How to roughly extrapolate artillery range for planets with different atmospheric pressure?

I'm looking for some rough way of extrapolating artillery range for planets with different atmospheric pressure:

Realistically, the data available:

1. Hypothetical range in vacuum: $$\dfrac{V^2}{g}$$ (V - muzzle velocity, g - gravity)
2. Range on Earth

So based on those data I could easily calculate which percentage of energy was "lost" on Earth atmosphere. Could I extrapolate from that how would it roughly work on a different planet?

EDIT: I also thought about simply applying drag equation. It would make perfect sense for calculating how a bullet (or an armor piercing projectile) would lose its kinetic energy. The relation would be proportional to air density.

Just there is one big problem with any artillery - it uses ballistic trajectory, so I have to incorporate BOTH gravity and pressure. I can not simply extrapolate from atmospheric pressure, because it would mean that for planet with vacuum the artillery range would be infinite.

• This may be both more complex and simpler than you first assume. Projectile size, shape, initial velocity and mass will all interplay with not only atmospheric pressure but also what the exact gas mix is. On the other hand if you have a fast, heavy projectile and your target isn’t miles away you can reasonably use projectile in a vacuum as a decent enough model since the energy lost prior to impact will be small relative to the projectile’s kinetic energy. Drift is a different issue though... – Joe Bloggs Jul 28 '19 at 9:12

For slow projectile, energy loss due to drag is proportional to the velocity of the projectile and to the density of the medium.

Density of atmosphere varies linearly with pressure, everything else constant.

In this regime, thus, if you increase the pressure you are proportionally increasing the density and the lost energy.

For higher velocities drag is proportional to the square of the velocity, but still goes linearly with density: same as above.

Drag equation.

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

You can use this to calculate the force slowing your projectile (𝐹𝑑): the drag force.

Here is the drag equation. $$F_d = 1/2 \rho u^2 C_d A$$

• $$F_d$$ drag force
• $$\rho$$ mass density of the atmosphere
• $$u$$ velocity of the bullet
• $$A$$ area of the bullet
• $$C_d$$ drag coefficient of the bullet

Here is an online calculator and you can tweak 𝜌 (atmospheric density) to see how that affects range. But it should be a straight multiplication if density is the only thing changing.

https://www.omnicalculator.com/physics/drag-equation

Interesting stuff in this question as well that you might find useful for your endeavor. They also ask about the impact of higher atmospheric pressure on projectiles.

What would be a reasonable caliber for an assault rifle and sniper rifle for a planet with pressure of 3 atm?

• Yes, my question concerning small arms. ;) Just from physical perspective the case was simpler as kinetic energy matter, while gravity could be mostly ignored. – Shadow1024 Jul 30 '19 at 8:35