Why Is Bullet Weight Not a Factor in Trajectory?

[Mark85304]

I've been playing around with ballistics programs for decades and just realized that the trajectory does not vary with bullet weight for a given caliber. It does, however, vary with changes in starting velocity and BC. I have not noticed this until today and am perplexed as to why bullet weight does not affect a bullet's trajectory. Looking at resulting ballistic trajectories from various programs shows the point of impact at 200 yards with a rifle sighted in at 150 yards is within an inch of each other.

This is a great learning opportunity. I know some of the more seasoned loaders/casters will be able to explain this.

[runfiverun]

take your calculator out to 300 and 400 yds.

[375RUGER]

Looking at resulting ballistic trajectories from various programs shows the point of impact at 200 yards with a rifle sighted in at 150 yards is within an inch of each other.



So it does vary.

[Mark85304]

So it does vary.

The results vary between programs, but not when changing bullet weights within the same program. Within each of programs, a .30 cal 125 grain bullet and a .30 cal 180 grain bullet show the same bullet drop.

[newton]

He's right. Ballistics calculators I tried did the same thing. Not a change from a 100 grain object to a 500 grain object. I guess I have never noticed this before.

However, I think the issue is that all objects drop the same rate. I just tried it with a very small pepper(size of a pea or little smaller, probably weights 3 grains) and a 515 grain boolit I have.

They hit the ground at the same time. I think it has to do with terminal velocity.

[newton]

To add. The B.C. changes the way a boolit reacts to air. If the B.C. does not change, for example testing a feather with a different B.C. than a boolit, the objects fall at the same rate. Its very interesting. Its one of those complex, yet very simple, principals.

I think that what your seeing is that most boolits change their B.C. when the weight changes. But, there are other things associated with B.C. so its possible to have two boolits that weigh different, yet still have the same B.C.

In order for a lighter weight boolit to have the same B.C. a heavier one does it has to be more aerodynamic than the heavier one. That is one thing that a lot of people do not get, a heavier boolit (of the same design of a lighter boolit) will travel farther than the lighter boolit. It retains its velocity.

The smaller boolit has to reduce drag to make up for the inability to retain momentum.

Man! I sound smart! Trust me, I'm not!

[atr]

velocity is the primary factor .....the amount of time the projectile is in the air is the amount of time gravity gets to work on pulling the projectile down...the slower the projectile the more time gravity works hence more drop,,,yes shape of projectile (ballistic coefficient) etc. does have a slight influence mainy due to skin friction as the projectile pushes it way throught the air mass, but its really all about velocity.
high velocity=less time in flight=flat shooting=slight drop
low velocity=more time in flight=rainbow shooting=more drop
atr

[Elkins45]

This is a proof of a classic idea of physics: the mass of a projectile doesn't impact how fast it falls (subject to the effects of air, of course). As a practical matter heavier bullets drop more at longer ranges, but that's only because the extra mass can't be moved as fast given the same cartridge. I can probably make a 200 grain 30 cal drop less than a 130, but I have to fire the 200 from a magnum and the 130 from a 300 Savage or similar. If you could magically take the ability to accelerate them out of the equation you would find that heavy long bullets are actually a better choice for long range due to higher BC and greater retained momentum.

[prs]

The drop of a boolit over distance is chiefly influenced by TIME. For the greatest part, the drop when fired at any given time in the "flight" of the boolit is equal to that amount of drop in the same time frame as if the boolit were simply dropped while stationary. The speed at which an object falls is not directly influenced by its weight. However, we do not shoot in a vacuum or outer space, so given the two boolits are made of same material, the "heavier" one is also often larger in frontal area, thus faces more resistance from the air and drops SLOWER than the lighter boolit given time as the measure. BUT, given distance as the reference measure, the boolit facing more resistance will slow more quickly and thus will take longer to reach that same distance as its smaller sample and thus will fall further in that longer period of time it will take to cover said distance. Clear as mud? Thanks, Professor Newton.

prs

[MBTcustom]

That's why Galileo dropped two different mass balls off the leaning tower of Pisa in 1589. To demonstrate that objects fall at a constant rate.
However, just like it is an oversimplification to think of lead alloy as being composed of only three metals (lead, tin, and antimony) it is an over simplification to think of trajectory as being affected only by mass verses time. Other factors include BC (how easily the boolit slips through the air), humidity, air temperature, wind, and angle of trajectory. Heck extreme marksmen say that the direction of the earths rotation in relation to the direction of the shot even matters (called the Coriolis effect). For most purposes though, the strait simplistic calculation will get you darn close, because theoretically, a mass that is moving at a certain rate of speed will only drop so far in a given amount of time and distance, and its hard for an object to be moved too far from that baseline in less than 200 yards, which is how far most of our shots take place.

[David2011]

One of my favorite shooting subjects! It's all about the time of flight. The weight of the bullet is but a single factor in a complex equation. The mass/weight of the bullet affects its momentum retention but the BC, a measurement of drag, affects the speed retention. The shape of the bullet is actually a huge factor in drop. As speed diminishes, the drop increases. A short, fat bullet like a 150 grain .308 bullet will have a relatively low ballistic coefficient and will shed velocity more quickly than a long, slender bullet like the 142 grain .264 (6.5mm). Here's a chart comparing similar weight Sierra MatchKings, a 150 grain .308 (BC .397) and a 142 grain .264 (BC .580) both launched at 2700 fps. The "drop" figure is the distance a bullet would drop from the time it left the barrel if fired perfectly level from a high platform. Calculations are from handloads.com and corrected for my area at 3700' above sea level. Drift was calculated as a 90 degree crosswind at 10mph and is pretty interesting as well.

54882

This shows why the long distance shooters are going to cartridges like the 6.5mm Creedmoor.

