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Altrunchen
2015-06-23, 09:50 PM
Introduction:
One thing that I have occasionally found myself wanting/needing to know is how to stat real-world situations or things based on the force that they exert. I rankled at trying to determine how many newtons of force are equivalent to a damage die for a long time. Then, whether by suggestion (and I just forgot), or whatever means it occured to me that it may be possible to determine the ratio of Newtons to Damage die by investigating what the draw force and/or impact force of a longbow was. Since the longbow is a weapon in D&D with a specific statistic referring to damage (1d8), I figured that if I could find a range of numbers then I could find either a median or a mean average that would tell me how many newtons of force equaled one d8.

Considerations:
There are many variables that go into determining how effective the impact of an arrow is in terms of damage. Indeed assigning a die roll to such a widely variable thing is a dramatic simplification. Since the damage can vary based on angle of impact and firing angle, wind, accuracy, the target's material properties, precisely how far the bow was drawn, gravity and probably many other variables, I decided to ignore this side of the equation and instead look at a number that has much more precision (relatively speaking) and fewer variables: the draw weight of a longbow. Admittedly this isn't likely to be as accurate as measuring the impact forces from a wide berth of tests from many angles with one set of tests done in a vacuum chamber (removing air resistance), some outdoors, and some indoors, using a mechanical drawing mechanism to ensure consistent bowstring draw position and so on. But I think that given the limitations it may be the best I can do under the circumstances to base my ratio on draw weight.

Investigation:
People debate the draw-stengths of longbows heavily and the results are only slightly consistent. The most persuasive case to be made for an average draw-force of a longbow for the purposes of a medieval fantasy rpg like D&D is one that refers to the many longbows recovered from the sunken ship named the Mary Rose. This rating, from the English Warbow Society (http://www.theenglishwarbowsociety.com/warbow_EN.html) puts the ratio at a 140 lb (622.75 N) per 1d8. But since the idea of the most common draw-weight of a longbow is one so heavily debated, I figured that I would make a list of every rating I found and would present each of them here for other DMs to consider when deciding what ratio of force to damage die they want to use.

Force to Damage Die Ratios:
Honestly, it seems that the longbow was inconsistently made in terms of their draw-weights and that as such this causes for so much debate. But I honestly like the Mary Rose concept of taking a very common (relative to other finds) draw weight and basing the statistic on the mode average from those findings. In the end I feel it's best to present what I found and to let people decide for themselves with this as a suggested guideline.

My Suggestion: 140 lbf. - 622.751024 N - 1d8

Other Ratios:


Force (lbf)

Force (N)

Damage Dice



60

270

1d8



67.8896

302.0000
1d8


79.9967
355.8577
1d8


99.9955
444.8200
1d8


119.9952
533.7865
1d8


139.9944

622.7510
1d8


149.9416
667.0000
1d8


179.9906
800.6700
1d8


199.9911
889.6400
1d8



Orange = Modern longbows (hunting)
Blue = Mary Rose mode average longbows

Sources:

https://en.wikipedia.org/?title=English_longbow
http://www.thebeckoning.com/medieval/crossbow/cross_l_v_c.html
http://history.stackexchange.com/a/8023
http://forums.taleworlds.com/index.php?topic=13567.130;wap2
http://blogs.indium.com/blog/an-interview-with-the-professor/who-holds-the-world-record-in-pulling-back-a-longbow
http://www.archers-review.com/magazine-articles/june-2010-longbows-of-the-mary-rose
http://www.theenglishwarbowsociety.com/warbow_EN.html

Maglubiyet
2015-06-23, 10:21 PM
Very impressive research.

I think it's difficult to quantify damage in newtons, though. The mechanism of injury is different for different weapons. A warhammer and a battleaxe each do 1d8 too, and both have x3 critical like the longbow.

But a warhammer causes blunt force trauma, a battleaxe wounds by causing severe lacerations and blood loss, and a longbow does its damage by puncturing vital organs. How can you quantify each of these using the same metric?

Altrunchen
2015-06-23, 10:30 PM
Very impressive research.

I think it's difficult to quantify damage in newtons, though. The mechanism of injury is different for different weapons. A warhammer and a battleaxe each do 1d8 too, and both have x3 critical like the longbow.

But a warhammer causes blunt force trauma, a battleaxe wounds by causing severe lacerations and blood loss, and a longbow does its damage by puncturing vital organs. How can you quantify each of these using the same metric?

In reality, you can't. But the game has already quantified them by giving them a common damage die. I admit it's probably an oversimplification if realism is desired, but I feel that sacrifice is made so that the game is more fun. At least to some people I'd wager.

The Evil DM
2015-06-23, 10:36 PM
I commend your effort, and it is a worthy attempt at defining damage but ultimately force in terms of bow draw strength is the incorrect approach.


Impulse Momentum Theorem is the appropriate methodology to consider real world damage effects from weapons or object impacts.

For an introductory reference use Wiki Impulse Momentum Theorem (https://en.wikipedia.org/wiki/Impulse_(physics))

Force alone does not indicate damage. Force applied over a discrete quantity of time (an impact) yields a change in momentum for both objects. That change in momentum can be equated to energy equations for resulting impact motion. Energy lost during the impact becomes heat and damage. A secondary key to impulse and momentum analysis is understanding how impact types, elastic and inelastic affect the before impact and after impact momentum conditions.

Momentum is Mass * Velocity. This is why a bullet does a lot of damage with very small mass. It has high velocity. We human beings have been throwing mass at each other since the stone age, we only find better ways to throw that mass faster.

So for an example - assume I bounce a billiard ball off of my friends head. Now my friend has a fairly thick skull and when the ball strikes, there is momentary compression of both ball and skull. Then the billiard ball deflects and bounces away.

The initial momentum of the Ball-Skull System is Mballinitial*Vballinitial + 0 (assuming my friend was sitting on the couch and engaged in negligible motion)

The ball impact time is very short, fractions of a second, but during that time dT, damage is occurring and momentum is being transferred.

The ball deflects from skull and has a new Mballfinal*Vballfinal. Momentum is a vector so change in direction is a change in momentum. In addition my friend's head deflects away from the ball and has its own momentum Mheadfinal*Vheadfinal.

Momentum is conserved through impacts so Mballinitial*Vballinitial + 0 = Mballfinal*Vballfinal + Mheadfinal*Vheadfinal

The average force applied to my friends skull during impact is his change in momentum divided by dT. or (Mheadfinal*Vheadfinal)/dT. Calculus is required to determine peak force within the span dT. (it is not linear constant)

Using energy equations on the ball, The difference in the kinetic and potential energy for the system before and after the impact

(1/2mvballi2 + mghballi + 0 + 0) - (1/2mvballf2 + mghballf + 1/2mvheadf2 + mghheadf) = dE or change in energy.

The change in energy provides damage

It is important to note, energy equations are scalar and v is the magnitude of the velocity vector.

This is why the ability to roll with a punch or a hit reduces injury and why trying to take an impact through tension and rigidity increases it. Increased movement as a result of impact decreases the amount of energy available to transform into heat and damage.

If you are interested with assistance in this sort of study let me know I can guide you through more detail.

Edit

The force work above is an important first step in momentum analysis because it can be used to determine the initial velocity of an arrow.

Altrunchen
2015-06-23, 11:14 PM
I commend your effort, and it is a worthy attempt at defining damage but ultimately force in terms of bow draw strength is the incorrect approach.


Impulse Momentum Theorem is the appropriate methodology to consider real world damage effects from weapons or object impacts.

For an introductory reference use Wiki Impulse Momentum Theorem (https://en.wikipedia.org/wiki/Impulse_(physics))

Force alone does not indicate damage. Force applied over a discrete quantity of time (an impact) yields a change in momentum for both objects. That change in momentum can be equated to energy equations for resulting impact motion. Energy lost during the impact becomes heat and damage. A secondary key to impulse and momentum analysis is understanding how impact types, elastic and inelastic affect the before impact and after impact momentum conditions.

Momentum is Mass * Velocity. This is why a bullet does a lot of damage with very small mass. It has high velocity. We human beings have been throwing mass at each other since the stone age, we only find better ways to throw that mass faster.

