TRAJECTORY MODEL FOR PROJECTILES

[rexford]

The following analysis is in response to a question regarding what equations were used to calculate hit % using range estimation and battlesight aim:

We have detailed info for German projectiles, flight time vs range, max trajectory height, firing and final descent angles, 50% dispersion (vert and lat), etc.

Max trajectory height = 1.336 (Flight time) raised to 1.919 power.

Max trajectory height occurs at following distance from gun: 0.4443 (range setting)raised to 1.0224 power.

Above two equations determined from regression equations through alot of data points for every German gun listing, and correlation R squared is over 0.99.

We have found that these equations work very well with 105mm APDS and 50mmL42 APC. They don't work especially well with slow rounds, like 75L24. But we're mostly interested in 88 and 76 anti-tank types, for most part. And slow rounds will have low hit % anyway.

We used above two equations to fit an equation thru trajectory and found that following equation predicts max trajectory height, point of max height and descent angle reasonably well and is fairly easy to use on spreadsheet, it is a good and quick and dirty shortcut to complex trajectory analysis:

Trajectory height at given range equals:

A x (distance to target)squared plus

B x (distance to target)

A = flight time to gun range setting divided by (C - D)

C = .4443squared times gun range setting raised to 2.0448 power

D = .4443 times gun range setting raised to 2.0224 power

B = A times -1 times gun range setting

For 88L56 aimed at 900m (battlesight range), equation results in following trajectory equation:

traj. height = -.00000952 x (target distance)squared + .000857 x (target distance)

This equation will closely predict max height of trajectory, location of max height and final descent angle, so should be good estimate for trajectory along path.

Estimated descent slope at 900m when 88L56 aims at 900m is 0.049° or .00857 radians or 8.57 mils. German charts list 9 mils as 88L56 fall winkel when gun is aimed at 900m.

Doing similar analysis for other guns aimed at 900m battlesight range results in following comparisons:

50L60 APCR

Trajectory equation predicts 9.26 mils descent, chart lists 10 mils

75L48 APCBC

Equation predicts 9.55 mils descent, chart lists 10 mils

88L71 APCBC

Equation predicts 7 mils descent for 1200m range setting, chart lists 7 mils

76.2L51.5 APCBC

Equation predicts 10.7 mils descent at 900m, chart lists 11 mils

75L70 APCBC

Equation predicts 7.7 mil descent when gun is aimed at 1100m, we don't have German data for this ammo

So above equation is reasonable model for wargame analysis, it may be off by close to 7% at times in predicted trajectory height, but to hit a target trajectory has to be very close to target and dispersion data might vary in field anyway (things start to loosen in field, or warp and bend under stress and bouncing), and guns wear so that muzzle velocity decreases with time which reduces accuracy.

What occurs in the field is not exactly what happens on a nice day in the firing range with carefully kept-up guns and ammo that has been proofed under non-stress and ideal conditions. So we can accept some slack in the trajectory model.

We'll use the model to predict battlesight accuracy when target is at 700m and is 2.2m high and 2.2m wide, fired at by 88L56 using 900m range setting with aim at bottom of hull.

Target is 2.2m high at 700m. Equation predicts that average trajectory height will be 1.33m above bottom of hull. This puts round right on the turret/hull meeting point on my 1/72 scale M4A1 Sherman.

So trajectory is 1.33m above bottom and 0.87m from top.

At 900m, doubled 50% dispersion of 88L56 equals 0.3m vertical and 0.2m lateral. Convert to 68.3% dispersion by dividing by .675, for .44m vert and 0.30m lat.

Trajectory breaks Sherman hull into 1.33m above and 0.87 below vertical aim point. Now we use bell shaped curve data for percentage vs # of standard deviations (1 standard deviation contains 68.3% of data points, 2 standard deviations contain 95.5%, etc.).

1.33m from trajectory to bottom contains 3 standard deviations, so 100% of dispersions fit into 1.33m. All of dispersions below trajectory hit target vertically although we still have to check lateral spread.

0.87m from trajectory to top of tank, which contains 1.98 standard deviations and contains about 95% of dispersion spread.

100% of dispersions above trajectory hit, 95% below hit, 97% of all dispersions hit target.

Lateral spread check. Tank is 1.1m from center to outside, lateral standard deviation is 0.3m, over 3 standard deviations from center to outside width equals 100% of dispersions within 2.2m lateral box dimension.

So when Tiger I uses battlesight aim at 700m Sherman target, 97% hit probability against fully exposed front hull.

What if Tiger uses range estimation vs. 700m target and average error is 35%. Standard deviation for 35% avg. error is about 28.6%.

