Hit Probability, Rate of Fire Muzzle Velocity

[bruno_]

Generally speaking, Target Distance (D), Rate Of Fire (ROF) and Muzzle Velocity (MV) are known as primary factors for hitting a moving/maneuvering target . Again, generally speaking, it is well known that the higher the ROF and the MV and the lower the D, the better the Hit Probability (HP).

As to D, since bullets have a dispersion, we can assume (with some simplifications) that bullet paths are contained within a "cone" whose "height" is D and with a base proportional to the square of D. Therefore statements like "doubling D reduces the hit probability by four times etc." can be easily and intuitively understood.

But what about a more quantitative analysis about the influence of ROF and MV?

Looking for "guru's statements" we can find:

"To find out the effective relationship between any two weapons systems, simply square the ratio between their rates of fire. IE, a plane that fires twice as many bullets as another will be four times as effective, ALL OTHER THINGS BEING EQUAL!"

or

"The USAF made many studies and determined that the third power of the difference between any two muzzle velocities would show the difference in probability to hit. In W.W.II the Germans determined that the square of the differences in muzzle velocity was equal to the difference in hit probability. The Russians came to much the same conclusions…".

Unfortunately no simple/clear explanation is "attached" there. I guess this could be an interesting point to be discussed here.
Thus my question is: is there anybody that can help with some clue/insight/documentation/ about a more quantitative analysis relating HP, ROF and MV?

As usual, thanks in advance!

 

[davebender]

But they rarely are.

For instance, what if weapons are mounted on the centerline rather then in the wings? I suspect that has a greater impact on WWII fighter weapons accuracy then muzzle velocity.

 

[davebender]

HoHun created this chart during 2007 and it's still the best explanation I have seen for hit probability. As the chart shows, Me-109 and P-38 centerline mounted weapons have a much greater hit probability then P-47 wing mounted weapons. That holds true even though the Me-109s Mk108 3cm cannon had a relatively low muzzle velocity. Factor in the Mk108 cannon's huge destructive power (1 shell hit kills almost any fighter aircraft) and the Me-109G is one aircraft you don't want shooting at you.

 

[Shortround6]

Very nice, except that it does not show "hit probability" but rather an abstract "hit power". And it doesn't seem to show that very well. A single MK 103 may very well have about 2.4 times the "hit power" of even four .40 cal guns but this chart is showing over 4 1/2 times the hit power which is really pushing things.
And if you think that a single MK 103 has almost TEN times the hit probability of four .50 cal guns at 500 meters I have got a great deal on a Bridge in New York city you would might be interested in.

To keep this from being a German vs American argument lets try comparing the .50 cal gun with the American 37mm gun, very powerful with a 608 grams shell at 610ms but with a rate of fire of 140-150rpm (2.5 per second) it's hit probability was rather low unless very close to the target. I don't think any pilot that used it complained about lack of effect IF it hit.
Now it's probability of a single round round hitting may actually be not all that bad but considering that for most pilots the hit percentage was in the single digits and usually low single digits at that a 3 second burst of 7-8 shells the chances of even a single hit per burst is rather low. The guns low velocity compared to the .50cal also meant a shorter effective range. Time of flight counts for a lot in air combat, un-like ground combat. a 300mph airplane is moving 440fps. Long range shooting (for aircraft) introduces a lot of possible errors.
So does target aspect, shooting at a target from the 6 o'clock position vs shooting from the 3 or 9 o'clock position. The higher the velocity the less 'lead' is needed on the crossing target. the higher the rate of fire the more projectiles are in the stream as the target crosses it. Using our 300mph fighter target and shooting from the 3 o'clock position (90 degrees) without turning the firing aircraft A 1200rpm gun will put a bullet every 22 feet along the length of the target aircraft, a 900 rpm gun every 29.3 ft, a 600rpm gun every 44 ft and our 37mm gun every 176 ft. Granted this is an extreme case but shows why high rates of fire were important.
The importance of velocity is also shown by this. IF fire is opened at a distance equal to one seconds time of flight of the projectile the target moves 440 ft before the projectile gets there. If the target is moving at 270mph the distance is 3896 and if 330mph the distance covered is 484. Misjudging the targets speed by 10% means a complete miss on a fighter ( or even a small bomber), even at 2/3 sec flight time range hitting a fighter is chancy. The higher the velocity the greater the range for the same flight time or the less lead is needed, less lead means a wider speed variation will still be in the 'covered zone'. Considering that against a single engine fighter ( about 30 ft long) crossing at 90 degrees and doing 300mph you have to aim about 14.6 plane lengths ahead of it at a distance of 1 seconds flight time we can see both the difficulty and the actual futility of long range (600 meters and up) air to air combat in WW II for most of the war.