So for an example - assume I bounce a billiard ball off of my friends head. Now my friend has a fairly thick skull and when the ball strikes, there is momentary compression of both ball and skull. Then the billiard ball deflects and bounces away.

The initial momentum of the Ball-Skull System is Mballinitial*Vballinitial + 0 (assuming my friend was sitting on the couch and engaged in negligible motion)

The ball impact time is very short, fractions of a second, but during that time dT, damage is occurring and momentum is being transferred.

The ball deflects from skull and has a new Mballfinal*Vballfinal. Momentum is a vector so change in direction is a change in momentum. In addition my friend's head deflects away from the ball and has its own momentum Mheadfinal*Vheadfinal.

Momentum is conserved through impacts so Mballinitial*Vballinitial + 0 = Mballfinal*Vballfinal + Mheadfinal*Vheadfinal

The average force applied to my friends skull during impact is his change in momentum divided by dT. or (Mheadfinal*Vheadfinal)/dT. Calculus is required to determine peak force within the span dT. (it is not linear constant)

Using energy equations on the ball, The difference in the kinetic and potential energy for the system before and after the impact

(1/2mvballi2 + mghballi + 0 + 0) - (1/2mvballf2 + mghballf + 1/2mvheadf2 + mghheadf) = dE or change in energy.

The change in energy provides damage

It is important to note, energy equations are scalar and v is the magnitude of the velocity vector.

This is why the ability to roll with a punch or a hit reduces injury and why trying to take an impact through tension and rigidity increases it. Increased movement as a result of impact decreases the amount of energy available to transform into heat and damage.

If you are interested with assistance in this sort of study let me know I can guide you through more detail.

Edit

The force work above is an important first step in momentum analysis because it can be used to determine the initial velocity of an arrow.

You're absolutely right, my science is way off and when it comes down to it, representing physical damage with a dice roll, regardless of so many variables, isn't realistic. I believe I noted in the beginning that I was aware that basing the damage on draw strength was not the ideal thing to do, and I admit the basis of this whole thing is a casual and ill-informed approach in the face of doing actual experiments with raw data. I also concede that I am taking a grossly non-mathematical approach to evaluating what can only be a situation that would be best represented with higher level physics. Unfortunately I do not have that data and decided to really just try to reverse-engineer the damage dice to the weapon to get a roughly estimated ratio, albeit a scientifically flawed one. But I appreciate your post and its intent.

The Evil DM
2015-06-23, 11:18 PM
You're absolutely right, my science is way off and when it comes down to it, representing physical damage with a dice roll, regardless of so many variables, isn't realistic. I believe I noted in the beginning that I was aware that basing the damage on draw strength was not the ideal thing to do, and I admit the basis of this whole thing is a casual and ill-informed approach in the face of doing actual experiments with raw data. Unfortunately I do not have that data and decided to really just try to reverse-engineer the damage dice to the weapon to get a roughly estimated ratio, albeit a scientifically flawed one. But I appreciate your post and its intent.

I am not trying to knock your effort - but you can do the same analysis using momentum and you will get something that will scale better for size because it includes mass in the system. My offer of assisting with a momentum based study of the problem was an actual offer.

Even using momentum - while getting closer - just assigns a value of energy change to damage while neglecting cutting and tearing effects and thus becomes a first order approximation of the effects of weapons.

Altrunchen
2015-06-23, 11:28 PM
I am not trying to knock your effort - but you can do the same analysis using momentum and you will get something that will scale better for size because it includes mass in the system. My offer of assisting with a momentum based study of the problem was an actual offer.

Even using momentum - while getting closer - just assigns a value of energy change to damage while neglecting cutting and tearing effects and thus becomes a first order approximation of the effects of weapons.

Oh I see, you actually want to help. Awesome! Thanks :).

And I see what you're saying, that it's the change in energy that causes the damage. Kind of how it isn't the fall that kills you, it's the landing.

My question is then: If mass is a part of the problem, how do you intend to represent this in a game with abstracted physics and separate size categories that are based on weight and dimensions? Doing these types of equations for each situation probably wouldn't be very conducive to a gaming atmosphere, so perhaps some means of scaling it with the size category might be in order?

If we're using energy change as the basis of damage, then how do you propose that we use mass to scale with it in terms of D&D's rules?

Another concern is how the mass of an arrow versus the mass of a longsword or war hammer, as well as their respective momentums (which varies all on the "human" user's input of energy) going to play into all this?

Feddlefew
2015-06-23, 11:28 PM
You'd also need to account for surface area of each weapon- 5 pounds of force spread out over one foot is very different from 5 pounds of force spread over one centimeter.

Edit: In Dwarf fortress, weapon damage is determined by how the weapon's physical properties, including size, density, flexibility, and compression verses those same properties in the materials they were being used against. This lead to such oddities as whips functioning like lightsabers, sand being the best throwing weapon, and most weapons becoming functional useless against Steel and Bronze armor.

The Evil DM
2015-06-24, 01:54 AM
Yes, I can help - Some of this stuff I have done before. I did an undergraduate thesis applying Impulse Momentum Theorem to Martial Arts, and some momentum and energy equations I already use for things in my own game. That said, I don't have unlimited time so there might be bits of digging or figuring left for you to go mess around with. Consider me an available tutor to help work through harder bits of physics - if you are really interested

Before I get deep into analysis I want to briefly answer questions.


And I see what you're saying, that it's the change in energy that causes the damage. Kind of how it isn't the fall that kills you, it's the landing.

That is exactly how it is. Momentum and Energy are similar constructs in Physics. Momentum is a vector quantity and energy is a scalar quantity. Both relate to Mass and Velocity, or its scalar equivalent speed. But Momentum also relates to Force - which you can never derive from a pure energy analysis. An object moving in three dimension has six terms of momentum (three linear and three angular) but only one Kinetic Energy. We will simplify by starting with single dimension impacts of thrown or fired objects. It is the easiest to start with.


My question is then: If mass is a part of the problem, how do you intend to represent this in a game with abstracted physics and separate size categories that are based on weight and dimensions? Doing these types of equations for each situation probably wouldn't be very conducive to a gaming atmosphere, so perhaps some means of scaling it with the size category might be in order?

If we're using momentum change as the basis of damage, then how do you propose that we use mass to scale with it in terms of D&D's rules?

First - Read the spoiler above, and go to the link on Impulse Momentum.

Kinetic Energy = 1/2*MassObject*SpeedObject2
Potential Energy = 1/2*MassObject*HeightObject*Local Gravitational Constant - However, unless you are firing up or down a large hill this will almost always be a negligible and we can leave it out for now.
Momentum = MassObject*VelocityObject - Velocity is a Vector, conveniently in 1d problems Velocity = Speed so long as sign convention is maintained for left or right motion.
Impulse = Force*dT
Impulse Momentum Theorem is Force*dT = Mass*dV reads, Force applied over a period of time equals mass of the object applying the force times change in objects velocity over period of time.

This can be arranged into Force = Mass*dV/dT which is one of Newton's Laws (force = mass * acceleration)

Mass is an inherent piece of any momentum, force or energy analysis.


Another concern is how the mass of an arrow versus the mass of a longsword or war hammer, as well as their respective momentums (which varies all on the "human" user's input of energy) going to play into all this?

It is mass * velocity that gives momentum. A sword is heavier than an arrow but travels much slower. The wielder of the weapon does input energy but for purposes of trying to get something useful out of this assume for the moment all users are equal without strength variation.


You'd also need to account for surface area of each weapon- 5 pounds of force spread out over one foot is very different from 5 pounds of force spread over one centimeter.

Actually, when using impact mechanics to calculate total momentum transferred this does not matter. However once the momentum transfer is known it is useful to use a term called sectional density which is the mass of the object / area applied as you have noted.

The sectional density is a factor in force penetration.


Edit: In Dwarf fortress, weapon damage is determined by how the weapon's physical properties, including size, density, flexibility, and compression verses those same properties in the materials they were being used against. This lead to such oddities as whips functioning like lightsabers, sand being the best throwing weapon, and most weapons becoming functional useless against Steel and Bronze armor.

I have never played dwarf fortress but the material properties are not at all part of momentum transferred. Like the size of object and force distribution they are important in determining what happens with the energy after it is transferred to a target.