700m times 28.6% is 200m, which represents standard deviation for range estimation error. 700m + 200m is 900m aim for 35% average error.

Standard deviation for Tiger I aim error is associated with 900m aim, target is at 700m. 2m is standard deviation for range error impact, which places trajectory 2m above or below aim point at center of Sherman (for discussion purposes we'll ignore aim at turret/hull point).

2m above target center is 0.90m above turret top. 0.90m divided by 0.44m vert error standard deviation equals just over 2 standard deviations, so about 96% of high dispersions miss target and 4% hit. Same thing for low dispersions.

All shots are within lateral dimension as previously determined.

So 35% average range estimation error against 700m Sherman target results in 4% hit chance. For 35% range estimation error we normally assume poor crews and increase dispersion, so hit % even lower.

Equation for trajectory height with 840m aim is -.00000959 (target range)squared plus 0.00805 times target range.

With 25% average estimating error, trajectory above aim point is 0.94m above aim point, which places trajectory on target. % of hits is found by dividing distances above and below trajectory by standard deviation of vertical dispersion and then checking to see that all lateral dispersion is within width.

0.26 meters from trajectory to top of tank divided by 0.44m standard deviation for vert dispersion yields 0.59 standard deviations, or 44% of high dispersions hit target.

There is 0.94m from aim point to center of target, which equals 2.14 standard deviations of vert dispersion or 97%. So average % of dispersions on target is (97 +44)/2, or 70%.

Battlesight increases hit % over range estimation by average crew from 70% to 97% against 2.2m high target.

This can be placed on a computer and math can be automated. Our spreadsheet goes thru the above steps with a few shortcuts but uses the same logic. We model dispersion with curves of best fit and randomly choose a range estimation error for each shot based on curve for crew quality, lateral and vertical error from aim is determined for each shot and we measure error on model to see where shot falls.

With above models and dispersion data one can calculate hit %'s. We had used ballistic analysis program in BASIC language but wanted something that would quickly predict trajectory with reasonable error. For Battlefield aim the trajectory equation predicts a height that is about 0.1m above maximum trajectory height, our spreadsheet reduces estimates near or above max height but it is okay to use it unrevised to model gun barrel wear or misaligned sights or whatever. Accuracy never improves with weapon age.

Regarding Clint Eastwood's advice on how to pick targets, he was asked how he could take on ten men at once and get them all. Luck, and knowing who was likely to aim and who would just pull the trigger.

Regarding limitations, it is early in the morning and I tried to be careful but some minor errors may have crept into the analysis. The equations were checked several times.

A man has got to know his limitations.

[rexford]

We have math models that predict second and follow-up shot range estimation errors and dispersion decreases. Also have math models for velocity vs range for all rounds so computer can calculate flight times. Dispersion data is also converted to math models vs. range.

Made one little goof in previous message, used dispersion at 900m instead of target range of 700m. Lower dispersion increases 25% avg range error hit prob. and lowers 35% avg range error prob.

88L56 with 25% average range est. error gets 75% hit probability, and 35% avg error actually goes up because we quadruple basic dispersion for poor crews (35% avg range error), which lowers % when trajectory is on target but increases hit % when trajectory misses target: big dispersion brings more rounds down on target when trajectory is high, raises half of shots when trajectory is low.

We compared the results of the simplified trajectory model with a complicated BASIC ballistic computer program for naval ammunition (great accuracy to 10 miles range, I guess), and our model was always within about 0.1m of the time consuming result for typical tank battle ranges.

If anyone wants it we can post German dispersion data on this site, say every 500m for lateral and vertical dispersion and convert 50% data to 68.3% (one standard deviation coverage).

Can also post flight time data and max trajectory height info.

[Ari Maenpaa]

Unbelievable posts I have to say. It will take some time to carefully analyze all you wrote

I would be very interested to see the dispersion data mentioned.

Anyway You have shown so much knowledge about tank warfare that I thought to ask your opinion about 88L71's penetration performance. We had quite a discussion recently, but without the final word. Here's a link: http://www.battlefront.com/discuss/Forum1/HTML/009258.html

Maybe you have something to add there.

All the best,

Ari

[Jeff Duquette]

<BLOCKQUOTE>quote:</font><HR>Rexford Said:

If anyone wants it we can post German dispersion data on this site, say every 500m for lateral and vertical dispersion and convert 50% data to 68.3% (one standard deviation coverage).<HR></BLOCKQUOTE>

I would love to look at this data.

[Forever Babra]

Outstanding. I bow before the mega-grog.

"We're not worthy... we're not worthy..."