These aspects are probably more important than the "drop" of a projectile. ALL projectiles drop at the same speed. they all drop 16 feet in the first second of flight and they all drop 48ft in the second second. What varies is how far they travel in that time. Against all but the smallest targets the other problems in hitting out weigh the drop at any practical WW II ranges. In a P-38 the trajectory of the guns (which were angled up a bit) were never going to be more than 22 in away form the line sight from the nose of the plane till over 600yds away.
The .50 cal guns 880ms velocity offers a 40-50% greater "effective" range than the 37mm cannon has for defection shooting. Or a much better chance of hitting a crossing or off angle target at the same range.

 

[Sagittario64]

This is immensely helpful information from all of you. Theres so many factors in this weapons discussion too though, including:
1. Caliber
1a. brand of gun in said caliber (italian vs. american .50cal for example)
2. Ammunition used
3. Muzzle velocity
4. Range
5. rate of fire
6. trajectory
7. location on aircraft
8. reliability of said gun
9. possibilities of the target aircraft being armored or not; radial engined or not

 

[tyrodtom]

Something is very wrong with that graph, I think someone took the graph for 4 Mk108 cannons on a Me-262 and substituted it as if it was for one Mk108 on a Me -109.

One Mk108 round has 42,100 joules of muzzle energy, with the .50 round having 17,800 joules, you can't have nowhere that much of a mismatch in muzzle energy with multiple guns on the allied aircraft and their higher rates of fire too. And that chart doesn't even address hit probability at all.

 

[Juha]

Hello Dave
MK 108 had reliability problems, especially early on and especially when under G-loads, so in a fighter combat which incl. manoeuvring, one can argue that 109G-6/U4 (G-6 with MK 108 ) was the plane one hoped was firing at him under those circumstances. This is of course only a half truth, when working and hitting MK 108 was very powerful gun. Düttmann said that he had never problems with MK 108, but that was because he never fired one unless the 109 was under 1 G condition.

Juha

 

[Sagittario64]

Was the Mk108(or any other 30 or 37mm weapon) really suited towards fighter armament? if it has problems under G-forces as Juha says(in addition to the slow firing rate), then won't a fighter which is far more likely to be pulling G's than any other aircraft be better suited towards a different weapon? The MG 151/20 seemed to do fine when the fw.190 was equipped with 4 of these

 

[davebender]

A 3cm M-Geschoß shell has 72 to 85g of explosive filler. Compared to zero for a machinegun bullet. That's why the 3cm Mk108 cannon was such an aircraft killer.

 

[tyrodtom]

It's sort of strange that it's hit power goes from it's best to nothing in 100 meters, at 500 to 600 meters.

 

[kettbo]

Slow RoF for the Mk 108? Really? 650 rpm cyclic for 30mm was very respectable back in 1943! Think about it, that is 10.8 rounds per second. Care to run between the projectiles? Most of the other large cannon were slow-firing, the large Mk 103 shot at 400 rpm nominal. For air to air, I would call this low rate of fire. The Mk 108 is light weight, has a good RoF, delivers lethal effect.....great bang for the buck. Price for this was it was rather short ranged due to low velocity which lead to a more arched trajectory than most other weapons. But are you going to shoot at another fighter at 600 yards/575m? Usually not...get in close.

Forget all about the bench rest testing of the weapons...
The MOST IMPORTANT factor is the ability of the pilot of Fighter A to get his plane in position to hit Fighter B. Very difficult to do if Fighter B knows that Fighter A is present. Small wonder that most kills were against unsuspecting victims. Think about it. Most fighters carried adequate weaponry to shoot down other fighters; fighters have engines, fuel, some have lots of fuel, control surfaces, pilots seldom have more that back armor, inline planes had cooling systems, OMG they are all Aluminum and/or sometimes wood, all being complicated pieces of machinery, and they are flying! Lots of stuff that when hit can cause termination of controlled flight. Some weapon combinations are better than others for the killing, but the more difficult task is to get those first rounds on target

Twenty years with good ol Army weapons; 25mm cannon, Mk-19 40mm grenade machinegun, .50 HBMG, the Gunner Skill is all critical.

