How do I calculate the range of a warship's guns based on different gravity and air density?

I'm poking through information and searches on calculating ranges for artillery weapons, and I'm finding a lot of simple ballistic calculators, which don't fit what I want, or calculators for small arms, which may be accurate enough but don't allow for the correct values and/or use values for which I don't have the answer to.

I want to be able to calculate the range of a warship's guns. Specifically, I need this information to include air drag based on air density, and range reduction based on gravity. The latter is fairly simple, the former... probably not.

Ideally, I'd be able to punch in the information from the guns of an Iowa-Class battleship, and see how far it'd be able to shoot on different worlds. I'm not asking for an online calculator, though they'd be nice, "just" the equation and an explanation for that equation. I'm not afraid to do things by hand if I have to, but I am not a mathematician. I can generally understand formulas and how they work, but they need to be disassembled and explained. Walls of formulas become quite unreadable.

Edit: In my further studies, the math is beyond my personal knowledge to decipher a formula dump. This Page, for example, seems to have all of the information required. I just can't make anything useful out of minimal explanations and walls and walls of formulas. A more expansive explanation of the formulas involved would be greatly appreciated.

• Can you explain what is wrong with L.Dutch's answer? He's got the basic math right there. – kingledion Mar 30 '17 at 13:24
• Usability. As I explained in my most recent edit, I have a passing proficiency in math, but that's about it. His answer may have all of the correct formulas, but without an explanation with significant further simplification, I cannot use it. – Andon Mar 31 '17 at 7:11

A useful overview of the problem: http://nigelef.tripod.com/fc_ballistics.htm

This guy gives you the individual formulas. I wouldn't count on it to hit a target, but it's probably good enough for a story. Technology is mostly round shot to WWII. https://grantvillegazette.com/wp/article/publish-581/

The computer used on the Iowa was a mechanical analog one. They looked at digitizing it, and figured that it wouldn't be any more accurate. Article here: https://arstechnica.com/information-technology/2014/03/gears-of-war-when-mechanical-analog-computers-ruled-the-waves/

• It'll probably take me a while to chew trough this, but it looks like just what I was looking for. – Andon Mar 27 '17 at 4:32

If, during the flight there is drag $$F = - β v$$ (F and v are vectors), with $$β > 0$$ Newton's law becomes: $$m g - β v = m a$$ Choosing $x$ horizontal and oriented toward the flight direction, and $y$ vertical pointing upright, by projecting on the axis we get: $$a_x = - (β/m) v_x$$ $$a_y = - g - (β/m) v_y$$ Since $$a_x = dv_x/dt$$ $$a_y = dv_y/dt$$ you get two differential equations in $v_x(t)$ and $v_y(t)$ with boundary conditions $$v_x(0) = V_o cosα$$ $$v_y(0) = V_o sinα$$ Once known $v_x(t)$ and $v_y(t)$ and integrating you get $$x(t) = \int_0^t v_x(t)\ dt$$ $$y(t) = \int_0^t v_y(t)\ dt$$ considering the launch to be happened at the coordinates (0,0).

Setting $y = 0$ you get the time of flight and putting that time in $x(t)$ you get the distance.

• Usually drag is proportional to the square of the speed. – EngelOfChipolata Mar 27 '17 at 8:16
• @EngelOfChipolata, the formula I reported can be widely found across various references, like here farside.ph.utexas.edu/teaching/336k/Newtonhtml/node29.html – L.Dutch - Reinstate Monica Mar 27 '17 at 9:22
• From what I have understood the formula you use is for low speed projectiles. The example given by OP shows a supersonic muzzle speed (on Earth at sea level with a standard atmosphere). Even the squared velocity is far from the drag that really occurs at supersonic speed (e.g. wave drag omitted). – EngelOfChipolata Mar 27 '17 at 9:36
• What do you mean with the paragraph starting 'Once known?' It seems like you missed some words in there. – kingledion Mar 27 '17 at 13:54
• @kingledion Once know vx and vy by solving the differential equations. – L.Dutch - Reinstate Monica Mar 27 '17 at 14:13

The one variable you cannot tweak with this simulator is gravity. But you can tweak your projectile mass. Fortunately for you, the calculation determining the force exerted by gravity between 2 objects combines the mass of the attractor (your planet; m1) and the attracted (your projectile; m2).

$$F=G{\frac {m_{1}m_{2}}{r^{2}}}\$$

You can set the mass of your projectile. So if you want to simulate a situation with half the gravity of Earth (and so half the mass of earth; 0.5 * m1) you can instead use full Earth gravity (m1) and make your projectile half the mass (0.5 * m2).

To be clear: reduce your projectile mass by the same percentage reduction you want below Earth gravity and the simulation will work.

The simulator allows you to enter drag coefficient and altitude.

https://en.wikipedia.org/wiki/Drag_equation

Drag coefficient has to do with the shape and aerodynamics of your projectile. Maybe leave that at 1.

Altitude is what you want to alter air density: air density varies according to altitude.

• There are a few tricky things about this - Drag Coefficient is pretty important, as otherwise plugging in the 16"/50 Mk7 gives a range of about half of what it should be. And, annoyingly but also useful is that it gives updates at the projectile's range in real time. – Andon Apr 2 '17 at 23:01