In general we are going to look for a change in momentum divided by the sectional density to compare benchmarks.

Starting Analysis.

Lets begin with the same starting point. Long Bow = 1d8 damage. It is a good benchmark and 1d8 is the average damage die. (mid point of d4 to d12 spread)

The first question to answer is what is the momentum of an arrow? This is the first dig into bow mechanics this can be computed based on the momentum transfer from bow string to arrow. Your draw strength forces are applied to the arrow as the string pushes the arrow forward. The forces are not constant for the whole stroke but if you can find the amount of time an arrow remains in contact with the string a linear force reduction approximation is ok for a first run at the problem.

Here are my assumptions for momentum of arrow....

Arrow remains in contact with bowstring for .02 second. Time = .02 (this may be way off - look around for data)
During that .1 second force applied drops from 622N (Peak Force) to 0N (Final Force) based on your data. - Linearly reducing force is an assumption
Initial Momentum of the Arrow is 0
Final Momentum = the Impulse from the Bow which equals the average force applied over time
Mass of the Arrow = 100 grams - a little over .2 lbs

MassarrowVelocityarrow = (ForcePeak - ForceFinal)*Time/2

Velocityarrow = (ForcePeak - ForceFinal)*Time/(2*Massarrow)

Velocityarrow = (622*.02) / (2*.1) = 62.2 m/s

The arrows momentum = 62.2 m/s * .1 kg = 6.22 kg*m/s

Assuming we leave off losses for air resistance (very short range fire) the arrow strikes the target with full momentum. Leaving aside deflected hits for a moment, when an arrow hits a body it stops moving. The result is 100% of the momentum calculated enters the target. But our sectional density helps with arrow penetration.

Assume the arrowhead is a roughly circular and 1.25 centimeters in diameter. Its cross sectional area is Pi*d2/4

Working in SI the area (in m2) = .0125*.0125*Pi/4 = .0001227

Sectional Density = .1/.0001227 = 815 kg/m2

momentum / sectional density = 6.22 / 815 = .0763 m3/s - This will be the number we look to compare for various weapons.

Play around with these numbers, do some research on this and tell me what you come up with regarding my assumptions.

Here are a couple links that might be useful for research in this topic

Wiki Momentum (https://en.wikipedia.org/wiki/Momentum)
Student Archaeologist Work (http://www.thudscave.com/npaa/articles/howhard.htm)

Watch out for the Physics Forums

Physics Forums Thread on Arrow Penetration and Momentum vs Energy (https://www.physicsforums.com/threads/momentum-vs-ke-which-determines-projectile-penetration.195119/)

More than 75% of the people on that forum are not physicists and they mix things up. That particular thread has a link to a very detailed, but hard to read file by Dr Ed Ashby and if you really want to dig into bow performance go there.

Straybow
2015-06-24, 03:26 PM
I have no idea why you are dividing by sectional density.

Think of it this way: you have two arrow designs with the same cross section. One is 30" for a short bow, one is 36" for a longbow. The long arrow has a higher sectional density. If you divide by sectional density, the long arrow is worse than the short arrow. I'm pretty sure that is wrong.

The Evil DM
2015-06-24, 04:25 PM
I have no idea why you are dividing by sectional density.

Think of it this way: you have two arrow designs with the same cross section. One is 30" for a short bow, one is 36" for a longbow. The long arrow has a higher sectional density. If you divide by sectional density, the long arrow is worse than the short arrow. I'm pretty sure that is wrong.

It also has a higher momentum. If you look at the units the mass cancels and turns this into Velocity*Cross section area of impact.

Independently sectional density is used to provide insight into projectile penetration. Momentum is used to provide insight into force applied. Momentum/Sectional density provides insight into kinetic energy distributed over impact area. Using KE=1/2*m*v2 we can extract from that a number for energy per unit area.

But - we are stopping at this point just to see if a connection can be derived between energy, momentum and damage die.

-edit-

Physics Note: We are manipulating fundamental physics units of mass, length and time. Force = kg*m/s2 Energy = kg*m2/s2 momentum = kg*m/s. Each of the three is a unique physical parameter and each provides a different piece of information.

In real world archery there is a balance between momentum and energy that provides maximum damage to a target. Range of shot is a major contributor to that equation. The farther the shot you want more KE and a lighter arrow the shorter the shot you want more mass (momentum).

noob
2015-06-24, 05:27 PM
I believe arrows hurt more at point blank range than at the limit of range.
But in base dnd there is no rules for that while the difference is really high.
Dnd does not try to have any realism with arrows nor with the strength for example there is the problem that your strength in newtons increase exponentially while your damage does not and you find yourself dealing more damage by letting a X tons rock fall on someone than by hitting them five times with it.
But if damage progression was not linear then life should also not be linear and so in the end someone higher of four level than an army of 10000 persons could kill them all easily.

The Evil DM
2015-06-24, 06:37 PM
I believe arrows hurt more at point blank range than at the limit of range.
But in base dnd there is no rules for that while the difference is really high.
Dnd does not try to have any realism with arrows nor with the strength for example there is the problem that your strength in newtons increase exponentially while your damage does not and you find yourself dealing more damage by letting a X tons rock fall on someone than by hitting them five times with it.
But if damage progression was not linear then life should also not be linear and so in the end someone higher of four level than an army of 10000 persons could kill them all easily.

Arrows lose velocity over flight to drag forces. For a melee weapon strength is not the only contributor weapon velocity at the point of impact.

Furthermore, starting out any form an analysis clouded by what you believe sets yourself up for bias.

Altrunchen posted a hypothesis


it may be possible to determine the ratio of Newtons to Damage die by investigating what the draw force and/or impact force of a longbow was.

I have revealed a more useful form of analysis and extended his hypothesis to:


it may be possible to determine the ratio of Momentums and Energies to Damage die by investigating what the draw force, impact force, momentum and energies of a longbow and arrows are.

In the end, the hypothesis may not be true, but whether or not the hypothesis is true, there can be value in the investigative process. To this end I am willing to help Altrunchen look at this problem from a more thorough perspective. Of course - if Altrunchen doesn't return to the forums, my work on it stops here. :)

System doesn't matter. This does not necessarily relate to D&D or any other system. In the end we are looking at bow and arrow physics. After which we could see if it fits any pattern in one or more gaming systems.

Incidentally, arrows are one of the easiest places to learn about these types of mechanics. They can easily be simplified to 1d problems. Their impact with a target is a pure inelastic collision where all energy and momentum in the arrow is absorbed by the target. It is a good learning tool that can then be transferred to more complex problems like sword thrusts, or axe swings.

Straybow
2015-06-25, 10:29 PM
It also has a higher momentum. If you look at the units the mass cancels and turns this into Velocity*Cross section area of impact.

Independently sectional density is used to provide insight into projectile penetration. Momentum is used to provide insight into force applied. Momentum/Sectional density provides insight into kinetic energy distributed over impact area. Using KE=1/2*m*v2 we can extract from that a number for energy per unit area.

But - we are stopping at this point just to see if a connection can be derived between energy, momentum and damage die.
Sorry, I still don't see the point. Arbitrarily manipulating units doesn't make for a coherent analysis, you need a theoretical model before you go unit-hunting. Most of the time, what you are looking for is a factor that cancels out all the units (e.g., the Reynolds number in fluid mechanics).

Cancelling out the mass doesn't seem to make any difference in the analysis, you're still left with "higher velocity = higher damage" and "bigger projectile = higher damage," both fully intuitive.

Mr Beer
2015-06-26, 12:16 AM
There is quite a lot of work in GURPS 4e which equates various energy loads to damage, including for bows/crossbows, guns and explosions. Probably energy weapons as well. If you dig around on the SJ GURPS forum you will likely find articles and/or links where someone else has already done this kind of thing in great detail. IIRC The Deadly Spring does crossbows and bows in great detail.

The Evil DM
2015-06-26, 01:59 AM
Sorry, I still don't see the point. Arbitrarily manipulating units doesn't make for a coherent analysis, you need a theoretical model before you go unit-hunting. Most of the time, what you are looking for is a factor that cancels out all the units (e.g., the Reynolds number in fluid mechanics).

