Range Increments & Projectile Speed

[MetaKirby06]

So the other day, I was looking at the "Longest shot possible" (http://www.giantitp.com/forums/showthread.php?t=166182) thread. Before I made it to the first mention of projectile speed, I thought, "Wait... that arrow has to hit a target several miles away in 6 seconds... so it must be going really fast... hmmm..." Then I read the first mention of speed, and read on to find that the required speed to hit a target 8 miles away would be about 2 km/h, or about 4,800 mph, or about mach 6.31. Farther on, even greater ranges were figured out, which meant greater speeds. Eventually, the maximum pre-epic distance (with magic & psionics) was found to be about 18.25 miles, which would mean the arrow is traveling at 10,950 mph, or about mach 14.39! The max pre-epic mundane distance is about 4 miles, which means the arrow is traveling at 2,400 mph, or about mach 3.15. Relative to light speed, the 4-mile travel is made at about 0.0000035788 c ("c" is "light speed"), the 8-mile travel is made at about 0.0000071621 c, and the 18.25-mile travel is made at about 0.000016328 c.

So now my question is this: Now that we have determined how to break the sound barrier 18 (*EDIT* 14) times over, what would happen to the arrow if it hit something? Would it even survive those speeds? This is where I need your help: determining if an arrow can be turned into a self-destructive arrow-of-death.
*EDIT* - The above question has been answered (sort-of). My new question is: What kind if impact forces will the arrow & target feel?

Here's the statistics I have:
Arrow's weight = 0.15 lb. (2.4 ounces, ~68.04 grams, ~0.06804 kg)
Speeds = 2,400 mph, 4,800 mph, & 10,950 mph (3,862.426 km/h, 7,724.85 km/h, & 17,622.317 km/h, respectively)
Time of deceleration = Varies by material impacted (I don't know the exact times, but if someone could find out the times for impact on flesh (average human thickness), metal (1 or 2 inches thick), wood (probably oak & pine, 6" thick & 1' thick), & brick (clay or stone), then that would be great)

If anyone can figure these out, I would most appreciate it. It would also be nice if someone gave comparison impact forces, like being hit by a car or falling from a great height. Thanks in advance for any answers, and happy computing!

[DeusMortuusEst]

The arrow would probably break when hitting the sound barrier.

Aside from that comment... It's 5 am here, and I've been awake for almost 24 hrs now, I might come back when I've had some sleep.

[White_Drake]

If you could somehow manage to get it to stay together though, it'd probably leave a crater the size of Manhattan when it hit.

[Kalaska'Agathas]

So the other day, I was looking at the "Longest shot possible" (http://www.giantitp.com/forums/showthread.php?t=166182) thread. Before I made it to the first mention of projectile speed, I thought, "Wait... that arrow has to hit a target several miles away in 6 seconds... so it must be going really fast... hmmm..." Then I read the first mention of speed, and read on to find that the required speed to hit a target 8 miles away would be about 2 km/h, or about 4,800 mph, or about mach 6.31. Farther on, even greater ranges were figured out, which meant greater speeds. Eventually, the maximum pre-epic distance (with magic & psionics) was found to be about 18.25 miles, which would mean the arrow is traveling at 10,950 mph, or about mach 14.39! The max pre-epic mundane distance is about 4 miles, which means the arrow is traveling at 2,400 mph, or about mach 3.15. Relative to light speed, the 4-mile travel is made at about 0.0000035788 c ("c" is "light speed"), the 8-mile travel is made at about 0.0000071621 c, and the 18.25-mile travel is made at about 0.000016328 c.

So now my question is this: Now that we have determined how to break the sound barrier 18 times over, what would happen to the arrow if it hit something? Would it even survive those speeds? This is where I need your help: determining if an arrow can be turned into a self-destructive arrow-of-death.

Here's the statistics I have:
Arrow's weight = 0.15 lb. (2.4 ounces, ~68.04 grams, ~0.06804 kg)
Speeds = 2,400 mph, 4,800 mph, & 10,950 mph (3,862.426 km/h, 7,724.85 km/h, & 17,622.317 km/h, respectively)
Time of deceleration = Varies by material impacted (I don't know the exact times, but if someone could find out the times for impact on flesh (average human thickness), metal (1 or 2 inches thick), wood (probably oak & pine, 6" thick & 1' thick), & brick (clay or stone), then that would be great)

If anyone can figure these out, I would most appreciate it. It would also be nice if someone gave comparison impact forces, like being hit by a car or falling from a great height. Thanks in advance for any answers, and happy computing!

Oh! The cat girls!

Essentially what you're describing is a kinetic kill vehicle. If we were to use real world physics, you would have one (a simple projectile at that velocity is going to have significant KE on impact). However, D&D doesn't really model this sort of behavior, and as far as I know, there is no bonus damage for having your arrow go really really fast.

