How to Determine Maximum Ordinate without Ballistic App

[ShooterGavon]

I have gone to various Forums and Sites looking for a Formula or Model that I could use in the field to determine my maximum ordinate and be fairly accurate. I don't like being dependent on apps, battaries, and electronics, and would love a Mathematical Formula. I have not had any luck thus far. One guy asked for a hypothetical situation with regard to data, I gave one, but he never got back, so here is what I gave him.

Caliber - .308 W
Bullet - 168gr BTHP
Muzzle Velocity - 2678fps
Muzzle Velocity Variation - 32fps
Ballistic Coefficient - 224
Sectional Density - 192~lbs/ft ^2
Target Distance - 1000yds
Elevation - 1000ft
Humidity - 0-10%

These are the parameters the fella said he would need, but like I said he never got back to me. I suppose it would be possible for other factors to play a role on the Maximum Ordinate (MO), Spin Drift, CANT, ammo Temp etc, but this is all the guy said he would need to show me how to set up the formula and work it our.

Thanks in advance, this stuff isn't easy.

 

[mikecr]

I don't see how you could do it without software.
You need BC at some standard, adjusted via local air density and velocities taken to a best match of a drag table.
You need scope height and any slope to the shooting. You need drops at range intervals calculated, subtracting from path along the way. Then you would look up the MO in your path.
Bunch of formulas to get there.

 

[chkunz]

Google "trajectory formula" and you will find the several references to the equation. The formula is in the Sierria loading manual also.

 

[dcali]

The only way to do it without software is to do what the software does by hand, which would take days. The math is what it is - a system of differential equations with no known analytical solution.

That, or do it the old school siachi method, or use an approximation like Pejsa. But there is no known “formula”.

 

[ShooterGavon]

I figured as much. However, if a trajectory can be plotted, that is (the parabolic arc/ path by which the projectile travels) would that not simultaneously indicate the MO to some degree? Or are there simply too many variables with which to account for? I know a laundry list of empirical data has to be measured and accounted for before trajectory can be accurately measured mathematically.

But merely accounting for all the meteorological conditions and environmental conditions just to accurately plot a bullets trajectory boarders on the aeronautical engineering level of mathematics.

 

[Puzzaz01]

I'm not trying to sound like a smart a*s but you could do the same thing the PRS shooters do and shoot and record your point of impact for various distances and attach it to scope cap. I know it sounds a little old fashioned but it sure works for those guys.

Darrin

 

[chkunz]

I don't see how you could do it without software.
You need BC at some standard, adjusted via local air density and velocities taken to a best match of a drag table.
You need scope height and any slope to the shooting. You need drops at range intervals calculated, subtracting from path along the way. Then you would look up the MO in your path.
Bunch of formulas to get there.

This is not a big deal. We went to the moon with slide rules and log tales, "Hell I was there!".

 

[chkunz]

The only way to do it without software is to do what the software does by hand, which would take days. The math is what it is - a system of differential equations with no known analytical solution.

That, or do it the old school siachi method, or use an approximation like Pejsa. But there is no known “formula”.

There is a formula and it takes a few minutes, not days.

 

[chkunz]

I figured as much. However, if a trajectory can be plotted, that is (the parabolic arc/ path by which the projectile travels) would that not simultaneously indicate the MO to some degree? Or are there simply too many variables with which to account for? I know a laundry list of empirical data has to be measured and accounted for before trajectory can be accurately measured mathematically.

But merely accounting for all the meteorological conditions and environmental conditions just to accurately plot a bullets trajectory boarders on the aeronautical engineering level of mathematics.

This is not that difficult, simply plug in the trajectory formula and turn the crank. Some of us lived before we had apps and we did the calculations and got the answers. This is no big deal.

 

[ShooterGavon]

"Simply plug in the trajectory formula and turn the crank".

What are you talking about? Are you referring to the common formula thats taught in Calculus? Because if you are, that only applies to a projectile thats launched or projected in a vacuum. A vacuum meaning (no MET or ENV conditions) as potential variables. If its raining, or if it's 105*F, or if there is a tail or head wind , if so, everything changes. Are you talking about a ballistics program? If you are, most only require few data inputs, such as MV, Bullet weight in Gr, Caliber, Wind and distance to TO (Target Objective).


I was asking about calculting the MO (Maximum Ordinate) with pen and paper. Do you know how many variables play a role in calculting and determining the answer for that? Everything from ammo temp, humidity, to barometric pressure, muzzle velocity, angle cosine, up to sectional density and ballistic coefficient, to G1 vs G7 form factor, gyroscopic stability averages, head wind vs tail wind (as that can drastically effect the point of impact, which in turn will affect the MO).

I guess were all just not as smart as you "oh great math lord". Was that what you were looking for?

 

[dcali]

There is a formula and it takes a few minutes, not days.

Please share it. I'm all ears.

 

[dcali]

I figured as much. However, if a trajectory can be plotted, that is (the parabolic arc/ path by which the projectile travels) would that not simultaneously indicate the MO to some degree? Or are there simply too many variables with which to account for? I know a laundry list of empirical data has to be measured and accounted for before trajectory can be accurately measured mathematically.