This is actually not entirely true. It is convenient to look for unitless numbers - like Reynolds Number, but it is not a requirement when considering ratios. In the end, we are looking for a number, that when calculated for two different objects and then again divided into each other produces a unit less ratio. That only means that we need to be certain to make sure that the numbers have the same units.

The original hypothesis was looking at force ratios for different object under the objective to determine if F(longbow)/F(shortbow) = 4.5/3.5 (average damage for these weapons in D&D)

As far as analytic versus numeric techniques, it is entirely valid to manipulate numbers, units and equations in order to seek patterns. Feigenbaum discovered the ratio of period doubling between differing non-linear systems in this manner. Feigenbaum quite literally manipulated ratios that appeared in iterative functions until he found a pattern.


Cancelling out the mass doesn't seem to make any difference in the analysis, you're still left with "higher velocity = higher damage" and "bigger projectile = higher damage," both fully intuitive.

I agree both are intuitive. Just as Higher mass and velocity gives more kinetic energy, gives more momentum and so on.

I am going to look at four projectiles.

Heavy Atlatl Dart - 1/2 lb, Diameter 3/4 inch - Shorter thicker dart for atlatl
Light Atlatl Dart - 1/4 lb, Diameter 1/2 inch - Longer thinner dart for atlatl
Longbow Arrow - .1 lb, Diameter 3/8 inch
Shortbow Arrow - .075 lb Diameter 3/8 inch

These will be converted to SI units in calculations

I assume that the human using the atlatl can impart 20 N*s Impulse Units to the Atlatl darts. The thrower uses a lower force for a longer period of time and leverage through the atlatl.

Having done some more research on transfer of momentum from bow to arrow, my calculation in the post above (assuming .2 second of contact between string and arrow) is off. A better number is a bow can impart 15 N*s Impulse units to the arrow and the short bow with about 60% the draw force imparts 9 N*s Impulse Units to the bow.

These correspond to the following velocities

Heavy Atlatl Dart - 18.18 m/s or 60 fps or 41 mph
Light Atlatl Dart - 36.36 m/s or 120 fps or 82 mph
Longbow Arrow - 68.18 m/s or 225 fps or 153.5 mph
Shortbow Arrow - 60.61 m/s or 200 fps or 136.4 mph

None of these are too unreasonable - however, I am making some assumptions and if anyone wants to provide more accurate data for these and other fantasy type projectiles I can recalculate.

I am going to leave out a lot of intermediate calculations. However it is an interesting point to make, if you determine the amount of impulse the weapon or thrower applies to a weapon and assume no loses to drag or other effects, velocity = impulse/mass which once velocity and mass are multiplied together again to get momentum it shows I = momentum. More generally Impulse equals changes in momentum but since we are considering objects that start at rest, are thrown or fired and then assume they come to rest once more in the flesh of their target the it allows for the Impulse = Momentum relationship.



Weapon
Kinetic Energy (kg*m2/s2)
Momentum (kg*m/s)
Sec Density (kg/m2)
M/SD (m3/s)


Heavy Atlatl Dart
181.82
20
3904.55
.0051


Light Atlatl Dart
363.64
20
4392.62
.0046


Longbow Arrow
511.36
15
3123.64
.0048


Shortbow Arrow
245.45
9
2342.73
.0038



Based on M/SD If I use 1d8 as a benchmark for damage on a long bow. A light Atlatl dart should be similar and historically these darts have proven effective in hunting. The heavy atlatl dart would be shorter and thicker, but might warrant a 1d8+1 damage. Short bow lines up to 1d6.

Longbow = 4.5 avg damage, short bow = 3.5 avg damage. 3.5/4.5 = .79 .0038/.0048 = .78

I am not claiming that this relationship is universal. There appears to be a pattern works for these objects, but it may not hold for others. This pattern does track with the results of numerous ballistic studies.

The trick is to strike a balance between momentum, energy and penetration.

Lightweight high kinetic energy objects lack the mass to penetrate (very low sectional density)
Heavyweight low energy objects have the mass to penetrate but require more impulse to accelerate(high momentum, low kinetic energy, high section density)

Straybow
2015-06-27, 04:59 PM
This is actually not entirely true. It is convenient to look for unitless numbers - like Reynolds Number, but it is not a requirement when considering ratios. In the end, we are looking for a number, that when calculated for two different objects and then again divided into each other produces a unit less ratio. That only means that we need to be certain to make sure that the numbers have the same units. Again, making a ratio doesn't have any validity as a mathematical model unless you have the physics to back it up... dividing by sectional density has no physics behind it.

On another point, the "clothyard" arrow was probably larger in diameter than the shortbow arrow for any given use, but in any case could vary widely depending on the intended use of the arrow. That would lower the longbow arrow's sectional density slightly but raise its mass and momentum.


Longbow = 4.5 avg damage, short bow = 3.5 avg damage. 3.5/4.5 = .79 .0038/.0048 = .78

I am not claiming that this relationship is universal. There appears to be a pattern works for these objects, but it may not hold for others. This pattern does track with the results of numerous ballistic studies.

The trick is to strike a balance between momentum, energy and penetration. On that note, there isn't any physics behind the longbow doing 1d8 and the shortbow doing 1d6, nor that the ratios extrapolate linearly, etc. You've shown a coincidental correspondence between a meaningless ratio of projectile statistics and a meaningless ratio of arbitrarily assigned damage values.

The longbowman's battlefield virtues were threefold: great range compared to other bows, high rate of fire compared to crossbows that can match or exceed the range, and the culture of martial training behind the yeomen that meant they were also effective light-to-medium footsoldiers if the enemy closed the distance. The longbow also penetrated light armor better, but nobody has conclusively shown better penetration of heavier mail or various forms of steel plate.

The problem is that DnD in various forms nerfs the rate of fire difference and any penetration difference but wants to make the longbow something special, so they've raised the damage even though the wound track of an arrow doesn't vary that much for the same type of arrowhead.

The Evil DM
2015-06-27, 10:01 PM
Again, making a ratio doesn't have any validity as a mathematical model unless you have the physics to back it up... dividing by sectional density has no physics behind it.

This statement is incorrect. What is being considered is terminal ballistics. Terminal ballistics is usually studied using computational models for fluid dynamics and something that would require several days of my time to fully describe here. And considering I cannot bring the calculus necessary into the forum format it is even more difficult to describe.

However, - Terminal Ballistics is the study of projectile impact to target. Most modern work on the subject is focused on very high energy collisions between bullets and other materials. In the case being spoken about we have relatively low velocity objects (arrows and darts) striking a fluid medium - Living target. Considering that a living target is neither a perfectly incompressible fluid (air spaces such as lungs allow for compression) nor is a living target entire fluid (bone and such) the actual physics of the impact is much more complicated. And I noted that I was just looking for something that is a simplification.

Regardless. Momentum transfer from object to fluid or fluid to object (either way) is proportional to the Velocity * the cross section Area through which the interaction occurs. When you divide linear momentum by sectional density the result is Velocity * Area. The units match and it supports the physics. If you take the first derivative of momentum divided by sectional density you get the acceleration across a unit of cross sectional area.

So an object, rock, arrow, spear, billiard ball has some velocity, and it has a cross sectional area. When it strike the fluid form of a living creature it applies forces and transfers both energy and momentum in relationship to those two parameters.

And regarding ratios as invalid mathematical constructs again totally wrong. There are many ratios in common use throughout engineering and physics. What is being considered is specifically comparing a calculable number for one object to another. Specific gravity compares the weight of an object to the weight of an equivalent volume of water. Just as I am comparing Velocity*Area for two separate objects under various conditions.


On another point, the "clothyard" arrow was probably larger in diameter than the shortbow arrow for any given use, but in any case could vary widely depending on the intended use of the arrow. That would lower the longbow arrow's sectional density slightly but raise its mass and momentum.

On that note, there isn't any physics behind the longbow doing 1d8 and the shortbow doing 1d6, nor that the ratios extrapolate linearly, etc. You've shown a coincidental correspondence between a meaningless ratio of projectile statistics and a meaningless ratio of arbitrarily assigned damage values.

I don't disagree with this. Give me numbers for a clothyard arrow and I will see where they fit. I have mentioned (three times now) that I may not have the most accurate performance data for a particular object and would be interested to see.