 

[davebender]

That range means very little as WWII era fighter aircraft typically opened fire at 200 meters.

 

[tyrodtom]

What i'm saying is the graph is suspect.

 

[Shortround6]

Slow RoF for the Mk 108? Really? 650 rpm cyclic for 30mm was very respectable back in 1943! Think about it, that is 10.8 rounds per second. Care to run between the projectiles?

650rpm isn't that hot for an entire airplane (OK the 109 also had a pair of MG 131s) a pair of MK 108s really ups the hit potential and the Me 262 with four is putting out 43-44 shells per second, running between the projectiles gets an awful lot harder. please remember that in that one second it takes to fire the 10.8 rounds a 300mph plane has moved 440ft. Now if the target plane is climbing, diving (even shallow) or turning the aim point for the end of the second and the beginning of the second are going to be a bit different.

Forget all about the bench rest testing of the weapons...
The MOST IMPORTANT factor is the ability of the pilot of Fighter A to get his plane in position to hit Fighter B. Very difficult to do if Fighter B knows that Fighter A is present. Small wonder that most kills were against unsuspecting victims. Think about it. Most fighters carried adequate weaponry to shoot down other fighters; fighters have engines, fuel, some have lots of fuel, control surfaces, pilots seldom have more that back armor, inline planes had cooling systems, OMG they are all Aluminum and/or sometimes wood, all being complicated pieces of machinery, and they are flying! Lots of stuff that when hit can cause termination of controlled flight. Some weapon combinations are better than others for the killing, but the more difficult task is to get those first rounds on target

Quite true, but multiple high velocity, high rate of fire weapons make getting getting hits early a lot easier. They often trade the killing power of a single large shell to do so.

 

[vanir]

MK108 and MG131 had to be set for vertical convergence or else they wouldn't sight properly. Because of the low relative-ME of the MK108 compared to the MG131 it had a much greater ballistic curve, so you only had two effective kill ranges where rounds passed upwards through the sight at 150m and back down through it at 450m. At about 300m or over 500m the rounds went either well below or well above the forward field, which were sighted for the MG131 to 600m or thereabouts.
So enemy a/c at 300m, you only got the MG131. a/c at 150m you hit him with everything. a/c at 450m you hit him with everything. a/c at 500-600m you only got MG131 again and over that you got nothing.

BMGs are like the 131 and the MK108 is like the Oldsmobile gun, that's all. The early series P-38 use the oldsmobile 37mm too, but the major production variants all had it swapped out in favour of the Hispano. I'm guessing because the Hispano has more complementary ballistics for better vertical convergence but I really wouldn't know why, I just read the initial P-38 was very much like the P-39 purpose built for the oldsmobile gun as far as the manufacturer was concerned, as an initial specification. As it turned out a lot of Hispano were used rather as a substitute.

I've hopefully, successfully attached the Rechlin ballistics handbook if anyone's interested. Note that tensile grades of plate steel or duralumin deflection for penetration testing vary and are specified (important note if cross referencing between gun/ammo and trying to correlate figures).

 

[vanir]

What i'm saying is the graph is suspect.

taken at face value it's just an individual's means of communicating an idea. It's clearly not hosted by a peer site which qualifies submissions based upon testable results of reproducible experimentation or primary source documentation. It's just a way for a guy to say what he was thinking, or figured out (and maybe quite right), in an illustrative fashion. I like the graph presentation sometimes, He also did a nice one exploring comparative high altitude performance between marqués of P-47 which I saved to disk it's so illustrative.

Of course it cannot account for historical accounts of pressure issues overspinning the impeller at 8000 metres in all the early series (or something to that effect, I'm not intimate with this plane), I think that didn't get solved 'till 44...and unsurprisingly this mirrors everybody else's experiences with high altitude piston fighters around the same time. The Germans couldn't even get a turbosupercharger working without an impeller grenade going off, they defaulted to a twin stage heavy driven unit that cost them how many hp I wonder. The British would never talk about the fact they were having the same problems with HF and recon Spits trying to roam at 40,000. They didn't.
It's very much a case of paper history and how the technology worked when you actually jump in it and do something very dangerous and very stupid in something built no better than a modern $300 lawnmower technology wise.

 

[Sagittario64]

30mm ammunition is bulky. the 109 could not even carry 100 rounds. at 10.8 rounds per second, a 65 round capacity would give you 7-8 seconds of firing if you let in little breaks. youd waste it

 

[bruno_]

I'd like to clarify that I know that Hit Probability is not equal to "kill probability". The overall result being of course given if one hits the target and each hit is as effective as possible. Thus only if all the fired bullet are the same and each of them is able to shot down the target whatever part is hit, then hit probability is equivalent to the kill probability.

That said, even if Hit probability is only a part of the whole picture, what I proposed here was a discussion about the relationship Hit Probability - ROF - MV. Of course everybody can bring here his feeling an enlarge the scope of my question but, in order to better clarify my point, please, consider that:

1) When comparing two weapons different just for ROF and or MV, all other things being equal (same mounting, same aircraft, same pilot, same target etc etc), the actual purpose of my question was again to understand if, doubling the ROF, the hit probability is doubled or squared or what else and thing like that. Same as for MV: doubling dhe MV will simply double the Hit Probability or will have an higher than linear effect? Just as for the Distance effect I mentioned in my first post of this thread.

2) Graphs like those shown in this thread (and elsewhere in this forum) are very nice and meaningful when discussing about the general problem of the gun effectiveness. They also tell many thing about hit probability but they cannot catch everything. This is why, I presume, the problem has four dimensions: the 3D of the space + the time. This is way, I presume, a burst occurs in a finite time so that the bullets aren't launched at the same time and, in the meanwhile, the attacking aircraft is moving so that each single bullet of any single weapon is shot at different times and at different points in the air, against a moving (and sometimes) manoeuvring target trying to avoid most of them.

Of course I don't want to bother people with "useless details" but, since it is apparent the air forces made a lot of studies about this subject and someone stated the Hit Probability is more than proportional to ROF and VM, and, beside many other factors gave guidelines in combat aircraft design and weapon development, I presume an explanation of these findings must exist and could be interesting to find out.

So please, forgetting all the many other factors determining the "kill", could someone kindly help me in working out such an explanation?

As usual, thanks in advance for any clue/document etc etc

 

[davebender]

One 30mm shell hit will fatally damage most single engine aircraft. I expect Me-109 pilots were taught to fire very short bursts except when attacking heavy bombers.

 

[Shortround6]

That said, even if Hit probability is only a part of the whole picture, what I proposed here was a discussion about the relationship Hit Probability - ROF - MV. Of course everybody can bring here his feeling an enlarge the scope of my question but, in order to better clarify my point, please, consider that:

1) When comparing two weapons different just for ROF and or MV, all other things being equal (same mounting, same aircraft, same pilot, same target etc etc), the actual purpose of my question was again to understand if, doubling the ROF, the hit probability is doubled or squared or what else and thing like that. Same as for MV: doubling dhe MV will simply double the Hit Probability or will have an higher than linear effect? Just as for the Distance effect I mentioned in my first post of this thread.

2) Graphs like those shown in this thread (and elsewhere in this forum) are very nice and meaningful when discussing about the general problem of the gun effectiveness. They also tell many thing about hit probability but they cannot catch everything. This is why, I presume, the problem has four dimensions: the 3D of the space + the time. This is way, I presume, a burst occurs in a finite time so that the bullets aren't launched at the same time and, in the meanwhile, the attacking aircraft is moving so that each single bullet of any single weapon is shot at different times and at different points in the air, against a moving (and sometimes) manoeuvring target trying to avoid most of them.

I would doubt that doubling the rate of fire would quadruple (square) the hit rate. Maybe somebody can come up with a mathematical explanation but why wouldn't an increase in rate of fire (wither by faster guns or more guns) simply increase the probability of hit in a linear fashion? 50% increase in rate of fire increases the hit probability by 50%, a 75% increase in rate of fire increases the hit probability by 75%, 100% increase in rate of fire increases the hit probability by 100% and so on? You are increasing the number of projectiles crossing the targets path by a given amount, why would each projectile have a higher chance of hitting?

MV does have a changing aspect to it. The US figured that an increase in velocity of 50% would increase the hit rate by 3 times. Increasing the MV decreases the time to target. Decreasing the time to target means range estimation is less critical, target speed estimation is less critical, target course estimation is less critical. The pilot (gunner) has a wider margin of error in estimating these factors before missing the target.

Close range (used by many/most aces) substitutes for high velocity, both give short flight times to the target. If someone is firing tracers at a moving target and the tracers have a flight time of 1 full second they are telling the "shooter" where he should have aiming ONE second AGO. At a 1/4 of a seconds "range" the tracers are somewhat more useful giving much faster feed back.