David

[44man]

I cant explain some things, poor at the math but I see things that baffle me with revolvers.
I shoot a 330 gr .44 WLN at near 1350 fps, it drops 35" at 200 yards.
I shoot a 420 gr WFN at the same velocity from my .475 and it drops 18" to 20" or so.
You would think the larger diameter boolit with a bigger meplat would drop more but it is not the case.
Both are sighted at 75 yards at the start.
I can only say it is the weight difference. The BC of the WFN must be awful but it sure gets there so it must retain velocity better. I shot the 420 gr, WFN from the .475 and kept them on a 6" swinger at 400 yards. The holdover is not near as much as the 320 gr 45-70 boolt from my BFR at 1630 fps yet the 45-70 only drops 16" at 200, it does not retain it to 400 yards or 500 meters, (547 yards.)
When I shot long range rifle I never went by figures in books, I actually shot the distances. Nothing ever matched. I do the same with revolvers, I actually shoot and measure.
Calculations and programs are like cell phones to me, never to be understood! :veryconfu

[MBTcustom]

Another factor I didn't mention is barrel harmonics. If your barrel imparts a little upward english on the boolit as it exits, then it will effect the trajectory as well.

[Silvercreek Farmer]

Here's a fun one, drop a boolit and shoot a boolit perfectly horizontally at the same time and according to the laws of physics they will hit the ground at the same time. Simple principal but pretty dificult to execute, Mythbusters came pretty close to it

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=2&cad=rja&ved=0CDgQFjAB&url=http%3A%2F%2Fdsc.discovery.com%2Ftv-shows%2Fmythbusters%2Fvideos%2Fdropped-vs-fired-bullet.htm&ei=qMq3UOf0AsaY2wXzu4AI&usg=AFQjCNGjVyTnFDK51TYeiG1gRH9KP9omWg

[David2011]

I cant explain some things, poor at the math but I see things that baffle me with revolvers.
I shoot a 330 gr .44 WLN at near 1350 fps, it drops 35" at 200 yards.
I shoot a 420 gr WFN at the same velocity from my .475 and it drops 18" to 20" or so.
You would think the larger diameter boolit with a bigger meplat would drop more but it is not the case.
Both are sighted at 75 yards at the start.
I can only say it is the weight difference. The BC of the WFN must be awful but it sure gets there so it must retain velocity better. I shot the 420 gr, WFN from the .475 and kept them on a 6" swinger at 400 yards. The holdover is not near as much as the 320 gr 45-70 boolt from my BFR at 1630 fps yet the 45-70 only drops 16" at 200, it does not retain it to 400 yards or 500 meters, (547 yards.)
When I shot long range rifle I never went by figures in books, I actually shot the distances. Nothing ever matched. I do the same with revolvers, I actually shoot and measure.
Calculations and programs are like cell phones to me, never to be understood! :veryconfu

The time a boolit spends in the barrel is a different influence. When I first shot my Contender this was a big issue. It has a 14" .44 Mag barrel and a 2x scope. All rounds used the RCBS 250 grain KT that actually weighed 255 gr. I sighted it in at 50 yards with some .44 special loads at around 840 fps. The first magnum loads completely missed the target. At 50 yards the magnum loads, which were running right at 1600fps, were almost 2 feet LOW. The heavy gun was not reacting as much to the motion before the boolits left the barrel while the slower Special loads had twice as much time in the barrel, thus twice the time to cause the barrel to rise.

David

[huntincowboy]

Kind of a cool side note here, but if you were to fire a bullet from a perfectly rigid prfectly level rifle and somehow drop a bullet of the same mass the instant the fired bullet leaves the barrel, both bullets would hit the ground at the same time assuming that the ground was also level and the bullet was dropped in the same orientation as the fired bullet i.e. not nose down. Gravity affects all bodies the same assuming similar drag profiles. What makes impacts of bullets different in reality is that we cannot perfectly rigidly hold a rifle or even allow the same recoil effect between different bullet weights. Some shooters come dang close but even a minute elevation change from recoil, say .001" at the barrel would result in a little under a fifth of an inch rise at 100 yds. Very fun to think about!

[44man]

Another factor I didn't mention is barrel harmonics. If your barrel imparts a little upward english on the boolit as it exits, then it will effect the trajectory as well.
I agree it can have an affect, mostly with a rifle. Maybe with my revolvers too, good catch but it is recoil rise in the revolver before boolit exit.
To make it confusing, the 45-70 has far less recoil then my .44 yet drops half as much.
It is a mix of all those things, something not added to calculations that might assume no recoil or harmonics.
Each different gun, gun weight and barrel length in the same caliber will change everything.
I prefer bootstrap calculations---shoot!

[Doc Highwall]

If you take two bullets of the same caliber and same shape like a 30 caliber 150gr. RN and a 180gr.RN, the 180gr. will have a higher B.C. and drop less with the same muzzle velocity.

If you take 2 bullets with the same B.C. and start them out at the same muzzle velocity they will have the same trajectory.

[fishnbob]

:popcorn:

[felix]

All true. The first shot I do at the "riva" is at a small rock at the same distance as the genuine target. I move the scope hairs to exactly where the boolit struck. Trajectory is now sighted in for. Repeat for other targets at another location, be that location up or down in elevation and/or sideways from the original target. Deer hunters must mark the scope values for each location where a shot is to be taken. Then any deviation from spec is overridden on the day of the shot simulating any drastic change in distance and/or wind. ... felix

[fecmech]

Trajectory and wind drift are functions of BC and velocity, weight is immaterial. A 30 cal. 125 gr spitzer started at the same velocity as a 220 gr RN will not only drop less than the RN but also drift less in the wind.

[subsonic]

+1 fecmech

BC *is* the measure of drop, weight, aerodynamics, etc. BC can almost be called "rate of drop or drift". Weight for those programs is only for you to use to identify the bullet or boolit you're looking at. BC is the "factor" for them.

Heavier projectiles for caliber generally have better BC because they are more streamlined and have more momentum to retain velocity. Problem is, heavier projectiles can't be pushed to the same starting velocity from the same cartridge as a lighter projectile without exceeding pressure maximums. Thus the belted magnum was born....

[MBTcustom]

A projectile with a bigger BC does not drop any slower than the one with less BC. If fired from parrelell barrels at the same speed but differnet BCs, they would both hit the dirt at the same time, but the higher BC projectile would be further down range. The one with less BC would hit closer to the shooter at the exact same time, because it encountered more resistance.
In contrast, two projectiles of equal BC and speed but different mass (50BMG Vs. 30-06. assuming you could get a projectile for 30-06 with the same BC) would fly exactly the same, and projectile drop would be equal (the recoil however, is much different).

BC can almost be called "rate of drop or drift"
Not really, it drops no faster than the other. It merely takes longer to get to the target thereby printing lower.