What I do have is several years of experience working in the ballistics industry on terminal effects of objects. The energy to momentum ratio of an object affects many things - not just the impact. All that is being considered in the formulation is impact. The length and form factor of an object affects flight trajectory. Spin provides gyroscopic stabilization and transfers some of the total momentum to an angular momentum component.

Nowhere did I state, that there was any real physics behind D&D or any other game system. Just extending the OP's original work on equating force to damage to something a little more inline with real terminal ballistics.


The longbowman's battlefield virtues were threefold: great range compared to other bows, high rate of fire compared to crossbows that can match or exceed the range, and the culture of martial training behind the yeomen that meant they were also effective light-to-medium footsoldiers if the enemy closed the distance. The longbow also penetrated light armor better, but nobody has conclusively shown better penetration of heavier mail or various forms of steel plate.

The problem is that DnD in various forms nerfs the rate of fire difference and any penetration difference but wants to make the longbow something special, so they've raised the damage even though the wound track of an arrow doesn't vary that much for the same type of arrowhead.

None of this is being argued. The Hypothesis is strictly looking at impact. Damage X to Parameter Y. None of this section of the comment has anything to do with the work above because when speaking about penetration in terms of terminal ballistics the term means - depth into target. How deep does that arrow plunge into flesh.

I could address armor penetration as well. But it dives even deeper into materials issues raising the complexity further. However at sufficiently high energies (achievable by the most powerful longbows) the mechanics dictating how arrows punch through metals and the physics is very fluid like and the relationships boil down to comparing the velocity and cross section area of impact against the relative densities of the materials.

Akodo Makama
2015-06-28, 03:56 PM
All this work has already been done.

3G3: Guns, Guns, Guns - Weapon design for any RPG (3rd Edition) Paperback – October, 1997 (http://paizo.com/products/btpy7nyo?3G3-Guns-Guns-Guns)

Includes:
Rules for edged, blunt, and other muscle-powered weapons.
Technology level conversion (from bronze age uzis to monomollecular rapier)
Convenient charts and formulas for converting to systems from Cyberpunk2020 to D&D 2nd ed. to GURPS
Rules for creating conversion tables for any system

Straybow
2015-07-01, 02:57 AM
Regardless. Momentum transfer from object to fluid or fluid to object (either way) is proportional to the Velocity * the cross section Area through which the interaction occurs. When you divide linear momentum by sectional density the result is Velocity * Area. The units match and it supports the physics. If you take the first derivative of momentum divided by sectional density you get the acceleration across a unit of cross sectional area. Except the cross-section area isn't what's doing the damage. They don't make arrows as flat-ended sticks for a reason. The pointed or bladed tip is what does the damage. It is cutting through the flesh, not beating against the surface.

Sectional density gives some idea of how well the momentum can be preserved as the projectile moves through resisting medium, whether air or flesh. In this case, preserving the projectile's momentum is desirable, as opposed to dissipating the momentum on the surface of the target. That means sectional density is a multiplier, not a divisor.

My old roommate in college was writing a program to process explosive shot records to interpret positions of rock layers, etc. His initial results were amazingly clear depictions. Success! Then he discovered that he had left out a factor of pi in one of the equations. When he put the data through the corrected program it came out looking much fuzzier, just like existing algorithms.

The first result is what is called an "artifact." It means something that was produced by the modeling process, rather than arising from the data being analyzed. It looked clear, but it didn't actually represent the existing rock layers.

There simply is no physics that justifies dividing by sectional density. Coming up with a ratio that happens to match the ratio of damage dice averages is an artifact, not a meaningful analysis of the data.

Sorry, I know this isn't Real World Weapons and Armor thread. But you are presenting something as a rigorous analysis, and then asking us not to be rigorous about evaluating your model.

The Evil DM
2015-07-01, 08:03 AM
I am taking your quote a little out of order in my reply.


Sorry, I know this isn't Real World Weapons and Armor thread. But you are presenting something as a rigorous analysis, and then asking us not to be rigorous about evaluating your model.

The problem I am having here is not that you are evaluating the analysis, but it is clear you don't understand the underlying physics well enough to provide a real assessment of its validity.

Case in point


Except the cross-section area isn't what's doing the damage. They don't make arrows as flat-ended sticks for a reason. The pointed or bladed tip is what does the damage. It is cutting through the flesh, not beating against the surface.

You clearly don't understand momentum and how it transfers from fluid to object or vice versa. In any physics problem the frame of reference can be reversed. That reversibility is one of the fundamental defining aspects of physics.

So - I can view this problem as an arrow striking a fluid body or a fluid body hurtling a long at 200ft/second and striking the stationary arrow. The momentum of the fluid heading striking the arrow is proportional to the column of fluid striking the head of the arrow and defined by the cross sectional area of the arrow as view from direction of impact.

Another way of viewing fluid to object momentum transfer is a columnar jet of fluid striking a large object. If I hold a nozzle spraying a stream of fluid and point it at a wall, it creates forces proportional to the cross sectional area of flow.

Regardless of the perspective the resulting momentum always includes velocity * cross sectional area.

Another point you fail to understand in the above statement is the distinction that momentum is a vector quantity. The shape of the object striking the fluid does not matter when determining total momentum transferred. The difference between a blunt strike, and the strike from a wedge shaped or edged object is the resulting direction of flow in the impacted fluid. An edged weapon will push the fluid sideways deflecting off the wedge while the blunt arrow pushes the fluid backwards.

Using the alternative perspectives I could imagine a fluid mass moving towards a wedge. The wedge splits the fluid mass, but the forces applied to the wedge are still proportional to the cross sectional area where fluid intersects with wedge.

Incidental Note - the edged arrow also has a more complete momentum transfer because it sticks into the target releasing all momentum rather than deflecting itself and retaining some momentum in the deflection. This is also why bullets with the most stopping power expand on impact to transfer energy and momentum more completely to a target. A bullet that bounces off a target (rubber bullet) does less damage, a bullet that has too much density and kinetic energy and passes through the target does less damage, but the bullet that stops completely in the target does maximum damage.


Sectional density gives some idea of how well the momentum can be preserved as the projectile moves through resisting medium, whether air or flesh. In this case, preserving the projectile's momentum is desirable, as opposed to dissipating the momentum on the surface of the target. That means sectional density is a multiplier, not a divisor.

As noted previously, I am not trying to model the exact effects of impact. I am merely looking for a number that is proportional to the momentum transferred to a fluid. Which in terms of units happens to be velocity * cross section area. I just as easily make that declaration and compute it directly but knowing momentum and kinetic energy of a projectile is also useful. It is convenient after having calculated momentum to get V*A by dividing by sectional density.


My old roommate in college was writing a program to process explosive shot records to interpret positions of rock layers, etc. His initial results were amazingly clear depictions. Success! Then he discovered that he had left out a factor of pi in one of the equations. When he put the data through the corrected program it came out looking much fuzzier, just like existing algorithms.

The first result is what is called an "artifact." It means something that was produced by the modeling process, rather than arising from the data being analyzed. It looked clear, but it didn't actually represent the existing rock layers

There simply is no physics that justifies dividing by sectional density. Coming up with a ratio that happens to match the ratio of damage dice averages is an artifact, not a meaningful analysis of the data.

You have gone from stating that I am arbitrarily unit hunting.


Arbitrarily manipulating units doesn't make for a coherent analysis, you need a theoretical model before you go unit-hunting.

To saying that forming ratios has no mathematical validity


Again, making a ratio doesn't have any validity as a mathematical model unless you have the physics to back it up... dividing by sectional density has no physics behind it.

To now calling what I am doing is an artifact.

Make up your mind - or are you just looking for things to argue over.

For your information, since you have been unable to notice. The ratio I am looking at is not Momentum over Sectional density or more conveniently Velocity * Cross Sectional Area. The Ratio I am considering is Momentum over Sectional Density for a Longbow arrow compared to the Momentum over Sectional Density for a Short Bow arrow.

Again, since Momentum over Sectional Density yields Velocity times Area this is proportional to the momentum transferred to a fluid target. The proportionality constant - if I were to extend to searching for actual effects on the target would require density of the target.

However, I am not trying to determine actual effects. I am comparing the relative momentum transfer of a long bow arrow to the momentum transfer of a short bow arrow. This is a method of using scaling to compare different objects and systems.