But merely accounting for all the meteorological conditions and environmental conditions just to accurately plot a bullets trajectory boarders on the aeronautical engineering level of mathematics.

Plotting the trajectory is exactly what you would do. But that is not easy. As you've noted, there are multiple inputs that need to be accounted for, as well as the inherent complexity of solving the equations of motion. There isn't a method I would use on paper. The closest would be Pejsa's method, which is a clever way to simply things, but it's still not something I would do by hand.

Why do you want to do it by hand?

 

[geo.ulrich]

There are a couple of missing pieces to the puzzle, such as ogive , nose length bt length and angle and meplat diameter with this you can calculate drops ,,,,

 

[ShooterGavon]

There are a couple of missing pieces to the puzzle, such as ogive , nose length bt length and angle and meplat diameter with this you can calculate drops ,,,,

I was thinking that as well. I'm certain the geometrics and angles of the bullet itself, play a crucial role. Meplat I imagine you would measure the diameter of the bullet tip.

The whole reason I would like to know, or am fairly interested in this, is because not many people know it, or are even that interested in it. Lots of us rely on DOPE which can be extremely helpful, but is far from a science. Others, well we use Electronics or Ballistics Programs - Anemometers w/ Applied Ballistics, Trimble Recon with Ballistics Program, various Phone Apps etc.

While these are cool, easy and fun to use, few of us out there (except for the edgy math lord who commented earlier) can get along with pen and paper. I would rather have the "know how" but perhaps not need it, rather than the other way around, i.e. "Needing it" but not having the knowledge.

 

[dcali]

I was thinking that as well. I'm certain the geometrics and angles of the bullet itself, play a crucial role. Meplat I imagine you would measure the diameter of the bullet tip.

The whole reason I would like to know, or am fairly interested in this, is because not many people know it, or are even that interested in it. Lots of us rely on DOPE which can be extremely helpful, but is far from a science. Others, well we use Electronics or Ballistics Programs - Anemometers w/ Applied Ballistics, Trimble Recon with Ballistics Program, various Phone Apps etc.

While these are cool, easy and fun to use, few of us out there (except for the edgy math lord who commented earlier) can get along with pen and paper. I would rather have the "know how" but perhaps not need it, rather than the other way around, i.e. "Needing it" but not having the knowledge.

If you really want to know, buy Bob McCoy's book, "Modern Exterior Ballistics". It describes in painful detail the methods used for this, and for far more advanced calculations. It is not for the faint of heart - the math is intense. Knowledge of calculus, differential equations, and numerical methods is assumed. But it's hard to find a better book on the subject. If you have the math background, you'll see that there just isn't a simple way to do it. For what it's worth, McCoy describes the methods developed and used by the military going back decades - the ones that form the baseline for pretty much every ballistic calculator out there.

And yes, the bullet geometry matters - there are ways to calculate their impact as well, and it's easier than calculating the trajectory, but still not what I'd want to do with paper. The earliest computers were basically created to calculate trajectories - otherwise, it's a long (long!) slog of simple calculations that add up to something useful. Before computers, methods utilizing tables (look up Ingalls' "Artillery Circular M" and the Siacci method) that were used. I don't think anyone ever did it strictly by hand.

 

[ShooterGavon]

If you really want to know, buy Bob McCoy's book, "Modern Exterior Ballistics". It describes in painful detail the methods used for this, and for far more advanced calculations. It is not for the faint of heart - the math is intense. Knowledge of calculus, differential equations, and numerical methods is assumed. But it's hard to find a better book on the subject. If you have the math background, you'll see that there just isn't a simple way to do it. For what it's worth, McCoy describes the methods developed and used by the military going back decades - the ones that form the baseline for pretty much every ballistic calculator out there.

And yes, the bullet geometry matters - there are ways to calculate their impact as well, and it's easier than calculating the trajectory, but still not what I'd want to do with paper. The earliest computers were basically created to calculate trajectories - otherwise, it's a long (long!) slog of simple calculations that add up to something useful. Before computers, methods utilizing tables (look up Ingalls' "Artillery Circular M" and the Siacci method) that were used. I don't think anyone ever did it strictly by hand.

Very interesting! Will definitely check out McCoys book, without a doubt. Thanks a bunch for the info!

 

[dcali]

More history: ENIAC, which was among the first electronic computers, was also the first ballistics calculator in 1945. https://en.wikipedia.org/wiki/ENIAC

"ENIAC calculated a trajectory in 30 seconds that took a human 20 hours"

 

[geo.ulrich]

Robert McCoys book is the book for ballistics, I will say its not for everyone very math heavy.I have recommended to a few people that could not get through it. I believe you can download it as a pdf and read it

 

[Ballisticboy]

The only thing I would add is that the McCoy book still appears to contain some errors in the maths from what I have seen in a recently bought copy. I would say check the maths carefully for typos before using any of the final equations.
Unfortunately I cannot remember where I spotted an error.

 

[dcali]

The only thing I would add is that the McCoy book still appears to contain some errors in the maths from what I have seen in a recently bought copy. I would say check the maths carefully for typos before using any of the final equations.
Unfortunately I cannot remember where I spotted an error.

Is that the 2nd edition? The first edition is riddled with errors. I haven't seen any in the 2nd, not that I've really looked carefully.