[fredj338]

Bullet wt changes BC & vel, that changes POI. At 200yds, there isn't much diff w/ 300-400fps. Go out to beyond 300& things change quickly. "Flat shooting" is a relative term. if the furthest you shoot is 200yds, they are all pretty much the same from 2600-3400fps.

[Catshooter]

Isn't it true though that when the projectile gets around two or three pounds (or more) that inertia starts to come into play?

I don't know the BC of a 20 inch cannon bullet, and they certainly aren't going 10,000 fps, but they will sure go further than an '06.


Cat

[Mark85304]

He's right. Ballistics calculators I tried did the same thing. Not a change from a 100 grain object to a 500 grain object. I guess I have never noticed this before.

However, I think the issue is that all objects drop the same rate. I just tried it with a very small pepper(size of a pea or little smaller, probably weights 3 grains) and a 515 grain boolit I have.

They hit the ground at the same time. I think it has to do with terminal velocity.



To add. The B.C. changes the way a boolit reacts to air. If the B.C. does not change, for example testing a feather with a different B.C. than a boolit, the objects fall at the same rate. Its very interesting. Its one of those complex, yet very simple, principals.

I think that what your seeing is that most boolits change their B.C. when the weight changes. But, there are other things associated with B.C. so its possible to have two boolits that weigh different, yet still have the same B.C.

In order for a lighter weight boolit to have the same B.C. a heavier one does it has to be more aerodynamic than the heavier one. That is one thing that a lot of people do not get, a heavier boolit (of the same design of a lighter boolit) will travel farther than the lighter boolit. It retains its velocity.

The smaller boolit has to reduce drag to make up for the inability to retain momentum.

Man! I sound smart! Trust me, I'm not!


That's the absolute best explanation yet. I had momentarily forgotten that physics principle (my DUH moment) :oops: . It all makes sense now. I was confusing weight with BC.

Like I said in my original post, a great learning opportunity. I learned I need to identify the right thing. A quote from someone I don't remember who, "The main thing is to keep the main thing the main thing." In this case the main thing was not bullet weight and time of flight, but BC and time of flight as the factors affecting trajectory.

Thanks for the DUH moment guys.

[youngda9]

Kind of a cool side note here, but if you were to fire a bullet from a perfectly rigid prfectly level rifle and somehow drop a bullet of the same mass the instant the fired bullet leaves the barrel, both bullets would hit the ground at the same time assuming that the ground was also level and the bullet was dropped in the same orientation as the fired bullet i.e. not nose down. Gravity affects all bodies the same assuming similar drag profiles. What makes impacts of bullets different in reality is that we cannot perfectly rigidly hold a rifle or even allow the same recoil effect between different bullet weights. Some shooters come dang close but even a minute elevation change from recoil, say .001" at the barrel would result in a little under a fifth of an inch rise at 100 yds. Very fun to think about!

I've heard this a lot...but I always thought that if the bullet was fired level and stayed perpendicular to ground the entire length of the shot then it would be a true statement.

A bullet's flight is similar to a football, as it comes down it points down. The forward motion of the bullet pointed downward would seem to me to result in some downward acceleration on it's own. Equivalent to the wing on the back of a drag race car at a downward angle going through the air...it pushes down.

Or am I all wet ??

[PuppetZ]

The rate of drop of a bullet will be chiefly affected by gravity, which is a constant in theory. Any object will have the exact same acceleration under gravity if ALL parasitic effects, like drag of lift, are taken out of the equation. Bullets are small and massive enough to be little affected by drag. BC will have a lot of effect on the forward movement of a bullet but very little on it's downward movement. Why? The velocity component downward is minimal and thus is not affected by air resistance much. And since the amount of lift they will produce is very very small due to their rounded shape, their vertical movement will only slightly be affected by lift. There is a lot of confusion among shooters as to what exactly "drop" means. When we say a certain round or bullet has more drop than another, the difference come chiefly from difference in muzzle velocity. I'll try to explain it clearly. Gravity will accelerate any object by 9.2 m/s/s. That is, it add 9.2 m/s to it's velocity for every second the object is in freefall. Now to translate this to the bullet world, we'll take an idealised situation in which we eliminate any air resistance and other parasitic effect such as drag, lift, BC, ect... Once the bullet leave the barrel, it keeps on going with the same velocity in this "thought experiment". imagine a bullet fired @1000 fps. After a flight time of 1 second, the bullet will have dropped by 9.2 m and will be located 1000 feet from the muzzle. Now take another rifle with a muzzle velocity of 2000 fps. After a flight of 1 second, the bullet will have dropped by 9.2 m and be located 2000 feet from the muzzle. Now imagine we look at the location of the second bullet at 1000 feet. To get there, it will have flown 0.5 second and will have dropped only 4.6 m. We will say the second rifle drop half as fast but it's not technically true. Since the bullet is twice as fast, it reach the same point in half as much time and thus the bullet will be half as low. Mass of the projectile has nothing to do with the amount of the acceleration gravity will impart on it. 2 bullets fired at the same muzzle velocity but with different mass will have very similar trajectory.


I've heard this a lot...but I always thought that if the bullet was fired level and stayed perpendicular to ground the entire length of the shot then it would be a true statement.

A bullet's flight is similar to a football, as it comes down it points down. The forward motion of the bullet pointed downward would seem to me to result in some downward acceleration on it's own. Equivalent to the wing on the back of a drag race car at a downward angle going through the air...it pushes down.

Or am I all wet ??

Technically true but will only have a large effect on the steeper angle. The downward component of the total velocity vector of the bullet will still be mainly dominated by gravity and the effect of the angle of the bullet in regard to the ground can be dismissed as small disturbance as far as small arms bullets are concerned. Also consider that the velocity vector of a stable bullet come straight out of it's nose. It will not yaw in the airstream like a race car aileron.

[youngda9]

Technically true but will only have a large effect on the steeper angle. The downward component of the total velocity vector of the bullet will still be mainly dominated by gravity and the effect of the angle of the bullet in regard to the ground can be dismissed as small disturbance as far as small arms bullets are concerned.That's what I figured...but I always figured that it had some effect.

Thanks.

[PuppetZ]

I just want to add that when you do factor in "parasitic effects" bullet trajectory will vary wildly, mainly because of their shape and resultant BC difference. Similarly shaped bullets will fly similarly regardless of mass.

[williamwaco]

The bullet weight IS a factor.