It is possible to define the parameters of the weapon: Impulse, projectile momentum, energy, sectional density. Those are calculable numbers (three are extensive, the other intensive see below). Then I am using these two properties to compare a ratio of the extensive to the intensive property for a Long Bow Arrow to the same numbers for a Short Bow Arrow.

Since the momentum transfer to a fluid is proportional to the Velocity times Area - assuming that both arrows strike same fluid I can compare this value for both objects and look at the ratio.

The question proposed by the OP referred finding a more (not necessarily entirely) realistic method to compare different weapons. In that regard what I have proposed is:

Not arbitrarily unit hunting
Supported by real underlying physics
Not mathematically invalid
and is Not an artifact from a computer model

If you still think looking at scaling of a parameter between two systems or objects is an invalid approach to physics here is another perspective.

Extensive and Intensive properties

All things can be divided into extensive or intensive properties. An extensive property is something that changes with scale. Momentum, Kinetic Energy are all extensive properties.

An intensive property is any defined value that does not scale with the scale of an object.

Some examples are:
Chemical Potential
Elasticity
Electrical resistivity
Hardness
Magnetic field
Magnetization
Malleability
Melting point and boiling point
Molar absorptivity
Pressure
Specific Volume
Specific Internal Energy
Specific Entropy
Specific Enthalpy
Specific Heat Capacity (At constant volume or pressure)
Temperature
Viscosity

From this perspective. What I have done is taken an extensive property of an arrow and divided by an intensive property to obtain a value proportional to momentum transfer to a fluid. Momentum is the extensive piece that is scaled to the intensive sectional density. I am then using that number to compare scaling of the extensive property relative to the intensive property across multiple projectile types.

I do agree that the selection of 1d8 damage for a long bow, and 1d6 damage for a short bow was not based on any physics and likely intuitively decided based on the idea that a long bow is larger and has more force. I do agree that the coincidental agreement between the values I calculated and the average damage for 1d8 and 1d6 does require better input data and measured values for long bow and short bow performance are likely to yield different - and better results.

But I don't actually care what the actual total momentum transferred is. I don't actually care what the real effects are. What I do care about is:


If I assume that a longbow is a benchmark. A standard against which I can compare other projectiles on the basis of ratios of momentum transferred to a fluid target. I does not matter if a long bow = 1d8 or 1d12. The important point is does the ratio of momentum transfer for longbow, divided by momentum transfer for short bow approximate to the ratio of average 1dx/ average 1dy

The method described does provide a means to look at the scaling of momentum transfer between different projectile and thrown weapons. It also reveals the severity to which D&D in particular is broke across creature sizes (relative to physics). Something I have explored elsewhere but have not included it above.

A similar method can be defined for hand held weapons but it requires more terms to the equation because, as they are swung through an arc, they usually have multiple momentum values (due to vector nature of momentum) and an angular momentum component. A sword thrust could be modelled in this manner if the swordsman and sword is being considered as a whole system.

Straybow
2015-07-01, 06:25 PM
You clearly don't understand momentum and how it transfers from fluid to object or vice versa. ... Regardless of the perspective the resulting momentum always includes velocity * cross sectional area. No, momentum ignores cross sectional area and only cares about velocity and mass. You have arbitrarily factored out the mass, and are comparing nonsensical measurements.


Incidental Note - the edged arrow also has a more complete momentum transfer because it sticks into the target releasing all momentum rather than deflecting itself and retaining some momentum in the deflection. This is also why bullets with the most stopping power expand on impact to transfer energy and momentum more completely to a target. Not so. The edged arrow can overpenetrate and pass completely through the body, as many of my bowhunting friends say happens on occasion. The blunt may have enough energy to penetrate the skin and not bounce off, thus transfer more energy to the target. But an arrow doesn't do damage based on energy transfer, it does damage based on cutting the flesh.


A bullet that bounces off a target (rubber bullet) does less damage, a bullet that has too much density and kinetic energy and passes through the target does less damage, but the bullet that stops completely in the target does maximum damage...

As noted previously, I am not trying to model the exact effects of impact. I am merely looking for a number that is proportional to the momentum transferred to a fluid. Which in terms of units happens to be velocity * cross section area. Now here you're again confusing tissue damage with momentum transfer. A rubber bullet that bounces off transfers more momentum and energy to the target than had it stopped dead. Do the math. A rubber bullet does less damage than a lead bullet because it is made of rubber with about 1/12th the mass of the lead bullet of the same size. But your method would factor out the mass and compare only the sectional area. By your model, the rubber bullet and lead bullet do the same damage. That isn't a reasonable result because the model is invalid.


You have gone from stating that I am arbitrarily unit hunting.
To saying that forming ratios has no mathematical validity
To now calling what I am doing is an artifact.
Make up your mind - or are you just looking for things to argue over. No, I said "you need a theoretical model, you need a theoretical model, you need a theoretical model." Your unit hunting was not derived from a valid theory, your ratios (and the damage ratios) have no origin in physics, and the resulting success is an artifact of your invalid mathematics, not the conclusion of a valid theoretical model. If I didn't think pointing out faulty physics was worth arguing about, I wouldn't be posting here.


For your information, since you have been unable to notice. The ratio I am looking at is not Momentum over Sectional density or more conveniently Velocity * Cross Sectional Area. The Ratio I am considering is Momentum over Sectional Density for a Longbow arrow compared to the Momentum over Sectional Density for a Short Bow arrow. Which would make sense if m/sd or v·area made sense. Please show one professional physics/ballistics page (say, college physics tutorial or ammunition/gun maker/seller) that uses m/sd or v·area, and you'll prove your point.


It is possible to define the parameters of the weapon: Impulse, projectile momentum, energy, sectional density. Those are calculable numbers (three are extensive, the other intensive see below). Then I am using these two properties to compare a ratio of the extensive to the intensive property for a Long Bow Arrow to the same numbers for a Short Bow Arrow. Just because they are calculable numbers doesn't mean the numbers are being used properly. For that you need a valid theoretical model.


Since the momentum transfer to a fluid is proportional to the Velocity times Area... You keep saying that phrase. I don't think that phrase means what you think it means. No, momentum transfer isn't proportional to v·area.


All things can be divided into extensive or intensive properties. An extensive property is something that changes with scale. Momentum, Kinetic Energy are all extensive properties. ...

From this perspective. What I have done is taken an extensive property of an arrow and divided by an intensive property to obtain a value proportional to momentum transfer to a fluid. Momentum is the extensive piece that is scaled to the intensive sectional density. Density is intensive, but sectional density is NOT intensive. It is strictly dependent on dimensions which are unarguably extensive. Second, just because a property is intensive or extensive doesn't mean you can arbitrarily divide one by the other. You need a valid theoretical model to tell you which to divide or multiply by what, else the number you get is meaningless.


But I don't actually care what the actual total momentum transferred is. True. In order to determine the momentum transfer you need a valid theoretical model. I keep using that phrase, and I do know what it means.
__________________________________________________
TL;DR

I've issued a challenge. "Please show one professional physics/ballistics page (say, college physics tutorial or ammunition/gun maker/seller) that uses m/sd or v·area, and you'll prove your point." I would consider it a courtesy to accept or decline before posting another lengthy reply.

The Evil DM
2015-07-01, 07:55 PM
I've issued a challenge. "Please show one professional physics/ballistics page (say, college physics tutorial or ammunition/gun maker/seller) that uses m/sd or v·area, and you'll prove your point." I would consider it a courtesy to accept or decline before posting another lengthy reply.
Accepted with one caveat. I am going to show you the fluid mechanics information. Because that is where this comes from and the ballistics companies don't go into fluid mechanics. Most ballistics information deals more with flight of the bullet, not the actual terminal impact. Shear Forces impart momentum in the fluid due to momentum of a solid object passing into or through a fluid. The shear forces are the forces that cut, crush, and separate the fluid. High kinetic energy provide hydro static shock, which is an additional damage phenomena.

These forces are damage. And they are proportional to the velocity and cross sectional area.
Gathering the links and information now.

The Evil DM
2015-07-02, 04:04 AM
Starting from accepting this challenge.

