range and atmospheric density

SteveS · 07-27-2018, 06:07 PM

I was studying 1/2D and Max range in Vehicles 2*, and came across a puzzle. How is a weapon affected by changes in atmospheric pressure? I can think of a lot of factors that might be involved.

1. In vacuum, friction is absent; the projectile is affected only by its initial velocity, gravity, and hitting something. Accuracy benefits from the absence of winds. But fin stabilization doesn't work.

2. At subsonic velocities, drag is proportional to density (and also to area, the square of the velocity, and the coefficient of drag for the object).

3. I know that drag works differently at subsonic, transonic, and supersonic velocities.

3a. At subsonic velocities in normal air, the speed of sound depends mainly on temperature (approximately proportional to the square root of the absolute [Kelvin] temperature).
3b. At transonic velocities, when part of the air flow around an object is subsonic and part of it is supersonic, things are different, but I don't know how.
3c. At supersonic velocities, things are different, but I don't know how.

4. Based on what I found and could understand, the effective coefficient of drag changes between subsonic, transonic, and supersonic velocities. The most complicated region is transonic.

5. Aiming is easier when the transonic region is avoided, either by staying within a range where the bullet is supersonic, or by firing bullets that start out subsonic.

- - -

More points:

A. Range depends on initial velocity, the arc determined by gravity, and the reduction due to drag.

B. Impact velocity depends on initial velocity and the reduction due to drag.

C. Damage depends on the square of the impact velocity and the characteristics of the projectile.

D. In normal atmosphere, game-calculated weapon characteristics model "A", "B", and "C".

E. In vacuum, range increases to the projectile's gravity-guided arc, because there are no losses due to drag. A projectile does full damage to anything it can hit, because drag doesn't slow it down. The game-calculated full-damage figures are correct as long as the projectile can hit, but the game-calculated range figures are too low. The game doesn't provide the math here, but it's pretty easy to calculate based on initial velocity, which is widely published for real projectiles, and can be derived for fictional weapons.

F. In a half-pressure atmosphere, both the 1/2D and Max ranges should be longer than the game-calculated distances.

G. In a double-pressure atmosphere, both the 1/2D and Max ranges should be shorter than the game-calculated distances.

H. Both "F" and "G" seem to be quite a bit more complicated than "E".

I. High relative velocity between shooter and target changes the velocity, as do substantial differences in altitude. In air-to-air combat, drag is higher for bullets fired by a pursuing gunner than those fired by a tail-gunner. But those effects are too complicated for gaming, even if they're substantial.

* If the corresponding weapon design system for GURPS 4th edition is published somewhere, I haven't found the right search term.

Fred Brackin · 07-29-2018, 12:50 PM

Was there a question in there somewhere or was it just an invitation to ramble on about stuff?

My simplifies for actual gmaing rules are:
Divide 1/2 by pressure
Divide Max by Gravity
In the absence of either adapt the "time in flight" rules from Tactical Shooting if it actually becomes necessary. Which it probably won't because the Speed/Range chart is extremely hostile to very long range shots.

Anthony · 07-29-2018, 01:09 PM

The basic problem is that GURPS 1/2D is fairly dissociated from the range at which drag actually reduces projectile velocity by 50% (which we would expect to halve GURPS damage). It's particularly pronounced for thrown weapons, most of which should have 1/2D exceeding their Max.

Phoenix_Dragon · 07-29-2018, 01:59 PM

If the deceleration of a projectile is roughly linear with pressure (I don't know if this is true, but it seems like a reasonable guess), it would mean dividing range by the square of the atmospheric pressure (So in 2 atmospheres of pressure you divide ranges by 4).

Anthony · 07-29-2018, 05:19 PM

Quote:

Originally Posted by Phoenix_Dragon

If the deceleration of a projectile is roughly linear with pressure (I don't know if this is true, but it seems like a reasonable guess), it would mean dividing range by the square of the atmospheric pressure (So in 2 atmospheres of pressure you divide ranges by 4).

It's typically linear in density, and no, there's no squaring involved.