Drop at any distance is determined by muzzle velocity and ballistic coefficient ( assuming the same angle of departure )
The weight is a very important factor in the calculation of the ballistic coefficient.

If you want to know more, check out the bullet's sectional density and how the sectional density affects the ballistics coefficient.

I have a degree in math and another in computer science and this math is beyond my capabilities so I am not going to attempt to explain it.

Visualize it like this. The drag on the bullet is related to the cross sectional area pushing through the atmosphere. The greater the area, the greater the drag that slows the bullet.

Since there is no force keeping the bullet going forward, the only thing that keeps it going is it's momentum. Momentum is directly related to the bullets mass. A 200 grain bullet will have twice as much momentum at any velocity as a 100 grain bullet and will loose velocity slower thus arriving at the target sooner.

Imagine it like this: Assume you have to arrows. Both are exactly the same in every detail except length. ( Implies the longer arrow is heaver. ) Both leave the bow at say 300 fps perfectly parallel to the ground which is flat as Area 51. One arrow is 36" in length and the other is 12" in length. Which one will make it to 50 yards first? Which one will have the higher velocity at 50 yards?

Which one will hit the ground first? ( Trick question )

[newton]

I like topics like these because they are a learning opportunity indeed. I did think of something further.

The only, and entire, reason the weight is an input into calculators is simply for energy. Nothing more. All the other numbers are not effected by that input. In fact, you can leave out the grain weight all together if the calculator will allow. I have found some calculators online that do allow you to leave it out and trajectory is not effected at all with a zero grain weight.

But that's a secondary point to the original question.

[Larry Gibson]

At a given velocity bullets of equal BC will have the same trajectory regardless of bullet weight; simple as that.

Larry Gibson

[fecmech]

Since there is no force keeping the bullet going forward, the only thing that keeps it going is it's momentum. Momentum is directly related to the bullets mass. A 200 grain bullet will have twice as much momentum at any velocity as a 100 grain bullet and will loose velocity slower thus arriving at the target sooner.
In a vacuum you might be right, in the air not so. In air there is such a thing as drag force and the drag force on a 220 gr RN .30 cal bullet is greater than the drag force on a 125 gr spitzer due to it's lower BC. Therefore the 125 gr. will have a flatter trajectory and arrive at the target sooner with less wind drift than the 220 gr RN and also at a higher velocity because of it's higher BC. All of this assumes they started at the same muzzle velocity.

[HangFireW8]

The bullet weight IS a factor.


Yes, of course it is, as weight (or more precisely, mass), figures directly into the BC of the projectile.

I suspect the OP is using software that keeps the BC constant while changing the weight of the projectile, which would not be very meaningful.

There are important things to remember when playing with projectile dynamics:
1. Ballistic Coefficient (BC) as used in ballistics is not really calculated, it is partly experimentally derived.
2. BC is not properly a single number. It actually is (or should be) on a sliding scale depending on velocity, with a discontinuity at the transonic transition point and increasing degredation somewhere above Mach 2. Some of the later references deal with this by having two BC's, one "above" a given velocity and another for "below".
3. Despite the above two points, good ballistics software gets "close enough" such that small variations in environment become more important than the calculation error.

My gripe with most modern exterior ballistics software is not the BC & trajectory, but the windage calculations. Someone wrote a good article on this with something in the title called "the big lie" or somesuch; wish I could find it. Until someone invents a device the cheaply measures wind across an entire trajectory, software will probably continue to get this wrong. It is rather straightforward to measure trajectory, anyone can do it, and write a letter to the programmers about their errors; witness the older Sierra handbooks and the later admission of how they got so much wrong. Setting up enough wind meters and catching their data in real time, at the correct altitude to match the trajectory, that's a lot more challenging, and expensive, most have not bothered to do it.

HF

[geargnasher]

"Weight" is actually a measure of Force, represented by the simple algebraic function of the variable Mass multiplied by the constant Gravitational Acceleration, which is 9.8 meters per second, per second, near sea level on earth. That's why the weight of something is a variable depending on relative position, and environment (for example, floating in salt water vs. air, or free-falling) and the mass of an object is constant no matter what.

When figuring a boolit's drop, you aren't figuring Force (weight), you're figuring distance over time in one direction (down), which involves the variable Time and the constant Acceleration, where the constant is "constant" in both direction and rate no matter any "sideways" movement. If you remove the factors of air density and curavature of the earth, a horizontally fired boolit and a dropped boolit will hit the ground at exactly the same time, no matter whether you drop a cannon ball and fire a .22 or vice-versa, because the downward acceleration is the same for any object in equal time intervals, and neither mass nor weight is part of the time/acceleration equation

The factors that affect this (air density, BC, trajectory angle, etc.) have been covered pretty well. There are some other things that affect those corrections, too, which are sometimes overlooked, such as the fact that the BC of a boolit as loaded into a gun can be significantly different after being fired out of the muzzle. Even jacketed bullets in some instances have been shown to change shape very significantly from launch forces, which explains a lot of the discrepancies between calculated and actual data. Also, as Goodsteel pointed out, barrel harmonics can make a huge difference in actual trajectory, as can barrel time.

Gear

[Gibson]

Such erudition from gun nuts. . .

:)

One of the redeeming features of this place.

Carry on.

[fcvan]

Ok, so we have talked coefficients, mass, trajectory, etc. I just want to know . . .

What is the airspeed velocity of an unladen swallow?

[PuppetZ]

500 ias

[Gibson]

Ok, so we have talked coefficients, mass, trajectory, etc. I just want to know . . .

What is the airspeed velocity of an unladen swallow?

If we're talking European, here, 11 m/s :)

[JLDickmon]

Ok, so we have talked coefficients, mass, trajectory, etc. I just want to know . . .

What is the airspeed velocity of an unladen swallow?
What do you mean? African or European swallow?

[JLDickmon]

This is a variant on DaVinci's experiment. He dropped two cannonballs off the Tower of Pisa. One ball weight two pounds, the other six.
Both balls hit the ground at the same time.

At some point in your 8th grade physics class, you should have put a feather and a steelie in a vacuum tube and drew it down to 29.5" and then flipped the tube over... both objects would have hit the other end at the same time. Gravity is what gravity is.

