Sniper Flash Cards: Ballistics for Homemade Mortars Firing Ball Bearings

This software assumes that drag is proportional to the square of velocity. A lot of software packages say this but, in the fine print, they add that they are going to “make some simplifying assumptions” or “drop the higher order terms.” Then they proceed to solve the much simpler case of drag being directly proportional to velocity. A word to the wise: If drag is proportional to the square of velocity, range can only be determined by an iterative method. If they are using table look-up (as I did in my pumpkin program) then they are rounding the results off in a way that is only acceptable for vegetables, not steel balls. If they have a formula in closed form – not a DO/UNTIL loop – then they are assuming that drag is proportional to velocity, which is dishonest unless they made that assumption clear.

This program uses an iterative method, so drag is indeed proportional to the square of velocity. This assumption is valid for velocities up to 240 m/s. From 240 m/s to 295 m/s, drag is proportional to velocity cubed. My program now takes account of this, which allows for higher initial velocities. It does not accurately model supersonic projectiles that decelerate through the speed of sound, 343 m/s, however, and should not be used for this purpose.

Air density is re-calculated at each step so, if the ball flies so high that the air is significantly thinner at its apex, this is taken into consideration. The coefficient of drag is also re-calculated at every step so it increases from 0.2 to 0.45 at the exact moment that the velocity falls below the critical point. Altitude, air pressure and temperature are all taken into consideration. Note that, so the isobars on weather maps are all in the same units, barometers are usually normalized to what they would read at sea level. This is also what you want because, since altitude is a separate parameter, if you reported the actual number of inches of mercury at your location, you would be over-estimating the effects of altitude.

Note that humidity is not taken into consideration. This is because it is negligible – really. Rifle matches are rarely called on account of rain and I can remember lying at the shooting line amusing myself by using my amplified earmuffs to listen to my competitor and his coach earnestly debating, like Twiddledee and Twiddledum, whether they should add one or two minutes of angle to their elevation dial to account for the “heavy, humid air.” This is foolish because humidity has a negligible effect at 600 yards and, anyway, they have got it backwards. Humid air is less dense than dry air. Clouds float, don’t they?

Ball bearings are assigned a grade to denote smoothness, starting at about 50 for “precision” bearings, 100 to 500 for “semi-precision” bearings and on up to 2000 or, in some cases, 5000. A polished surface is a mixed blessing. It lowers the coefficient of drag slightly, which increases range. But it also raises the critical point where, when velocity falls below it, there is a dramatic increase in drag. Thus, for balls launched at such great velocities that they stay well over their critical point, smoothness increases range. For balls launched at velocities only slightly above the critical point, smoothness reduces range. This is why golf balls have dimples; because they are driven at velocities near what the critical point would be for smooth balls. For most people, this discussion is academic because they could not afford to launch pricey precision bearings into a lake anyway, even if it were unequivocally a good thing. Bottom line: Buy bearings with a grade of 2000ish and don’t sweat it.

Large bearings are normally sold in 5mm increments and a 25mm bearing will fit in a 1” pipe. It is hard to obtain a bearing that will fit in 1½” pipe. I do not recommend 2” pipe because the volume of gunpowder is great enough to grenade the mortar. Also, 50mm balls are many times more expensive than 25mm balls. Thus, at least in America and England, I would recommend a caliber of no more than 1”. I don’t know what sizes of plumbing pipe are available in other countries.

This software is intended for use with steel ball bearings, but it can be adapted for other projectiles.

Pumpkins: If you are using a trebuchet to launch pumpkins, use this software to determine the initial velocity and the ballistic coefficient of your pumpkin. Multiply the ballistic coefficient by 0.1095 to get the diameter of an equivalent steel ball. Muzzle height is the height of the axle plus 1.7 times the length of the throwing arm. With this information, come back here to create a range table for your particular launch site and wind conditions.

Unstabilized Cylinders: The Stokes Mortar, widely used during World War One and the first mass-produced mortar in history, launched an unstabilized cylinder. Homemade mortars intended to be used as weapons (which I do NOT recommend) are typically knock-offs of the Stokes Mortar. The ballistic coefficient is between 0.87 and 1.22 times the mass divided by the cross-sectional area. The longer the cylinder is, the higher its ballistic coefficient. Multiply this by 0.1095 to get the diameter of an equivalent steel ball. Of course, once the cylinder starts to tumble, then any further prediction of its trajectory is thrown to the wind – literally.

Modern Mortar Shells: Spheres are just pushed sideways by the wind while projectiles with fins may turn their nose into the wind if the fins are too big. This makes windage adjustment vastly more complicated than it is for spheres. Also, such projectiles typically go high enough that the wind is stronger at their apex. There are software packages available to the designers of model rockets that may be helpful in finding the smallest fins needed to stabilize a projectile in still air. One can then use the software on this page for launching mortars short distances in mild winds.

Software for fin-stabilized projectiles that considers their tendency to turn into high winds and considers long-range projectiles that go high enough to encounter different wind conditions than are present near the ground is expensive and its purchase will attract the attention of law enforcement.