Fred Brackin · 07-29-2018, 06:20 PM

Quote:

Originally Posted by Anthony

It's typically linear in density, and no, there's no squaring involved.

Yes, at 2x density you're pushing twice as much mass out of your bullet's way.

vicky_molokh · 07-30-2018, 02:04 AM

Quote:

Originally Posted by Fred Brackin

Yes, at 2x density you're pushing twice as much mass out of your bullet's way.

Of course, with gyrocs and other powered projectiles, things get trickier.

SteveS · 07-31-2018, 10:14 AM

I suppose I could run a bunch of projectiles through Douglas Cole's ballistics spreadsheet, and try to understand the spreadsheet enough to know where atmospheric density comes in. And from that maybe try to come up with a reasonable game approximation rule.

But my guess is that it's a square-cube relationship. Double the bore size and leave the shape and density the same and the cross section area increases by a factor of four, and the mass by a factor of eight, so the drag per unit mass is halved.

Anaraxes · 07-31-2018, 10:50 AM

The drag equation is Drag = Cd * (rho * V^2) / 2 * A

Cd: coefficient of drag, a number that comes mostly from the shape of the moving object
rho: density of the air (so yes, this is linear with density, which is to say linear with pressure)
V: velocity of the object
A: reference area (usually the cross section perpendicular to the direction of motion)

So if you keep the same shape of object, then its cross-sectional area should indeed go up with the square of the length increase, and so the drag on that projectile.

Range is a function of the projectile's velocity. How far can it travel before it falls far enough to hit the ground? (Note that the falling time has nothing do with the velocity, though that does affect how far the projectile goes in that amount of time.) If gravity doesn't matter, then how far do you travel before velocity is reduced to zero by the drag?

Velocity is reduced by drag, so the actual velocity you want to use can be written as a function of pressure.

Anthony · 07-31-2018, 12:20 PM

Quote:

Originally Posted by Anaraxes

Range is a function of the projectile's velocity. How far can it travel before it falls far enough to hit the ground? (Note that the falling time has nothing do with the velocity, though that does affect how far the projectile goes in that amount of time.) If gravity doesn't matter, then how far do you travel before velocity is reduced to zero by the drag?

The answer to the latter is, unfortunately, 'forever'.
Drag = Cd * rho * V^2/2 * A.
KE = m * V^2/2.
Drag = KE * Cd * rho * A / m.
ΔKE/Δd = -Drag
If we assume Cd*rho*A/m is a constant, this corresponds to a simple exponential decay, with energy being reduced by a factor of e in a distance of m/(Cd*rho*A), or by a factor of 4 (half velocity) in a distance of ln(4) times that.

07-27-2018, 06:07 PM	#1
SteveS Join Date: Nov 2013 Location: near Seattle WA USA	range and atmospheric density I was studying 1/2D and Max range in Vehicles 2, and came across a puzzle. How is a weapon affected by changes in atmospheric pressure? I can think of a lot of factors that might be involved. 1. In vacuum, friction is absent; the projectile is affected only by its initial velocity, gravity, and hitting something. Accuracy benefits from the absence of winds. But fin stabilization doesn't work. 2. At subsonic velocities, drag is proportional to density (and also to area, the square of the velocity, and the coefficient of drag for the object). 3. I know that drag works differently at subsonic, transonic, and supersonic velocities. 3a. At subsonic velocities in normal air, the speed of sound depends mainly on temperature (approximately proportional to the square root of the absolute [Kelvin] temperature). 3b. At transonic velocities, when part of the air flow around an object is subsonic and part of it is supersonic, things are different, but I don't know how. 3c. At supersonic velocities, things are different, but I don't know how. 4. Based on what I found and could understand, the effective coefficient of drag changes between subsonic, transonic, and supersonic velocities. The most complicated region is transonic. 5. Aiming is easier when the transonic region is avoided, either by staying within a range where the bullet is supersonic, or by firing bullets that start out subsonic. - - - More points: A. Range depends on initial velocity, the arc determined by gravity, and the reduction due to drag. B. Impact velocity depends on initial velocity and the reduction due to drag. C. Damage depends on the square of the impact velocity and the characteristics of the projectile. D. In normal atmosphere, game-calculated weapon characteristics model "A", "B", and "C". E. In vacuum, range increases to the projectile's gravity-guided arc, because there are no losses due to drag. A projectile does full damage to anything it can hit, because drag doesn't slow it down. The game-calculated full-damage figures are correct as long as the projectile can hit, but the game-calculated range figures are too low. The game doesn't provide the math here, but it's pretty easy to calculate based on initial velocity, which is widely published for real projectiles, and can be derived for fictional weapons. F. In a half-pressure atmosphere, both the 1/2D and Max ranges should be longer than the game-calculated distances. G. In a double-pressure atmosphere, both the 1/2D and Max ranges should be shorter than the game-calculated distances. H. Both "F" and "G" seem to be quite a bit more complicated than "E". I. High relative velocity between shooter and target changes the velocity, as do substantial differences in altitude. In air-to-air combat, drag is higher for bullets fired by a pursuing gunner than those fired by a tail-gunner. But those effects are too complicated for gaming, even if they're substantial. If the corresponding weapon design system for GURPS 4th edition is published somewhere, I haven't found the right search term.