So to answer your question, 1d8+str+enhancement+whatever else (assuming a composite long bow of appropriate strength rating) damage to your target.

[Slipperychicken]

The arrow is (IIRC) magically-enhanced, so it survives unharmed, until it strikes its target, whereupon it breaks and deals its damage normally. If it missed, then it has a 50% chance of being broken or lost.

[MetaKirby06]

Thank you, Kalaska'Agathas, for your answer, but that's not exactly what I was looking for. I guess I worded it wrong (by not even asking it), but what I really wanted to know was about how much force would be experienced by the arrow and what it hits on impact. But, I was tired when I wrote the original post, so I guess I must have forgot to. Thanks again, though!

[Spiryt]

If you could somehow manage to get it to stay together though, it'd probably leave a crater the size of Manhattan when it hit.

Well, not really, with those stats provided it gives roughly the energies of WWII era light tank cannons.

Obviously in different, rather weird proportions (ridiculous velocity, much lower mass).

So if it by some magic, managed to stay intact, instead of turning into smoke and some rather melted metal, it could probably pierce light vehicle armors, concrete walls and stuff.

[whibla]

It is not possible to calculate either the pressure exerted by the projectile, or its force, because there are too many unknowns: the area of the arrowhead, the fate of the arrow (did it pass through the target, stop, how quickly, etc. - and asking for realistic estimates went right out the moment you suggested that a wooden shafted arrow could either reach mach 18, or be propelled to this velocity from a 'normal' bow.)
There's also a significant assumption in the scenario, which is that the projectile has a constant velocity. This is unlikely. It's maximum velocity would be reached the moment it leaves the string, and from that point on it would be decelerating. Who's to say that it has any kinetic energy left at all when it reaches its target?
Sorry to be a kill joy. :smallwink:

[Kalaska'Agathas]

Thank you, Kalaska'Agathas, for your answer, but that's not exactly what I was looking for. I guess I worded it wrong (by not even asking it), but what I really wanted to know was about how much force would be experienced by the arrow and what it hits on impact. But, I was tired when I wrote the original post, so I guess I must have forgot to. Thanks again, though!

Ah, a question of terminal ballistics then. I'll leave that to those who know enough to feel comfortable talking about it.

[MetaKirby06]

Reply time:

whibla: Yeah, I guess I did leave out some info, so here it is:
Area of arrowhead = Probably 1 to 3 sq. mm.
Fate of the arrow = While the arrow will most likely pass through anything it touches, there will be a (tiny) moment upon initial contact with an object where the arrow will come to a complete stop while it exerts its KE upon the object, at which point the energy will either continue to carry the arrow through the object, or not be strong enough to penetrate the object, causing the arrow to stop. (At least I think that's how works. I may be wrong)
Lack of realism = Well, the arrow is enchanted, so it may survive from the magic. Or it will simply disintegrate the moment it is fired.
The assumption of constant velocity = Wow, I never would have thought about that as factor, but your right. The arrow will always be losing speed after the moment its fired, but it should have enough speed at the end of the 18.25 miles to at least cause as much damage as a normal arrow (so we would hope). So, for the purpose of calculation, I think its safe to assume the arrow is going at about mach 13 or so within the first 3 or 4 miles, mach 11 in the next 3 miles or so, and so on (-2 mach per 3-4 miles of travel).
The "killjoy" part = Don't worry about it! I want a (somewhat) logical answer to this (somewhat) illogical question anyway.:smallwink:

[Amidus Drexel]

If we are assuming a non-constant velocity, wouldn't the arrow need to start at a faster speed to still reach the target in 6 seconds?

[whibla]

I just know I'm going to regret this, but the question did at least get me thinking.

Unfortunately, every time I try to find a way to reply the seemingly inherent illogical nature of the situation brings me up short. I'll try and explain:

The original statement was that the projectile flies its maximum distance. Now, all unpowered projectiles experiencing the force of gravity fly in a parabolic arc (ignoring the effects of air resistance). Their maximum range is obtained by firing upwards at a 45 degree angle (at least if I remember my school maths correctly). If the projectile flies for 6 seconds it will be climbing for 3 of those. The vertical element of it's velocity is given by the formula v = u + a x t where v is its velocity at any given point (in metres per second), u is its initial velocity (in metres per second), a is the acceleration due to gravity (let's round off to 10 metres per second squared), and t is the time taken at any given point (in seconds). So, after 3 seconds v = 0, which gives us a value for u of 30 metres per second. Since we fired the projectile at an angle of 45 degrees I would expect the horizontal component of velocity to be the same...erm...this is way short of the 5000 metres per second we require in order for it to travel the 18 miles in 6 seconds.

