Estimating Maximum Ranges of Ammunition for Tabletop War Games

[Twit42]

I am working on a simplified spreadsheet to estimate both the maximum useful ranges, and the maximum theoretical ranges of various ammunition. I plan to use this to establish some baseline limitations establishing scenarios for playing various table top war simulation / role playing games, such as Warhammer, Cyberpunk, D20 Modern, Combat Storm, etc.

I apologize in advance if this is not the most appropriate place to post this question. I figured this would fall under "self study", since I have a decent understanding of algebra and trigonometry, but little exposure to calculus and physics. I would beg for as much assistance with the basic principles as you would be willing to provide.

Homework Statement

I: What is the maximum realistic/practical range of a given bullet in combat?
(Launch angle of 0 or 1 degree, known muzzle velocity and mass)

- At which distance does the bullet "rise" or "drop" more than three inches?
- At which distance does the bullet drift more than 3 inches "left" or "right"?
- At which distance does the bullet travel time exceed 3 seconds? (if ever)

II: What is the maximum realistic/practical range of a given bullet in sniping?
(Launch angle of up to 15 degrees, known muzzle velocity and mass)

- At which distance does the bullet drift more than 3 inches "left" or "right"?
- At which distance does the bullet travel time exceed 3 seconds? (if ever)
- At which distance does the bullet no longer have enough velocity to cause injury?

III: What is the maximum realistic/practical range of a given artillery shell?
(Launch angle of up to 45 degrees, known muzzle velocity and mass)

- At which distance does the round drift more than 5 feet "left" or "right"
- At which distance does the travel time exceed 3 (or "X") seconds?

IV: What is the maximum theoretical range of a given bullet?
(Launch angle of up to 45 degrees, known velocity and mass)

- At which distance would the projectile be traveling too slow to cause injury?
- How long would it take the projectile to travel this distance?

V: How significant are wind, temperature, and humidity to the above calculations?

- With standard meteorological conditions as a starting point, how much of a change (+/-) in temperature, pressure, humidity, and windage (alternately) would it take to alter the calculated ranges by more than 5 feet?

Homework Equations

This is the core of the problem. So far, the best calculations I have been able to find and understand are all here: http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html I would like to be accurate to within +/- 5 feet if possible, but would be willing to accept a higher margin of error for the sake of simplicity in calculations.

I can calculate the ideal range in vacuum, but do not yet understand how to incorporate Ballistic Coefficient into the calculation for something that approximates accurate loss of range due to atmospheric resistance. I assume that once I am able to approximate the effect of wind resistance using BC, I would then be able to account for windage by using vectors.

The Attempt at a Solution

So far, I have been able to calculate range based on launch angle and gravity, in vacuum. Range = (Initial Velocity Squared * sin(Launch Angle * 2) / Gravity ) I am also able to calculate time to target assuming vacuum.

A .22 LR bullet launched at an angle of 45 degrees with a 0C of 0.1 and a Muzzle Velocity of 1265 fps. With the above calculation, this bullet would have a theoretical range in vacuum of 8.4 miles.

I am able to find resources that will give a calculated Ballistic Coefficient for a given caliber of bullet. Unfortunately, none of these resources also explain how to use BC in calculation. Is it possible to use BC as a reducing factor, to 'correct' for atmospheric drag after plotting the ideal range in vacuum? Example: 8.4 miles * 0.1 BC = 0.84 miles or approximate 4,435 feet?

If not, what is the appropriate equation to use?

I am also at a loss for calculating for maximum deviation from "center". I see the equation at: http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tra8 I see the equation at the site I linked, but have to confess I am more than a bit rusty when it comes to vector calculations.

I am also concerned with how accurate it would be to perform the calculation, then attempt to correct for atmospheric conditions afterward. Is there a better method to calculate this that would be relatively easy to follow?

I am assuming that the easiest calculation to make (I might be, and probably am, incorrect on this) would be the point at which the bullet loses too much velocity to be dangerous. Since Force = Mass * Acceleration, I am assuming that I can find a given force needed to penetrate a given material, and simply divide this by the weight of the bullet in order to determine minimum effective velocity.

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At this point, I would be grateful for any hint in the right direction. I have been reading what the forum has to offer, but haven't quite been able to follow as many of the equations as I would like to understand.

I am most concerned with being able to plot I, II, and III to establish some practical limits on the range of shots in simulation. I would like to plot IV, because excessive ranges have come up from time to time in certain games, and because I think it would interesting to see how far a given bullet could theoretically travel and still do damage. As far as weather conditions (V) are concerned, I realize that this can be a bit complex, but I would like to have some understanding as to how much this can impact the range.

I realize that this is a bit ambitious of a project to take on, especially when my own understanding of the mathematics involved is rather limited. I appreciate whatever assistance and direction that you would be able to provide.

Thanks in advance.

 

[Twit42]

I think I posted a bit too much of a problem, without enough understanding of the fundamentals, to be able to expect a useful response.

I'm going to attempt to break my original post down into several individual questions, so I can get help with one obstacle at a time. At the moment, the most pressing issue I have is with calculating maximum range with atmospheric resistance.

I understand how to calculate ballistic trajectory, in a vacuum environment per:

$$r_{max}=\frac{v^2}{g}Sin(2\theta)$$

With a .22LR bullet that has a muzzle velocity (MV) of 1265fps and a launch angle 1°

$$\frac{1265^2}{32}Sin(2°)=45506.4ft$$

According to this calculation, in vacuum, the .22LR bullet should have somewhere around 8.6 miles. How do I graduate from this, to something a little bit more realistic?

I have come across this (below) equation, in reading through other similar posts, but am unsure how to apply it to this application. https://www.physicsforums.com/showthread.php?t=347862

$$R_{max}=r*(1-\frac{4kV}{3g})$$

I have also read a post that refer to "quadratic drag", as a relatively efficient means of calculating ballistic trajectory for computer simulation. https://www.physicsforums.com/showthread.php?t=457357&highlight=ballistics

I have also been looking into the use of ballistic coefficient (since I understand BC to be an amalgamation of several variables into a single factor), but cannot find an equation that actually makes use of BC, only several on how to calculate BC itself.

Where do I go from here?

 

[Twit42]

I was lucky enough to be given access to some additional resources that are helping me bridge some of the gaps in understanding that I had.

When I manage to work through the new equations a little bit more, I'll make a more targeted post here to see if anyone would be willing to help check my work, once I have a little more to show for it.

Thanks to everyone who at least took a look, even though the original post was a bit too broad to start with.