07-29-2018, 12:50 PM	#2
Fred Brackin Join Date: Aug 2007	Re: range and atmospheric density Was there a question in there somewhere or was it just an invitation to ramble on about stuff? My simplifies for actual gmaing rules are: Divide 1/2 by pressure Divide Max by Gravity In the absence of either adapt the "time in flight" rules from Tactical Shooting if it actually becomes necessary. Which it probably won't because the Speed/Range chart is extremely hostile to very long range shots. __________________ Fred Brackin

07-29-2018, 01:09 PM	#3
Anthony Join Date: Feb 2005 Location: Berkeley, CA	Re: range and atmospheric density The basic problem is that GURPS 1/2D is fairly dissociated from the range at which drag actually reduces projectile velocity by 50% (which we would expect to halve GURPS damage). It's particularly pronounced for thrown weapons, most of which should have 1/2D exceeding their Max. __________________ My GURPS site and Blog. Last edited by Anthony; 07-29-2018 at 01:25 PM.

07-29-2018, 01:59 PM	#4
Phoenix_Dragon Join Date: Apr 2005	Re: range and atmospheric density If the deceleration of a projectile is roughly linear with pressure (I don't know if this is true, but it seems like a reasonable guess), it would mean dividing range by the square of the atmospheric pressure (So in 2 atmospheres of pressure you divide ranges by 4).

07-31-2018, 10:14 AM	#8
SteveS Join Date: Nov 2013 Location: near Seattle WA USA	Re: range and atmospheric density I suppose I could run a bunch of projectiles through Douglas Cole's ballistics spreadsheet, and try to understand the spreadsheet enough to know where atmospheric density comes in. And from that maybe try to come up with a reasonable game approximation rule. But my guess is that it's a square-cube relationship. Double the bore size and leave the shape and density the same and the cross section area increases by a factor of four, and the mass by a factor of eight, so the drag per unit mass is halved.

07-31-2018, 10:50 AM	#9
Anaraxes Join Date: Sep 2007	Re: range and atmospheric density The drag equation is Drag = Cd * (rho * V^2) / 2 * A Cd: coefficient of drag, a number that comes mostly from the shape of the moving object rho: density of the air (so yes, this is linear with density, which is to say linear with pressure) V: velocity of the object A: reference area (usually the cross section perpendicular to the direction of motion) So if you keep the same shape of object, then its cross-sectional area should indeed go up with the square of the length increase, and so the drag on that projectile. Range is a function of the projectile's velocity. How far can it travel before it falls far enough to hit the ground? (Note that the falling time has nothing do with the velocity, though that does affect how far the projectile goes in that amount of time.) If gravity doesn't matter, then how far do you travel before velocity is reduced to zero by the drag? Velocity is reduced by drag, so the actual velocity you want to use can be written as a function of pressure. Last edited by Anaraxes; 07-31-2018 at 11:00 AM.