[Monte99]

Oh, for God's sake...

[rexford]

German projectile data, doubled 50% dispersion and flight time to range, dispersion rounded to nearest 0.1m in many cases.

Divide 50% dispersion by .675 to obtain standard deviation (68.3%).

50L60 APC

100m:.03m vert and .03 lat

.12 sec

500m: .15m vert anc .15 lat

.65 sec

1000m: .33m vert and .30m lat

1.43 sec

1500m: .58m vert and .50m lat

2.35 sec

75L48 APCBC

100m: .1m vert and 0.04m lat

.13 sec

500m: .3m vert and .2m lat

.68 sec

1000m: .6m vert and .5m lat

1.44 sec

1500m: 1.0m vert and 0.9m lat

2.27 sec

2000m: 1.6m vert and 1.3m lat

3.16 sec

2500m: 2.4m vert and 1.8m lat

4.12 sec

3000m: 3.3m vert and 2.3m lat

5.20 sec

76.2L51.5 APCBC

100m: .1m vert and .1m lat

.14 sec

500m: .4m vert and .3m lat

.73 sec

1000m: .8m vert and .6m lat

1.52 sec

1500m: 1.3m vert and 1.om lat

2.38 sec

2000m: 1.8m vert and 1.4m lat

3.32 sec

2500m: 2.5m vert and 1.9m lat

4.33 sec

3000m: 3.3m vert and 2.4m lat

5.43 sec

88L56 APCBC

100m: .1m vert and .1m lat

.13 sec

500m: .2m vert and .2m lat

.65 sec

1000m: .4m vert and .2m lat

1.35 sec

1500m: .6m vert and .3m lat

2.09 sec

2000m: .9m vert and .5m lat

2.91 sec

2500m: 1.0m vert and .5m lat

3.08 sec

3000m: 1.3m vert and .8m lat

4.66 sec

88L71 APCBC

100m: .1m vert and .04m lat

.10 sec

500m: .2m vert and .2m lat

.50 sec

1000m: .5m vert and .3m lat

1.04 sec

1500m: .7m vert and .5m lat

1.60 sec

2000m: .9m vert and .7m lat

2.20 sec

2500m: 1.1m vert and .9m lat

2.83 sec

3000m: 1.4m vert and 1.0m lat

3.50 sec

3500m: 1.6m vert and 1.2m lat

4.21 sec

4000m: 1.8m vert and 1.4m lat

4.96 sec

75L70 APCBC

No dispersion data available.

Comparison of listed dispersion hit %'s in tables from various sources suggests using 50L60 APC up to and including 1000m, 88L56 after that.

Panther flight time estimated at:

100m: .11 sec

500m: .56 sec

1000m: 1.17 sec

1500m: 1.82 sec

2000m: 2.52 sec

2500m: 3.27 sec

3000m: 4.06 sec

For 88L56 vs. 2m high x 2.5m wide at 2000m, 50% dispersion is .9m vert and .5m lat. This corresponds to 68.3% dispersion of 1.33m vert and .74m lat (68.3% of dispersions will be within stated distance from center of box).

1m/1.33 = .75 standard deviations = 55%

For lat dispersion, 1.25m/.74m = 1.69 standard deviations = 91%

Lat % x Vert % = .91 x .55 = .50 listed in table for 88L56 at 2000m.

[Conall]

What I would be interested to know is how your model deals with the following factors:

1. The manner in which it treats aspects of fluid dynamics (drag etc), principally calculation of mach numbers & the transition from subsonic to transonic region & from there to supersonic & hypersonic (typically above Mach 5, equating very approximately to projectile speeds above 1000m/s). This area may help explain the difficulty you've had with low velocity guns the 75mmL24 for example.

2. Projectile shape (length, weight, ogive, skin etc), especially with regard to the calculation of the drag coefficient & therefore the ballistic coefficient.

3. How does it treat velocity loss, as a constant or otherwise?

4. Spin induced Yaw & aerodynamic lift.

5. External factors, notably crosswinds, air density & temperature.

6. Gun jump & barrel droop.

7. Treatment of subcalibre rounds - APDS etc.

Given that you are trying to produce a simplified model from a very complex subject I'd be fascinated to know how you've incorporated these factors, or alternatively felt confident that for your purposes you can safely ignore them. Having read your posts my greatest concern would be with a) the treatment of different shaped rounds (especially sub-calibre) & the impact of external factors.

Anyway thanks for all your material, it's been a really interesting way to start the New Year. I look forward to hearing from you.

Best regards,

Conall

[Maximus]

Well I guess I could speak for a lot of us that I have just one question...

Who really gives a ****?

That's what the CPU is for...to calculate all this crap so we don't have to.

This post was meant to have a humorous tint to it, so don't get all groggy on me.

------------------

"Live by the sword, live a good LOOONG life!"-Minsc, BGII

"Boo points, I punch."--Minsc, BGII

[B M Deleted]

May I please be the second one to say, "Who gives a ****????"

[aka_tom_w]

Some of us actually come here to discuss and read about this "stuff".

Rexford's comments and insight may actually make CM2 EVEN better!

I'm not sure why others here don't just disregard this "stuff" and post in other threads where perhaps there is a more constructive and positive discussion taking place. In the meantime some of us really enjoy all this technical data and ballistics stuff.

Keep up the good work Rexford!

-tom w

[Mace]

<BLOCKQUOTE>quote:</font><HR>Who gives a ****?<HR></BLOCKQUOTE>

The game designer!

You can't model any system or event without understanding how it works in the first place.

No understanding ---> no model ---> no rules/algorithms ---> no CM.

Simple as that.

Mace

[rexford]

I put a detailed explanation of the spreadsheet on the tanknet site, in response to the inquiry from Conall.

The Peng stuff is never opened by me because it is obvious where it is headed, and we don't want to go there. Others can simply avoid our road.

[Lacky]

I second mace's opinion.

More information is good. Less information leads to a departure from reality. I enjoy accurate simulations, not the one's based on false information carried forward after the fact. Far too many wargames/simulations have diluted reality.

[Jeff Duquette]

I'm with Tom regarding his sentiments. I find the information interesting. If your bored by this stuff its perfectly within your rights to avoid these threads.

[The Commissar]

Whoa...I just woke up after passing out when I took a look at all those purdy numbers.

Jesus, I just realised people build proffesions calculating this stuff!

And I though posts about the intricate dates and details of some meaningless battle was groggy!

Keep up the good work though, this could be useful to some, specially BTS!

------------------

"...Every position, every meter of Soviet soil must be defended to the last drop of blood..."

- Segment from Order 227 "Not a step back"

[hnh3_cm]

anti-spam bump

[Vanir]

<BLOCKQUOTE>quote:</font><HR>Originally posted by Ari Maenpaa:

Anyway You have shown so much knowledge about tank warfare that I thought to ask your opinion about 88L71's penetration performance. <HR></BLOCKQUOTE>

Done

------------------

You've never heard music until you've heard the bleating of a gut-shot cesspooler. -Mark IV

[This message has been edited by Vanir (edited 01-08-2001).]

[tss]

rexford wrote:

Max trajectory height = 1.336 (Flight time) raised to 1.919 power.

This is pretty close to the formula that we were taught in the artillery weather school. We used 1.5 * FT^2 that overestimates the actual ceiling but is close enough for artillery weather report purposes.

We used the data to estimate the time when a sounding had to be started. If the gunners wanted to shoot 40 second flight times, the ceiling was ~2000 m, and the sounding would take a little over 10 minutes with the time to fill the balloon counted in. (The balloon will take ~6 minutes to raise to 2 km, IIRC).

The more rough formula has the advantage that it can be calculated in head for most flight times. It certainly isn't accurate enough for probabilistic analysis of hits and misses.

- Tommi

[tss]

I read the formula again, and add few more comments:

rexford wrote:

What occurs in the field is not exactly what happens on a nice day in the firing range with carefully kept-up guns and ammo that has been proofed under non-stress and ideal conditions.

That is certainly true. I have been posting examples from the Kuuterselkä battle for few days and found one that illustrates battlefield accuracy:

At one point, alik. Vartio's stug (-7) was advancing on the point and noticed a T-34-85 on the side of the road, ~20 meters away. The driver quickly turned the vehicle towards the tank and the gunner, korpr. Leppänen fired. The shot either missed or hit a spot where it didn't cause any serious damage. The T-34 then fired back, and it too missed. Leppänen got the next shot, hit, and knocked out the T-34.

So here we have one or two misses at 20 meters, which, according to theoretical accuracy models, shouldn't be possible at all.

Gross aiming errors like that are pretty difficult to model using any theoretical models.

Gross errors sometimes happened also with field artillery using indirect fire. I have read of one occurrence where the gun crew (it was a harassment mission with only one gun) mixed full and half powder charges, and the result was that the rounds fell in the pattern of the British artillery motto.

What if Tiger uses range estimation vs. 700m target and average error is 35%. Standard deviation for 35% avg. error is about 28.6%.

The range-finding equipment of the day had limited precision. I have some data on range-finders that were used by the Finnish army.

First, the gunsight of the 45 mm gun of the T-26 tank had granularity of 4 mils on horizontal axis and 2 mils on the vertical axis. The firing range had to be set on a separate "elevation drum" and it wasn't shown on the lens. This caused many aiming errors. Also, the sight was attached to the gun and it moved up and down so the gunner could end in very uncomfortable positions. The final flaw was that there was only one scale that had to be used both with AP and HE rounds, even though the weight of an AP round was only 2/3 of the weight of a HE round.

The 76.2 mm gun of T-34-76 had a better sight but it too was far from optimal. The horizontal granulation was again 4 mils and vertical 2 mils. However, it was possible to get one mil accuracy with it. There were separate scales for AP and HE rounds as well as for the MG. The range was again set using an elevation drum but this time it was shown on the lens, also.

The sight of the 85 mm gun was otherwise similar to the one of the 76.2 mm but it didn't have cross-hairs but the vertical hair extended only to the horizontal mil indicator. According to my source ("Punaiset panssarit"), this was almost a direct copy of the Pz-IV's sight.

I have also data on field artillery range-finders, but it isn't available just now.

- Tommi

[Jeff Duquette]

Firstly: interesting post TSS. Do you have access to schematics or photo through the T26 or T34 primary gunsight. Its always kind of interesting to see how different engineers laid the stadia lines out in there optics. What was sight magnification in T26 and T34 gunsights, and were they variable magnification in either case.

=======================================

Something I wanted to clarify in Rexfords original post…

<BLOCKQUOTE>quote:</font><HR>

Trajectory height at given range equals:

A x (distance to target)squared plus

B x (distance to target)

A = flight time to gun range setting divided by (C - D)

C = .4443squared times gun range setting raised to 2.0448 power

D = .4443 times gun range setting raised to 2.0224 power

B = A times -1 times gun range setting<HR></BLOCKQUOTE>

the A constant is actually equal to:

A = (Max trajectory height) divided by (CminusD)

Remember that (Max trajectory height) = 1.336 times (projectile flight time) raised to the 1.919 power. This was a source of confusion to me as the original A indicated "flight time to gun range setting".

A couple questions regarding the derivation of the Trajectory Height at given range (i.e. A x (distance to target)squared plus B x (distance to target). Does this assume an initial height of the firing tanks main gun…in the case of the Tiger I 88L56 example this may be 2 to 2.5 meters from ground level to centerline of barrel. As the equation is generic I would have to assume that initial main gun height would be some average value for all main gun heights. Or is your frame of reference zeroed along the axis of the firing tanks barrel? Frame of ref assumes y1 or y2?

The flight time vs range you included in the dispersion numbers. Where did you come by this information? Is it calculated or based upon field test data?

[rexford]

The equation for max trajectory height versus flight time matches actual data to within 0.1m, it has been checked against detailed German data and a sophisticated naval ballistic program in BASIC language. It is considered adequate for hit/miss estimation due to the small errors that result.

The trajectory height is taken above the line from gun barrel to target aim point, the dispersion is measured from the trajectory curve height.

If a Tiger is using battlesight aim and aims at the hull bottom on a T34, the trajectory height is taken from gun barrel to hull bottom.

Since most shots are at a good range, it might be okay to assume that the gun barrel is at the same elevation as the target point on the target, on level ground. This will be checked into.

[tss]

Jeff Duquette wrote:

Do you have access to schematics or photo through the T26 or T34 primary gunsight.

I have a drawn schematics. They are from the book "Punaiset panssarit" ("Red Armor") by Pekka Kantakoski. He commanded the Finnish Armored Batallion in 1974-5 and before that he was an instructor (1949-60), teaching conscripts to use T-26, Vickers 6 ton, T-34-76, T-34-85, Stug-IIIG, Pz-IVJ, T-54, BTR-50, and PT-76.

A little addition about the T-26 sight, the 2 mil scale for MG use.

There is no mention about magnification on gunsights. However, nearly all early-war Soviet tanks had one or two observation periscopes that had a fixed 2.5 magnification. This caused more problems than it solved, since it narrowed the field of vision and range estimation was very difficult.

In T-34-85 and IS tanks the tank commander had also a adjustable unmagnified mirror that could be used to look out of the hatch. It was a copy of British mirror system that was used in lend-lease tanks.

Kantakoski also mentions that the Soviet observation cupola was much worse than German because its slits were narrower.

I may go to the scanner and get some pictures of them. Though I think I wait until tomorrow and bring also the field artillery regulations manual and get its pictures, too.

- Tommi