Damage in the fluid mass is caused by shear forces deflecting the fluid as the solid object passes through it. The shear forces impart momentum upon the fluid being displaced by the solid object. At lower velocities, the flow can be laminar at higher velocities the flow turbulent. As with all dynamics problems there are many ways to formulate the same material. Each with its own advantages and disadvantage.

The momentum formulation has simpler individual terms but requires reconciliation with the vector properties, while the energy formulation reduces to more complex scalar terms. Each formulation can yield different insights.

The starting point for any discussion of fluid mechanics in the context of momentum transfer and resulting forces is fluid shear, which ultimately relates direction to velocity.

Sophomore Year Physics on Fluid Shear (http://physics.info/viscosity/)

Engineering Tool Box on Dynamic Viscosity (http://www.engineeringtoolbox.com/dynamic-absolute-kinematic-viscosity-d_412.html) - for this one on dynamic viscosity, shearing stress is equal to Force / Area Moving area to the right hand side of the equation gives shear force = Area * Velocity * target material specific constants.

The text book definition of Viscosity is Shear Stress/Strain Rate. This link shows the typical text formulation where two solids are separated by a layer of fluid and the solids are pulled in opposite directions.

PDF on viscosity and fluid flow" (wps.aw.com/wps/media/objects/877/898586/topics/topic04.pdf)

In this document you can see the formulation for Shear Force. And even though the formulation usually involves a fluid moving along a stationary surface, it can also be used to model a moving object through a stationary fluid.

Shear forces = total Force/Area. When applied to fluids calculating the lower velocity Shear transfers the Area to the right hand side of the equation and results in = constants * Area * Velocity.

My statement is and still remains that the ability to cause damage is proportional to velocity and area. Not that they are equal to velocity and area.

Since the form for Shear Forces is an equation that looks like C*A*V when I make the comparison between long bow arrows and short bow arrows I divide C*A*V for one by C*A*V of the other. The collected constant terms C cancels to one because it is a set of constants based on properties of the target.

I am not looking for actual shear forces, nor am I looking for actual amounts of momentum transferred. I am looking for a ratio that allows me to make a reasonable approximation of the relative potential to cause damage between two different scales.

A statement such as - If a long bow arrow does x damage, a short bow arrow should do approximately .8x. I do understand that this is a very gross simplification, but for the context of gaming it is a functional approximation.

While the momentum formulation is sufficient for relatively low velocity damage. When bullets come into play and much higher velocities and energies, the cumulative effects of cavitation and hydrostatic shock are best approximated with the energy formulation.

In this formulation, impact energy is proportional to Area * Speed^2. The proportionality constants consists of material properties for the target fluid, density, viscosity etc, plus drag coefficient for the projectile. This means the energy formulation can only be used to compare projectiles with similar drag characteristics, so that the terms cancel in the ratio.

With that I again choose a benchmark and I can describe one bullet in terms of another bullet as a relative percentage compared to the standard.

By the way, I cannot link the document because it is not in a linkable format - but I can email you a book called

Wound Ballistics - Basics and Applications.

It is a text book on bullet wounds and terminal ballistics. Due to being primarily focused on bullets it focuses on the energy method. In this book on page 94. It states - and derives the theory behind…


Energy transfer per unit distance is hence directly proportional to the instantaneous
energy of the bullet and inversely proportional to its sectional density.

Inversely proportional = divide by sectional density. If I wanted a quick way to compare two bullets of approximately the same drag I can divide kinetic energy by sectional density and then look at the ratio of that term for each bullet. It won't tell me that I am meeting a particular would potential, but it will tell me that I can expect bullet a to have some percentage value like bullet A has 1.1 time the wound potential of bullet B (with all other things being equal).

The same book, on the same page even. Follows with


Penetration depth is hence directly proportional to sectional density and inversely
proportional to energy transfer.

This is used to give results about the length of the wound track. Which is something I am ignoring for this. Because I just want to look at damage potential, but maybe you are confusing the penetration depth - where you do divide by sectional density - with the actual wound potential through shear forces and energy.

In either case, when I am looking at momentum and kinetic energy in the context of gaming. sectional density is a convenient means to massage the data down to just Area*Velocity or Area * Velocity Squared for high kinetic energy and mechanical shock based projectiles under the assumption that all the other constants in a real world computation for wound ballistics are essentially the same when used to comparing two separate projectiles damage potential relative to the same target.

Your reply has added issues.


No, momentum ignores cross sectional area and only cares about velocity and mass. You have arbitrarily factored out the mass, and are comparing nonsensical measurements.

The factoring out is not arbitrary for reasons mentioned above.


Not so. The edged arrow can overpenetrate and pass completely through the body, as many of my bowhunting friends say happens on occasion. The blunt may have enough energy to penetrate the skin and not bounce off, thus transfer more energy to the target. But an arrow doesn't do damage based on energy transfer, it does damage based on cutting the flesh.

This is true and I did limit my analysis to the assumption that the arrow sticks and stops completely. If I want to broaden the assumption I need to add terms for residual velocity. In terms of the blunt arrow, yes blunt objects can penetrate flesh, they just require high enough energies.


Now here you're again confusing tissue damage with momentum transfer. A rubber bullet that bounces off transfers more momentum and energy to the target than had it stopped dead. Do the math. A rubber bullet does less damage than a lead bullet because it is made of rubber with about 1/12th the mass of the lead bullet of the same size. But your method would factor out the mass and compare only the sectional area. By your model, the rubber bullet and lead bullet do the same damage. That isn't a reasonable result because the model is invalid.

This is incorrect because you neglect the fact that the rubber bullets have much lower velocities. You are making an invalid assumption of equal velocities and then calling the method invalid because your invalid assumption provides nonsensical results. Also momentum on a deflection only provides an increase to momentum in the body beyond 100% transfer if and only if the deflection has velocity components that are directly opposite the direction of impact. Most of these will hit flesh and decelerate (even if they don't penetrate) and then fall or bounce at a very low velocities in another direction.


No, I said "you need a theoretical model, you need a theoretical model, you need a theoretical model." Your unit hunting was not derived from a valid theory, your ratios (and the damage ratios) have no origin in physics, and the resulting success is an artifact of your invalid mathematics, not the conclusion of a valid theoretical model. If I didn't think pointing out faulty physics was worth arguing about, I wouldn't be posting here.

As noted for the 3rd time now - I have a model, based on fluid dynamics, it is valid, and if you want a text book fine. Just because you don't know the model doesn't mean it is invalid. So my challenge to you in return is prove my model incorrect. You have stated it is incorrect four or five times, back up your assertion.


Which would make sense if m/sd or v·area made sense. Please show one professional physics/ballistics page (say, college physics tutorial or ammunition/gun maker/seller) that uses m/sd or v·area, and you'll prove your point.

Have it if you want it but can't link it.


Just because they are calculable numbers doesn't mean the numbers are being used properly. For that you need a valid theoretical model.

You keep saying that phrase. I don't think that phrase means what you think it means. No, momentum transfer isn't proportional to v·area.

Shear forces due to momentum transfer are proportional to V*Area. At this point I am done arguing that


Density is intensive, but sectional density is NOT intensive. It is strictly dependent on dimensions which are unarguably extensive. Second, just because a property is intensive or extensive doesn't mean you can arbitrarily divide one by the other. You need a valid theoretical model to tell you which to divide or multiply by what, else the number you get is meaningless.

Wrong. Sectional Density is an intensive property. It is derived using two separate extensive properties. Mass divided by Area. Any time you divide two extensive properties the result is an intensive property. This is a defining characteristics of these properties. Just like Density (intensive) is mass divided by volume. (both extensive)

If you want to continue to argue here are a few more examples of very high energy proportional to velocity squared.

Machines Tooling to punch holes in metals. The model (using energy model for very high velocity impact) for plug penetration of a tool through a plate is the proportional to the area * velocity squared with a whole bunch of constants regarding material properties. Which if my goal is to compare two separate punches for machine load the two main numbers I need are …. Area and Velocity.

Another method of looking at impact problems and damage potential and yielding the same factors as proportional is to look at the high energy problems from the perspective of Jerk. (yes - I am a physics Jerk) In some formulations of shock mechanics jerk - the first order time derivative of force - is used to describe kinetic energy change - which is defined on V(t). Since kinetic energy is proportional to Area times Speed squared when the derivative is taken, Jerk for kinetic energy change due to shock is proportional to Area times Velocity.