[44man]

All true. The first shot I do at the "riva" is at a small rock at the same distance as the genuine target. I move the scope hairs to exactly where the boolit struck. Trajectory is now sighted in for. Repeat for other targets at another location, be that location up or down in elevation and/or sideways from the original target. Deer hunters must mark the scope values for each location where a shot is to be taken. Then any deviation from spec is overridden on the day of the shot simulating any drastic change in distance and/or wind. ... felix
I do that too when sighting but my problem has always been to keep the gun still while a friend or I turns the dials without moving the stinking gun. Need a vise on the bench!
I get so disgusted with scopes and red dots today that don't move POI the right amounts so using a ruler on a target can dial in a gun. In the old days, even a Weaver would be exactly 1/4" at 100, let alone a B&L.
Now you need a $1500 scope.

[popper]

Read the source code for the ballistic calc. and you will find weight (mass) IS part of the equation. The result from a difference in mass is not exactly what our minds would expect.

[Doc Highwall]

Here is a book that is very well written and I would suggest you get a copy.
The 2nd edition is out.

http://www.accurateshooter.com/ballistics/applied-ballistics-for-long-range-shooting/

[Bob Krack]

Isn't it true though that when the projectile gets around two or three pounds (or more) that inertia starts to come into play?

I don't know the BC of a 20 inch cannon bullet, and they certainly aren't going 10,000 fps, but they will sure go further than an '06.

Good implied question. Seems to me the naval guns in the 16" range had muzzle velocities of around 3,000 FPS (probably wrong) but I am certain it was NOT 10,000 FPS. So how did we get 10-20 mile ranges from them?

Bob

[MBTcustom]

So how did we get 10-20 mile ranges from them?

Aim higher? The BC of a projectile that large would be impressive, so you would not have the problems with air resistance that small arms have.
That's my SWAG.

[PuppetZ]

Much higher mass to surface ratio. The surface of an object increase by the square. Volume by the cube. So for a big projectile, air resistance from it's higher surface is more than compensated by it's increased mass and therefore inertia.

A 1 inch ball has a surface area of 4.188 square inch and a volume of 12.566 cubic inch for a mass/surface ratio of 3:1. a 16 inch round ball has a surface area of 3216.99 square inch for a volume of 17157.3 cubic inch for a mass surface ratio of 5.33:1. The larger ball generate less drag for it's weight.

[fecmech]

I have been unable to find the BC for a 16" naval round but I do know that it is over 1 by quite a bit and velocity is in the 2400-3000 fps range. Max range for the 2700lb. projectile is 42,345 yds(about 24 miles!)fired at an angle of 45 deg.

[Willbird]

What is really confusing is that drop/wind drift is not purely about time of flight. To illustrate this take one bullet which weights 200 grains and has a BC of .250. Take another bullet which weighs 200 grains and has a BC of .50. For simplicity lets look at 500 and 1000 yards with a 100 yard zero, 10mph wind.

200 grains, .25 BC, velocity 1335fps
500 yards, velocity 838.7, traj -315', drift 66.1", flight time 1.493 seconds
1000 yards, velocity 647.7, traj -1878.5, drift 231.4", flight time 3.550 seconds

200 grains, .500 BC, velocity 1000fps
500 yards, velocity 845.6, traj -397.2', drift 25.3", flight time 1.642 seconds
1000 yards, velocity 741.6, traj -2014.4, drift 97.3", flight time 3.549 seconds

If we fired both rifles at the exact same instant, the bullets would both arrive at the exact same time, both exposed to the exact same time of flight, but the 1000 fps bullet would only drift roughly 1/3 as much in the wind as the 1335 fps bullet which had a lower BC. Also the velocities are fairly close at 500 yards, and again drop and wind drift are different.....and again the wind drift on the slower velocity higher BC bullet is a lot less.

[prs]

Well, if I take all of this into consideration, I will be bear poop before I can get the trigger pulled...


prs

[HangFireW8]

Yes, it is well known that projectiles that decelerate quickly, have greater wind deflection than those that don't.

I have to chuckle about all this, I took a deer last week, the wind was blowing at 20MPH. The two shots (both to the chest) were at ~ 20 ft and 15 feet. I didn't have to worry about windage!

[popper]

So how did we get 10-20 mile ranges from them? 4-5 bags of powder! 8" used 3 bags.

[Tom Myers]

I have been unable to find the BC for a 16" naval round but I do know that it is over 1 by quite a bit and velocity is in the 2400-3000 fps range. Max range for the 2700lb. projectile is 42,345 yds(about 24 miles!)fired at an angle of 45 deg.

The G1 Ballistics Coefficient was based on the Gavre resistance tables which were derived from a small artillery round weighing 1 pound, with a diameter of 1 inch and a pointed nose ogive that had a curve with a radius of 2 projectile diameters (2 inches). The ballistic coefficient of this round for the G1 ballistic resistance table is " 1 ".

54958

The ballistic coefficient of a projectile with the same nose configuration changes in direct proportion to the weight of the projectile.
If your 2,700 lb artillery round has a nose profile with an ogive of 2 calibers then it has the same form factor of the G1 projectile and a ballistic coefficient of 2,700. That is why it will shoot to such a great distance with a velocity that is nearly the same as a rifle projectile with a ballistic coefficient of 0.400 or there abouts.

Here is some more information concerning drag tables (http://www.tmtpages.com/Ballistics_Ver-4_Help/hs13.htm)

More Information concerning Ballistic Coefficients (http://www.tmtpages.com/Ballistics_Ver-4_Help/hlp_bx_cx.htm)

[Mark85304]

I like topics like these because they are a learning opportunity indeed. I did think of something further.

The only, and entire, reason the weight is an input into calculators is simply for energy. Nothing more. All the other numbers are not effected by that input. In fact, you can leave out the grain weight all together if the calculator will allow. I have found some calculators online that do allow you to leave it out and trajectory is not effected at all with a zero grain weight.

But that's a secondary point to the original question.

Yep, Strelok is one of the programs that does not allow bullet weight as an input, and incidentally the one that caused me to make this inquiry. I see thing more clearly now and yes, bullet weight is used in ballistic calculations for retained energy calculations as well as recoil calculations.

I stumbled on Strelok as I was looking for a ballistic calculator for my Android phone and iPad for field work. It is not always possible to tale a laptop into the field, but I always have my phone and have my iPad most of the time.