Ok, so, maybe we're not talking about the maximum maximum range of the projectile per se, just the maximum accurate range. Let's ignore the rise and fall, the parabolic path (which only occurs in an ideal world, with no air resistance anyway), of the missile, and instead simply state that it has a horizontal velocity of (roughly) 5000 metres per second.

For comparison, a modern day sniper rifle has a muzzle velocity of roughly 1000 metres per second (depends obv. on the rifle and the ammo used), and an effective range of not much more than 1 mile. The rifles are usually equipped with a tripod which serves two purposes; to steady the shot, and to mitigate the recoil as the 25 gram projectile (again, varies by rifle and ammo) is fired. Initial energy of the bullet (and hence the energy of the recoil) is thus 12.5 kilojoules (roughly the same as the kinetic energy of an 'average' human body travelling at 40 miles per hour. Gonna hurt if you stop suddenly!).


The assumption of constant velocity = Wow, I never would have thought about that as factor, but your right. The arrow will always be losing speed after the moment its fired, but it should have enough speed at the end of the 18.25 miles to at least cause as much damage as a normal arrow (so we would hope). So, for the purpose of calculation, I think its safe to assume the arrow is going at about mach 13 or so within the first 3 or 4 miles, mach 11 in the next 3 miles or so, and so on (-2 mach per 3-4 miles of travel).

The actual velocity of a standard round drops by a factor of somewhere between a quarter and a half in the first 200 metres of its flight (depends on its original muzzle velocity and the friction on the projectile. Faster, fatter, and longer projectiles slow down more, due to the nature of air resistance). I suspect I'm, again, going to have to suspend my disbelief, and assume that air resistance plays no part in affecting the flight of this extraordinary magic arrow.


Fate of the arrow = While the arrow will most likely pass through anything it touches, there will be a (tiny) moment upon initial contact with an object where the arrow will come to a complete stop while it exerts its KE upon the object, at which point the energy will either continue to carry the arrow through the object, or not be strong enough to penetrate the object, causing the arrow to stop. (At least I think that's how works. I may be wrong)

Um. I'm not quite sure why you would think there would be a moment when the projectile instantaneously stops, and in that moment impart all its energy. Kinetic energy can be expressed as force x distance, but unless we're talking about the mythical unstoppable force hitting the mythical immovable object that distance is never going to be zero. However, I think I can see where you're coming from, and yes, I guess in the instant that the arrow hits its target the various forces are resolved. In reality, and in D&D, the arrow will be destroyed when it hits, and a large proportion of its energy will be dissipated / redirected at this point. Since there's no real way of determining just how much energy will be lost in this virtual disintegration it's probably best just to ignore this 'little' detail.


Lack of realism = Well, the arrow is enchanted, so it may survive from the magic. Or it will simply disintegrate the moment it is fired.

I didn't read the original thread. After this thread I'm pretty sure I do not want to, now or ever! However, I'd probably guess that a significant amount of (broken) magic is involved, along with a fair amount of rules abuse, something along the lines of a range doubling magic arrow, a range doubling spell, and a range doubling psionic ability. So, taking that factor of 8 out of any calculations means that the initial input energy will be 8 times smaller than would usually be required to propel the projectile to such ludicrous speeds. But how much energy does the arrow carry, with all our simplified assumptions? KE = .5 x m x v x v
So, it's kinetic energy is roughly 850 kilojoules, and, from what I said above, the character firing it would be required to impart maybe 100 kilojoules of that energy (This is roughly the equivalent of the energy 'contained' in your 'average' human travelling at 110 miles per hour). I am impressed by the strength of the archer, and the quality of his bow!

In terms of what the arrow will do to the target...
It would be the equivalent of being hit by a small car (500 kilos) travelling at roughly 120 miles per hour.
It would pierce several feet of wood (assuming the arrow stayed in one piece for long enough), or at least 1 foot of solid metal or a couple of feet of concrete at least.


The "killjoy" part = Don't worry about it! I want a (somewhat) logical answer to this (somewhat) illogical question anyway.:smallwink:

My head hurts! :smallredface:

[Spiryt]

It would pierce several feet of wood (assuming the arrow stayed in one piece for long enough), or at least 1 foot of solid metal or a couple of feet of concrete at least.



3000 and more kJ 'traditional' tank shells were just coming close to piercing foot of armor, so I guess that with some 'metal' not actually designed to be armor, it would be possible.

Obviously our magical arrow would have much higher KE/cm2, assuming that it's magic and in some impossible way it's able to actually transfer it trough target. :smallbiggrin: Still, it would have somehow small momentum for it's energy.... Probably would go trough few dozens of mm of armor at most.