In particular the Shock wave effect described is of importance in body armor. Body armor distributes the momentum to stop penetration, but the shock effect can still penetrate the armor. This shock effect is related to the Jerk. A drawback of the Jerk formulation is deriving vector equations from a scalar quantity (kinetic energy)

All of this is due to fluids mechanics formulations defining forces in terms of speeds and velocities rather than acceleration of a rigid bulk material.- edit -

There is no clear consensus on an energy level where the cutoff between kinetic energy and momentum formulation is a preferred method of estimating wound potential. In general once hydrostatic shock effects begin to occur kinetic energy is the key. This typically occurs when impact velocities reach the speed of sound in the impact material. Thus bullets clearly do damage through kinetic energy. Thrown stones and other high mass low velocity objects do damage through surface damage compression through mass and their wounding potential is better measured through understanding of momentum transfer and shear forces. (the impact velocity of a thrown stone doesn't set up a shockwave that resonates) Low to mid velocity cutting is a shear force effect, which is easily related to momentum.

I did find a link to the book I have on wound ballistics.

Wound Ballistics (https://books.google.com/books?id=q4jzcfLhBcYC&pg=PA203&lpg=PA203&dq=sectional+density+cross+sectional+area&source=bl&ots=B5o83WijST&sig=V4thGb0Z4lYlnYbf9ymF2N_qsrs&hl=en&sa=X&ei=K3AOVbDJK8eVNoPYgig&ved=0CFQQ6AEwBw#v=onepage&q=sectional%20density%20cross%20sectional%20area&f=false)

Page 94. Equation 3.1:3 Wound Potential of Bullet is proportional to energy divided by sectional density.

Straybow
2015-07-03, 12:57 PM
Sophomore Year Physics on Fluid Shear (http://physics.info/viscosity/)

Engineering Tool Box on Dynamic Viscosity (http://www.engineeringtoolbox.com/dynamic-absolute-kinematic-viscosity-d_412.html) - for this one on dynamic viscosity, shearing stress is equal to Force / Area Moving area to the right hand side of the equation gives shear force = Area * Velocity * target material specific constants.

The text book definition of Viscosity is Shear Stress/Strain Rate. This link shows the typical text formulation where two solids are separated by a layer of fluid and the solids are pulled in opposite directions.

PDF on viscosity and fluid flow" (wps.aw.com/wps/media/objects/877/898586/topics/topic04.pdf)

In this document you can see the formulation for Shear Force. And even though the formulation usually involves a fluid moving along a stationary surface, it can also be used to model a moving object through a stationary fluid.

Shear forces = total Force/Area. When applied to fluids calculating the lower velocity Shear transfers the Area to the right hand side of the equation and results in = constants * Area * Velocity.

My statement is and still remains that the ability to cause damage is proportional to velocity and area. Not that they are equal to velocity and area. No, arrow damage is not related to fluid mechanics. That "area" appearing in fluids formulae isn't cross section. It is wetted surface area, which would be the sides of the arrow shaft rather than cross section. It would generally be simplified to area/unit length so that what we calculate is shear drag per unit length.

I did find a link to the book I have on wound ballistics.

Wound Ballistics (https://books.google.com/books?id=q4jzcfLhBcYC&pg=PA203&lpg=PA203&dq=sectional+density+cross+sectional+area&source=bl&ots=B5o83WijST&sig=V4thGb0Z4lYlnYbf9ymF2N_qsrs&hl=en&sa=X&ei=K3AOVbDJK8eVNoPYgig&ved=0CFQQ6AEwBw#v=onepage&q=sectional%20density%20cross%20sectional%20area&f=false)

Page 94. Equation 3.1:3 Wound Potential of Bullet is proportional to energy divided by sectional density. Too bad we aren't talking about bullets. This is exactly what "unit hunting" means: looking for a formula that uses the unit(s) without matching the theoretical model.

I'm well aware of bullet cavity mechanics, etc. A bullet wounds by tearing through flesh by blunt force. Lower sectional density actually means the bullet loses energy more quickly, and therefore deposits more of the energy into the flesh. For a bullet, increasing area at impact ("mushrooming" = reducing sectional density) increases damage because it slows the bullet faster.

This has no application for arrows, spears, throwing axes, and other weapons that do damage by cutting. It isn't even applicable for sling stones, because they don't go fast enough to penetrate the flesh like a firearm slug. They just smash against the target. Likewise a blunt arrow doesn't do damage by the same mechanism as a comparatively high velocity bullet. Blunts are sometimes used for hunting birds or vermin whose small ribs and wings/legs are easily broken. Totally different math.

Straybow
2015-07-03, 01:59 PM
Wrong. Sectional Density is an intensive property. It is derived using two separate extensive properties. Mass divided by Area. Any time you divide two extensive properties the result is an intensive property. This is a defining characteristics of these properties. Just like Density (intensive) is mass divided by volume. (both extensive)
[emphasis added] Area and volume are not intensive, they are the veritable definition of extensive. You can't create an intensive property by arbitrarily dividing extensive values. Mass divided by volume = density, which is intensive. Sectional density is mass divided by area, which is density [intensive] times length [extensive]. Whenever you multiply an intensive times a (relevant) extensive measure, the result is extensive. You are making it depend on length, the amount of matter present.


Extensive - Properties that do depend on the amount of matter present.
Mass - A measurement of the amount of matter in a object (grams). Weight - A measurement of the gravitational force of attraction of the earth acting on an object. Volume - A measurement of the amount of space a substance occupies. Length[emphasis added]
Link (https://www.chem.tamu.edu/class/majors/tutorialnotefiles/intext.htm)

Straybow
2015-07-03, 02:32 PM
Machines Tooling to punch holes in metals. The model (using energy model for very high velocity impact) for plug penetration of a tool through a plate is the proportional to the area * velocity squared with a whole bunch of constants regarding material properties. Which if my goal is to compare two separate punches for machine load the two main numbers I need are …. Area and Velocity. Unit hunting again. It makes no difference if you have the right numbers and use them wrong.

This theory is based on kinetic energy distributed over the shear area. You are claiming to measure arrow momentum, not kinetic energy. The shear area is not the sectional area, but the thickness of the metal plate times the circumference of the punch. This theory uses velocity squared, not simple velocity. So if you are using this example to justify your math, you are using the wrong measurement for the area term and you are missing the power (square) of the velocity term.

Straybow
2015-07-03, 02:48 PM
Another method of looking at impact problems and damage potential and yielding the same factors as proportional is to look at the high energy problems from the perspective of Jerk. (yes - I am a physics Jerk) In some formulations of shock mechanics jerk - the first order time derivative of force - is used to describe kinetic energy change - which is defined on V(t). Since kinetic energy is proportional to Area times Speed squared when the derivative is taken, Jerk for kinetic energy change due to shock is proportional to Area times Velocity.Brother, you really need to review your basic mechanics. You integrated from force to momentum rather than took derivative from force to yank (m·jerk).

x(t) = velocity = m/s
v(t) = acceleration = m/s²
a(t) = jerk = m/s³

Straybow
2015-07-03, 02:57 PM
In particular the Shock wave effect described is of importance in body armor. Body armor distributes the momentum to stop penetration, but the shock effect can still penetrate the armor. This shock effect is related to the Jerk. A drawback of the Jerk formulation is deriving vector equations from a scalar quantity (kinetic energy)

All of this is due to fluids mechanics formulations defining forces in terms of speeds and velocities rather than acceleration of a rigid bulk material. No, you definitely cannot model the shock of arrow impact against solid armor (much less compressible cloth armor) as fluid mechanics. Again you are confusing your basic terms, jerk is not based on scalar kinetic energy. You calculate the vector quantities based on conservation of momentum and elastic and plastic material mechanics, then you back-calculate change in energy.

Straybow
2015-07-03, 03:02 PM
Introduction:
One thing that I have occasionally found myself wanting/needing to know is how to stat real-world situations or things based on the force that they exert... My apologies to the OP, don't mean to threadjack.

Akodo Makama has given an answer to what you need.