[newton]

Yep, Strelok is one of the programs that does not allow bullet weight as an input, and incidentally the one that caused me to make this inquiry. I see thing more clearly now and yes, bullet weight is used in ballistic calculations for retained energy calculations as well as recoil calculations.

I stumbled on Strelok as I was looking for a ballistic calculator for my Android phone and iPad for field work. It is not always possible to tale a laptop into the field, but I always have my phone and have my iPad most of the time.

I forgot about recoil. The calculators I use do not have that. I wish I could find a good, yet cheap, app for my iPhone. I have not looked in a while though.

[fecmech]

Wow! A B.C. of 2700. Thanks Tom

[MBTcustom]

Recoil is totally separate and based more on personal opinion than science. I don't see that it has anything to do with what we are talking about here, does it?

[popper]

Good going Tom. Saw the Jean Bart (french BS) years ago. 2 turrets of 4 x 15"x 60', 2700fps, 1950# shell(BHN 555), with bbl steel ~ 7" thick. 4 bags for 640# charge. 450K psi. chamber pressure. Recoil 52". Bbl life of 200 rnds. It wasn't a very good gun(french design). And we think we have problems.
reason the weight is an input into calculators is simply for energy True,and that energy comes from internal ballistics, as VELOCITY. As far as our calculators are concerned, weight is NOT needed except to calculate terminal energy, but it is in there and DOES effect trajectory.

[Mal Paso]

A 1 inch ball has a surface area of 4.188 square inch and a volume of 12.566 cubic inch for a mass/surface ratio of 3:1. a 16 inch round ball has a surface area of 3216.99 square inch for a volume of 17157.3 cubic inch for a mass surface ratio of 5.33:1. The larger ball generate less drag for it's weight.

Both volumes are incorrect unless the inside is bigger than the outside.

1" sphere Volume .522" Surface 3.14"

16" Sphere Volume 2138" Surface 803"

So ya except for the math.

[PuppetZ]

Silly me, I meant a 1 inch diameter ball but inputed it as radius in the formula.... so the numbers would be for a 2 inch and a 32 inch sphere. Good catch!

[geargnasher]

Good implied question. Seems to me the naval guns in the 16" range had muzzle velocities of around 3,000 FPS (probably wrong) but I am certain it was NOT 10,000 FPS. So how did we get 10-20 mile ranges from them?

Bob

Simple. F=Ma, and a LOT more F is applied to a LOT more M to get about the same "a" in the big guns compared to a sporting arm projectile, while the resistance of air is the same. Then there's the lower M/BC relationship as was mentioned above (remember, it's all relative), so the result is much more INERTIA and lower BC for the same air resistance and thus a much longer range. Don't forget about trajectory angle differences, either.

When talking drop, it will be the same rate as a .22 rimfire if fired on the level, but that is in the DOWN direction. The HORIZONTAL direction is totally different, and that's where the various forces, resistances, inertia, BC, trajectory, etc. come into play.

Gear

[sthwestvictoria]

This is a variant on DaVinci's experiment. He dropped two cannonballs off the Tower of Pisa.


The pedant in me would offer that this was Galileo

[JLDickmon]

The pedant in me would offer that this was Galileo

could be. I wasn't there. :coffee:

[L Erie Caster]

Actually the ballistic coefficient has no effect on the rate a bullet drops toward earth, only on the rate at which a bullet decelerates. Thus a bullet with a high BC will have more time to fall before it reaches a distant target. Than one with a lower BC, if fired at the same mussel velocity, regardless of the bullet‘s weight. And a heavier bullet fired at the same velocity with the same BC will hit the target at the same place as a lighter bullet, everything else being equal.

[sthwestvictoria]

Thus a bullet with a high BC will have more time to fall before it reaches a distant target.

I disagree. If you shoot a very low BC boolit horizontally and drop a very high BC bullet from near the muzzle at the same time as firing, they will both hit the ground (have the same amount of time to fall) at the same time.

Likewise, firing a low BC and high BC boolit will result in the same time in the air. What will change is the horizontal distance traveled.

[MBTcustom]

I disagree. If you shoot a very low BC boolit horizontally and drop a very high BC bullet from near the muzzle at the same time as firing, they will both hit the ground (have the same amount of time to fall) at the same time.

Likewise, firing a low BC and high BC boolit will result in the same time in the air. What will change is the horizontal distance traveled.

I completely agree with the way you put it!
However, I think L Erie Caster was saying the same thing.
The way I see it is this:
You go to the range and set up a target at 600 yards. You fire two projectiles, one with a high BC and one with a low BC. The high BC prints the target higher than the low BC does even though they left the barrel at the same speed.
Why?
Simply because the higher BC projectile got there faster. The low BC projectile had to have coffee with wind resistance along the way to the target. Its the turtle and the hare thing.
The higher BC is not as effected by wind resistance, therefore it arrives at the target sooner, with more retained energy.
So Galileo rolls the two balls off the leaning tower of Pisa, and they both hit the ground at the same time. All that proves is that gravity has the same effect on both massive, and less massive objects. This has nothing to do with wind resistance.
How do you think it would have turned out of Galileo had a parachute attached to one of the balls?
Well of course it will hit the dirt a little later.
Likewise, a projectile that has a less aerodynamic shape, in a small way, is like having a parachute attached that slows it down. Surely most of you have driven in a pickup truck and a small car? You can literally feel the increased drag on the cab of the pickup truck as apposed to the small aerodynamic car.
Any of you fellers ever watched The Dukes Of Hazard? Ya know the creek that they are all the time jumping over with Rosco on their tail? Well what if they had a full load of moonshine? would it matter when it came time to jump the creek? Hmmm, only if they weren't able to get the car up to the same speed as they did before. If they jumped empty doing 75MPH then they can jump loaded if they can get the car up to 75MPH.
Now, let's attach a parachute to the back of the General Lee. Lets say that with the parachute, the General Lee would slam into the opposite bank 6 feet lower than it did without the chute or the load. Now here's the cool part: If the General was loaded and had the chute attached, it would hit the oposite bank higher (let's say 4 feet low) than the empty car did. The reason for this is that it has mass that will help drag the chute.
So, we apply this to projectile theory.
We go to the range with 10 rounds each of 308 winchester loaded with 168 Sierra MKs, 200 Sierra MKs, and 168 grain flat nosed gas checked cast lead boolits, and 200 grain flat nosed cast lead boolits.
All of these have been loaded to shoot at the exact same speed.
The 200 grain Sierra MK will be the highest group.
Then the 168 grain Sierra MKs
Then the 200 grain FNGC (probably right on top of the 168 Sierra MKs)
and finally the 168 grain FNGC.
All of them were fired at the same speed, but the 200 grain Sierra MK had both mass and BC on its side, therefore it printed highest.(this is like the General Lee with a full load doing 75MPH)
Next the 168 MK, because even though it has less mass, its BC is very good for its mass. Slightly better than the 200 grain FNGC.(like the General Lee empty going 75MPH)
The 200 grain FNGC has mass, but lousy BC for its size, therefore it will be very close to the 168MK. (this is like the General Lee loaded with moonshine and with a parachute on its bumper.)
The 168 FNGC has neither mass, nor BC on its side so it will be effected the most by the wind resistance. (This is like the General Lee, unloaded, with a parachute attached to its bumper.)