[MetaKirby06]

More replies!

whibla: Wow, that's a long answer! Thank you very much for taking your time to write all that (not to mention all the calculations!). So yeah, obviously the "real-world-arc" thing didn't really work out well. For the speed drop, I agree that we can leave that out of the equation (the arrow is being partly propelled by magic and psionics anyway, so that would probably keep the speed up).

Originally Posted by whibla http://www.giantitp.com/forums/images/buttons/viewpost.gif (http://www.giantitp.com/forums/showthread.php?p=13456630#post13456630)
Um. I'm not quite sure why you would think there would be a moment when the projectile instantaneously stops, and in that moment impart all its energy.
Yeah, I'm not exactly "well informed" about that kind of stuff (hence this post). But thanks for the correction.:smallwink:

About the details for how this all works, you're pretty close to it:

Originally Posted by cd4, "Re: Longest Shot Possible" http://www.giantitp.com/forums/images/buttons/viewpost.gif (http://www.giantitp.com/forums/showthread.php?p=9248614#post9248614)
I was looking around other sites and found this thread (http://community.wizards.com/go/thread/view/75882/19866454/Ok_so_what_is_the_maximum_distance_you_can_get_usi ng_a_bow?)

It basically asks the same question and ended up with a pre-epic shot of ~18.25 miles. The relevant stats of the final build are here:


Wilder (XPH/SRD) 15 / Cragtop Archer (RoS) 3 / Deepwood Sniper (MotW) 2

Race: Goliath (RoS) or Half-Giant (XPH/SRD)

Feats:
Expanded Knowledge [Extend Range] (XPH/SRD)
Practiced Manifester (CPsi)
Far Shot (PHB/SRD)

Equipment:
Large Dragonbone (Drac) Composite Great Bow (CWar) of Distance (DMG/SRD)
Kerrenderit Arrows (FR:RoF) + some way to keep them frozen
Scroll of Accuracy (OA, CArc)
Scroll of Giant Size (CArc) @ 19th level
Scroll of Hawkeye (CAdv, SpC)
Scroll of Wind Tunnel (MotW, FR:Mag, SpC)
Torc of Power Preservation (XPH/SRD)

Note: For this to work, you need to have a high Use Magic Device in order to be able to use the 3 scrolls.

The computations:

Base Weapon:
Composite Greatbow (130 ft.) + Dragonbone (+20 ft.) + Kerrenderit Arrows (+25 ft.) = 175 ft. range increment
Size increase:
Large weapon (Goliath or Half-Giant) = 1 size increase
19th level Scroll of Giant Size = Colossal = 4 size increases
Each size increase = x1.25 range (A&EG, SavS)
175 ft. x 1.25^5 = ~534.06 ft.

Multipliers:
x1 [base weapon]
x1 ...of Distance
x7 Extend Range - 25 psi points (15 Wilder + 5 Wild Surge + 4 Practiced Manifester + 1 Torc of Power Preservation)
x½ Far Shot
x1 Scroll of Accuracy
x½ Scroll of Hawkeye
x1 Scroll of Wind Tunnel
= X12 = ~6408.69 ft.

+ 20 ft. (Deepwood Sniper 2)
= ~6428.69 ft. (range increment is over a mile!)

Maximum Range:
15 range increments (Cragtop Archer 3)
~6428.69 ft. x 15 = ~96,430 ft. (~18¼ miles)

And here's the non-magic 4 mile shot:

Originally Posted by cd4, "Re: Longest Shot Possible" http://www.giantitp.com/forums/images/buttons/viewpost.gif (http://www.giantitp.com/forums/showthread.php?p=9248614#post9248614)
Oh and they also worked out without any magic or psionics it was only just under 4 miles!


Goliath Fighter 5/Deepwood Sniper 10/Cragtop Archer 4

Composite Greatbow Range: 130ft
Dragonbone Enhancement: +20ft
Flight Arrows: +25ft
Ranged Weapon Mastery: +20ft
Deepwood Sniper +100ft
--------------------------------------------- = 295ft
Distance Weapon x2
Far Shot x1.5
Large Weapon: x1.25
--------------------------------------------- = x4.75

295ft x 4.75 = 1401.25ft

So without Arcing shot our maximum range with our weapon is 14,012.5ft.

With Arcing shot our maximum range with our weapon is 21,018.75ft.


Originally Posted by whibla http://www.giantitp.com/forums/images/buttons/viewpost.gif (http://www.giantitp.com/forums/showthread.php?p=13456630#post13456630)
I am impressed by the strength of the archer, and the quality of his bow!
Me too! Although, the archer is a Goliath, and the bow is made of dragonbone.

Also, thanks for all the comparisons, they were really helpful for grasping this concept on a semi-real-world basis.


Spiryt: You do bring up an interesting point, so I'd guess the arrow would be nearly disintegrated after the few mm of piercing you said it could do. But on wood and flesh... :smallamused: