momentum is more complicated and unnecessary to boot.

Your logic is funny archery logic, and I've seen it all over the place: "We increase mass, which increases momentum, which increases penetration". The real logic: "We increase mass (inertia) of our system, which increases penetration". No middleman (momentum) needed.

#### GameWarden

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@Carboniac Im enjoying this discussion, refreshing to read

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#### Carboniac

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I have to start with this. And its not meant as an argument. Much of what you say is just semantic dancing. You alternate between acting pompous and making pointless references - and yourself taking short cuts.

At the core your point is simple - you feel momentum is a derived property and in turn not relevant.
But this is demanding sort of logic standard is oddly in contrast to you original long "treatise" that never actually says why the increased mass was was useful. You just sort of pronounce that its good. You also included the sentence "The truth is, for a soft tissue shot with a fixed, sharp, COC, efficient broadhead, it takes almost no energy for an arrow to penetrate an animal, assuming the arrow is heavy enough" - you must be real proud of that doozy.

So why dont you cool the attitude.

#### Carboniac

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Mass doesn't have a "tendency" to move in any direction unless coupled to some sort of potential field, like a gravitational field. Momentum isn't a "tendency" to do anything; its a designation of the state of a system in a given reference frame.
Once again semantics. I can find definitions that agree with me. I think you understand and are pretending you dont.

You essentially just repeated what I've already stated. Yes, the total momentum of an isolated system is conserved. Our archery system is not isolated, so the momentum of the arrow is not a conserved quantity... clearly, it starts with no momentum, has a large momentum in flight, and ends with no momentum. There isn't anything deep in this, other than pointing out the fact that momentum is conserved for an isolated system doesn't really play into this discussion since the system isn't isolated and does in fact interact with its environment in such a way that we can't keep track of all of the exchanges of momenta in any useful way.
I only pointed it out because you had said "Actually, in this analysis, momentum and velocity are not conserved. They are only conserved for isolated systems in inertial reference frames " in an attempt at a spurious argument.

What point? Your making an argument that we know how much momentum the arrow has before it hits the animal and if we know that it has no momentum afterwards, then than momentum had to "go" somewhere. You've nowhere made a cogent argument indicating why this is an indicator of penetration potential/lethality.
I was not making a point there other than showing how silly your argument about needing to keep track of all the molecules on earth was. And i suceeded.

What conditions? The conditions of drag linear in velocity? I thought I made it clear that we most certainly are not dealing with those conditions in archery. Those conditions are for very tiny systems moving very slowly in a viscous fluid. An example is the Millikan oil drop experiment, where the drops experience essentially drag linear in velocity and the drops have a diameter of approximately one micrometer and a speed of about 10^(−5) m/s. That's clearly not what is going on in an archery shot, so what is your point?
Honestly we would need to do actual to know just how velocity corresponded to drag as an arrow is driven through a deer. I suspect it would not be linear - butIi dont know.

Hah. You just circled back to my main point. You change the arrow mass (independent variable) and this results in a change of the arrows momentum (dependent variable). It's not the momentum increase that causes the penetration potential to increase—it's the increase in the arrows inertia making it harder to stop that increases penetration potential.
Or I circled back to my original point.
Inertia isnt doing much if it isnt moving.

I know why more mass helps; more mass means more inertia and so the arrow is harder to slow down... therefore more penetration. You say I make things more complicated, but the truth is introduction of momentum is more complicated and unnecessary to boot.
or the momentum keeps it moving.

Your logic is funny archery logic, and I've seen it all over the place: "We increase mass, which increases momentum, which increases penetration". The real logic: "We increase mass (inertia) of our system, which increases penetration". No middleman (momentum) needed.

Anyway momentum is going to be way more predictive of penetration than inertia/mass.

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