Ie, 200 grain Sierra MK = YeeeeeHaaawwww!!!!!
but
168 grain FNGC = Yeeeeeeee(crunch!!!!)

[prs]

Way back early in this thread I gave credit to Newton, not the fig cookie; but rather the Sir Issiac dude. Check out his Second Law of Gravity here:

http://sciencedemonstrations.fas.harvard.edu/icb/icb.do?keyword=k16940&pageid=icb.page80669&pageContentId=icb.pagecontent277503&state=maximize&view=view.do&viewParam_name=indepth.html

[geargnasher]

Galileo did the feather and hammer drop thing. It was David Scott who proved him irrefutably right several hundred years later at Hadley Rille.

Gear

[10x]

Gravity is a constant. A bullet starts to fall the instant it leaves the muzzle. A bullet shot with the barrel horizontal will fall 16 feet in the first second of flight no mater what weight it is.
The apparent "flatness" of flight is a result of the distance the bullet travels in that second, and how fast the bullet sheds speed due to air resistance.
Drop is a function of time of flight from the muzzle to the target. D=1/2G(TxT) where G=32, D is drop an T is the time in seconds.

[sthwestvictoria]

The elements fo trajectory are average velocity throughout bullet travel, bullet mass, and bullet configuration. Any projectile tends to hold its velocity in proportion to the square of its mass, something which I never understood until I began shooting projectiles much larger than those of sporting rifles. For example, I first assumed that the old 37mm antitank gun would shoot about like a 270, since its initial velocity was supposed to be the same. To my amazement that little 37 shot flat "way out past Fort Mudge" with minimal initial elevation. That was because its projectile wieghed 2.5 pounds, rather than 130 grains.

Jeff Coopers Art of the Rifle, page 44

Can anyone tell me what this principle or rule of thumb is called? Obviously the important part of the statement is hold its velocity, as Goodsteel pointed out so neatly with moon-shining car jumping escapades.
My google fu was not powerful enough to answer this.

[sthwestvictoria]

I would also like to thank Mark85304 for starting this terrifically interesting thread.

If you do not listen to Michael Bane's podcast Downrange Radio then I heartily endorse it. This weeks episode is about mastery and in it he states that if you want to learn a new skill you need be humble and put on the white belt. He also states it in the way - if your cup is full, how can you learn more?

Although I may get above my white belt in casting I hope I still find everything interesting.

[MBTcustom]

Isn't that Newtons second law? The third law is recoil.
I quote:
First law: If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).[2][3][4]

Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma.

Third law: When a first body exerts a force F1 on a second body, the second body simultaneously exerts a force F2 = −F1 on the first body. This means that F1 and F2 are equal in magnitude and opposite in direction.

I really like the third law, because it explains recoil to a small degree. It all comes down to the mass of the rifle vs. the mass of the projectile and how much inertia is imparted to blow the two away from each other. What makes it hard to calculate, is that there is a second reaction happening between the shooter's shoulder and the stock of the rifle. That's why little fellers lose their fillings when they try to shoot guns like Gibson's, heh heh. A mans chest is a shock absorber for all that inertia that is generated by the separation of the projectile from the rifle.
but I digress.

[10x]

Trajectory and wind drift are functions of BC and velocity, weight is immaterial. A 30 cal. 125 gr spitzer started at the same velocity as a 220 gr RN will not only drop less than the RN but also drift less in the wind.

In an equal amount of time (ie. one second time of flight) the distance both bullets travel over that time is different - the total drop and total wind drift over that time will be the same for both bullets.

[10x]

I disagree. If you shoot a very low BC boolit horizontally and drop a very high BC bullet from near the muzzle at the same time as firing, they will both hit the ground (have the same amount of time to fall) at the same time.

Likewise, firing a low BC and high BC boolit will result in the same time in the air. What will change is the horizontal distance traveled.

BC ( ballistic coefficient) is a variable that changes slightly as the velocity of a bullet changes. (near and under the speed of sound it changes dramatically).
A bullet with a BC of 1 has a measured standard trajectory, one that was established using artillery firing a bullet and actually measuring the drop and velocity at various distances. This is how the British 1909 ballistic tables were established ( a bullet of 1 inch in diameter weighing 1 pound and having an ogival head of 8 calibers radius [caliber being 1 inch the radius of the ogive will be 8 inches] - this bullet has a BC of 1 for the 1909 tables).
BC changes with air density and can be regarded as the efficiency of a bullet compared to a "standard bullet" .

BTW: Boat tail bullets only become really efficient below the speed of sound. Boat tail bullets were developed prior to WWII when the machine gun was sometimes used in a manner like artillery at very long ranges.

[44man]

Isn't that Newtons second law? The third law is recoil.
I quote:
First law: If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).[2][3][4]

Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma.

Third law: When a first body exerts a force F1 on a second body, the second body simultaneously exerts a force F2 = −F1 on the first body. This means that F1 and F2 are equal in magnitude and opposite in direction.

I really like the third law, because it explains recoil to a small degree. It all comes down to the mass of the rifle vs. the mass of the projectile and how much inertia is imparted to blow the two away from each other. What makes it hard to calculate, is that there is a second reaction happening between the shooter's shoulder and the stock of the rifle. That's why little fellers lose their fillings when they try to shoot guns like Gibson's, heh heh. A mans chest is a shock absorber for all that inertia that is generated by the separation of the projectile from the rifle.
but I digress.
This I like! How much velocity of the boolit is lost from the gun moving in the opposite direction?
How much POI change will you see with a hard kicking rifle shot off hand compared to one shot from a lead sled?
So much knowledge shown in these posts, stuff I no longer remember, stuff I never learned too. My hat is off to all of you.
I still like simple and actual shooting. It is as simple as holding a revolver exactly the same or you will crease a deer across the chest or top of the back.
It is still your gun in the end. Short barrels have shorter barrel time but more barrel rise, same with light and heavy guns that change it.
We see things that math and figures do not help because of the human element.
Even the bullet drop at firing reaching the ground at the same time as the one fired does not work unless fired from a fixed barrel. You would need to take recoil and barrel rise into account and release the dropped bullet at the exact time as bullet exit which means you need to drop the bullet at a higher height.
Shooting light bullets to heavy bullets at the same velocity also changes recoil and rise with the heavy bullets hitting higher. Only a fixed barrel will give readings for BC. If you think you can shoot your rifle from bags to get actual BC readings for a bullet, you might be wrong. Recoil is beating you out of figures.
Us dumb fellas that do not own slide rules really do see things. We live with Newton's third law.

[fouronesix]

Galileo did the feather and hammer drop thing. It was David Scott who proved him irrefutably right several hundred years later at Hadley Rille.

Gear

Gear,
Spot on! So few remember much about the Apollo missions much less know the difference between energy, inertia and momentum or a clue about f=ma or an inkling of the gravitational constant.

[44man]

Gear,
Spot on! So few remember much about the Apollo missions much less know the difference between energy, inertia and momentum or a clue about f=ma or an inkling of the gravitational constant.
It seems to me it was proved in a vacuum, no air resistance. Not what we live with either. The constant does not change but outside influence will. Like air on the feather. Flight of a bird is not possible in a vacuum.
A bullet fired in space will go straight forever unless a body with gravity bends the path.
None of this pertains to our boolit flight.
We need to fold space and jump. We need anti gravity.
Will a boolit shooter find it so we can jump to CA and seed the state with lead? :bigsmyl2::bigsmyl2:

[fouronesix]

It seems to me it was proved in a vacuum, no air resistance. Not what we live with either. The constant does not change but outside influence will. Like air on the feather. Flight of a bird is not possible in a vacuum.
A bullet fired in space will go straight forever unless a body with gravity bends the path.
None of this pertains to our boolit flight.
We need to fold space and jump. We need anti gravity.
Will a boolit shooter find it so we can jump to CA and seed the state with lead? :bigsmyl2::bigsmyl2:

All that is correct except, it all pertains to bullet flight! Gravity acceleration is a constant on the Earth at about 32ft/sec/sec. If you take a 100 grain 30 cal bullet and drop it from a set height, then take a 200 grain 30 cal bullet and drop it from the same height. The difference in air resistance between the two will be so near zero that it is of no consequence- so the time for each to drop the distance will be the same. Since the gravity acceleration is a constant for both bullet weights and the air resistance differential between the two is not significant as it pertains to the gravity acceleration constant, the ONLY variables, within the gross parameters being discussed here, that matter for ballistic or fight trajectory prediction are the velocity and the ballistic coefficient.

[MBTcustom]

In conclusion......let me repeat myself.:kidding:

[fecmech]

In an equal amount of time (ie. one second time of flight) the distance both bullets travel over that time is different - the total drop and total wind drift over that time will be the same for both bullets.
In the real world we shoot at targets or game at a given distance and equal time does not come into play. As long as my bullets (the .30 125 gr spitzer vs the 220 gr RN) are started at the same velocity and can reach the target the 125 gr will have less drop and less wind drift.

[10x]

In the real world we shoot at targets or game at a given distance and equal time does not come into play. As long as my bullets (the .30 125 gr spitzer vs the 220 gr RN) are started at the same velocity and can reach the target the 125 gr will have less drop and less wind drift.

Can you tell us why the 125 will have less drop and why it will have less wind drift?

[MBTcustom]

In the real world we shoot at targets or game at a given distance and equal time does not come into play. As long as my bullets (the .30 125 gr spitzer vs the 220 gr RN) are started at the same velocity and can reach the target the 125 gr will have less drop and less wind drift.

I don't get that at all. More massive projectiles buck the wind better, and that is a fact.
It is possible that the 125 has a better BC than the 220RN, but if you fired them at exactly the same speed, they should be very similar. Of course, like we said before, there are a lot of variables that come into play here. It could be that your barrel harmonics throw the 125 higher than the 220.

[fecmech]

Can you tell us why the 125 will have less drop and why it will have less wind drift?
My statements were based on my old Sierra manual ballistics table which had the 125 gr spitzer with a BC of .340 and the 220 RN with a BC of .255. With those bullets my statements were correct according to Sierra's ballistic tables. Before answering your question I went to Sierra's web site and discovered their 220 RN of today has a BC of .335 and their 125 a BC of .338 so using those bullets drop and drift would be the same for both a 125 spitzer and 220 RN. Sierra has evidently changed the design of their 220 .30 cal RN.

[10x]

My statements were based on my old Sierra manual ballistics table which had the 125 gr spitzer with a BC of .340 and the 220 RN with a BC of .255. With those bullets my statements were correct according to Sierra's ballistic tables. Before answering your question I went to Sierra's web site and discovered their 220 RN of today has a BC of .335 and their 125 a BC of .338 so using those bullets drop and drift would be the same for both a 125 spitzer and 220 RN. Sierra has evidently changed the design of their 220 .30 cal RN.

Works for me.

[HangFireW8]

Sierra has evidently changed the design of their 220 .30 cal RN.

Maybe. See my other comment- Sierra admitted that a lot of their old ballistic tables were just plain wrong.

HF

[10x]

Maybe. See my other comment- Sierra admitted that a lot of their old ballistic tables were just plain wrong.

HF

I wrote a ballistics software program in 1984 using the British 1909 tables and later on the Ingals tables. I found a number of typos in the tables. Some numbers were out by a decimal place. I can understand that as this stuff was type set with linotype or electrotype, and the proof reading of the tables must have been almost as bad as watching paint dry.
If you are interested in how ballistics started this is a rather dry way to start.
http://openlibrary.org/books/OL6613101M/Ingalls%27_ballistic_tables.