Wednesday, December 19, 2018

Caliber Selection – Part 3 – The Math and Science

Finally, after months of delay and by popular demand (not) by no one in particular, here is the nerdy math and science stuff for those who are interested. To those of you not interested in the nerdy stuff, you are welcome for not inflicting this upon you in part two as was the original plan. Now, do yourself a favor and read this anyway. You might learn something.

There are two basic concepts that get brought up anytime gun people start talking about a particular caliber: muzzle velocity and muzzle energy. We will talk about velocity first since it’s a necessary component in understanding energy (it’s also critical to understanding things like trajectory which is a subject for another time).

Velocity is a fairly simple concept. It’s basically just a measure of how much distance an object can cover in a given amount of time. The equation is equally straight forward: Velocity = distance divided by time. In mathematical notation, it’s V=d/t. You get a result that is rendered in miles per hour or feet per second or any other unit of distance and time. Gun people are generally interested in feet per second. So, that’s the way we will think about things going forward.

Now, for the really nerdy types, you can dust off your algebra skills (yay, math!) and run the math several different ways to get different results that might or might not be meaningful in a given situation. For instance, let’s say you want to know how long it will take a bullet to reach its intended target (this might actually be necessary to hit a moving target at longer range). If you know the muzzle velocity of your bullet and the distance to your target, you can solve the equation for time. For instance, let’s assume that your .357 magnum round traveling at 1300 feet per second needs to hit a moving target at 100 yards (3 feet per yard = 300 feet). The math works out as follows (ignoring the decrease in velocity due to drag, wind, etc. for the moment):

1300 feet per second = 300 feet

Now, multiply both sides of the equation by time because we need time to be in the left side of the equation. That gets us:

time x 1300 feet per second = 300 feet
(because the “time” cancels itself out on the right side of 300 feet times time / time)

Next, divide both sides of the equation by 1300 feet per second which looks like this:

Time x 1300 feet per second = 300 feet
1300 feet per second                 1300 feet per second

On the left side of the equation, that leaves us with just time (on our hands) after “1300 feet per second” cancels itself out. On the right side of the equation, “feet” cancels itself out leaving just the unit of time (seconds).

Time = 300                 = 0.230769 seconds
            1300 seconds

So, now you know that your bullet traveling at 1300 feet per second takes 0.231 seconds (give our take few 10,000ths after rounding) to reach its target 100 yards away. Chronographs work on the same basic principle by measuring how long it takes a bullet to pass over a set of sensors separated by a known distance and doing the math.

I kinda glossed over distance conversions in the middle of that discussion of the velocity equation, but it bears brief discussion on its own. You can convert any unit of distance and time into any other unit of distance and time by knowing how much of each unit is involved. For example, there are 5280 feet in a mile and 60 seconds in every minute, etc. So, how much is 70 miles per hour in feet per second? I’m so glad you asked.

First, convert miles into feet:

1 miles = 5280 feet


70 miles x 5280 feet = 369,600 feet

Next, convert hours to seconds:

1 hour = 60 minutes = 60 seconds times 60 minutes = 3600 seconds in 1 hour

369,600 feet = 102.67 feet per second.
3600 seconds

So, why does this matter? I dunno. Maybe you need to shoot out the tire of the bank robbery getaway car and it’s 100 yards away on a crossing street traveling at 70 miles an hour. How far do you need to lead the tire to hit it using your .357? That gets into some geometry and other voodoo that even I’m not going to tackle with you here (because I didn’t major in math or physics…I just think about them…a bit), but multiplying 0.231 seconds times 102.67 feet per second will get you in a rough ballpark of needing about 24 feet of lead

One last example before moving on: how does 900 feet per second covert into miles per hour?

900 feet           x          1 mile              x          3600 (60 sec/min/hour) = 613 miles per hour
1 second                      5280 feet        

But Daddy Hawk, why would a gun person want to convert feet per second into miles per hour? Short answer: subsonic vs. supersonic. Long answer: because this is fun with math day, and you’re  going to sit there and read this and like it. Just kidding. Sort of. Where was I?

Oh yeah. Sub vs. supersonic. The speed of sound is somewhere in the neighborhood of 761 miles per hour (or 1116 feet per second) at sea level. You should be aware that the speed of sound decreases as altitude increases due to the changes in the density of the atmosphere. It’s not a significant enough change to matter to most of us unless you have a cabin or outhouse sitting above 10,000 feet somewhere. So, what if you don’t have an outhouse on Mt. Everest that you want to protect from Yeti’s or wayward thrill seekers? Why should you care?

One word: Suppressors. Subsonic ammunition is easier to suppress than supersonic ammunition. You fat, slow and happy .45ACP is naturally suppressor friendly since  even the hottest +P rounds stay in the 1000 feet per second ballpark. You can still run a supersonic bullet through a can, and it will muffle some of the muzzle blast and noise from the hot gases escaping the end of the barrel. However, the suppressor will not do anything about the loud crack created by the bullet’s sonic boom as it breaks the sound barrier after it leaves the end of the suppressor.   

Enough about the ins and outs of velocity and unit of measure conversions.  Let’s talk POWER!!! or at least muzzle energy. What we are talking about when we discuss muzzle energy is really kinetic energy of a moving object. In other words, the amount of energy a bullet carries as it leaves the barrel at a given velocity. The classical physics/math formula is: Kinetic Energy (K.E.) =  ½ mass times velocity squared.


KE = mass x velocity x velocity

In our case, mass is the bullet weight in grains divided by 7000 (which happens to be the number of grains in 1 pound [so, a 230 grain .45ACP weighs 0.03286 pounds])  times the rate of acceleration due to gravity (9.8 meters/sec. or about 32.1739 ft./sec. sq. depending on whose cheat sheet you use [and also noting that there are known variations in the figure due to altitude and latitude of about 0.5% {just roll with it, okay? <parentheticals inside parentheticals ROCK!!>}]). Velocity is measured in feet per second here in the US. The rest of the world will need to convert this mess to metric to make sense of it…you bunch of heathens.

ME (or KE) = 1 x bullet grains x velocity x velocity = bullet grains x velocity x velocity
                       2     7000 x 32.1739 ft. sec. sq.                        450435

So, for example, our .45 ACP load would have a ME calculation as follows:

230 x 900 x 900 = 186,300,000 = 413 ft. lbs.
450435                    450435

Bullet Area is just the simple calculation for the area of a circle which is the number Pi times the square of the bullet radius. Using our .45 again, the area is Pi (3.14andabunchofothernumbers) x .226 x .226 = 0.16046.

Another concept important to shooting is momentum which is just the relation between an object's mass and velocity. So, that fat, happy 230 grain .45ACP scooting along at 900 feet per second has a momentum of: 230 / 7000 x 900 = 29.57. So, what? Isaac Newton, that’s what. Old Isaac’s laws of motion tells that 1) if it’s moving, it’s not stopping unless acted upon by another force, 2) momentum is a thing, and 3) for every action there is an equal and opposite reaction. Bottomline, bigger, faster bullets carry more momentum which makes them harder to stop, and .25ACP bounces off cheap Hyundai doors.

If you poke into the dark recesses of the gunternet long enough, you will come across the “Taylor Knockout Factor”. It is, unsurprisingly, named after a guy who was a big game hunter who was not satisfied with just using muzzle energy as a predictor of how well a caliber would perform. To figure TKF, you take the mass of the bullet x velocity of the bullet x bullet diameter divided by 7000. Take a 230 grain .45ACP vs. a 147 grain 9mm for example:

230 x 900 x .452 / 7000 = 13.37 TKF

147 x 1100 x .356 / 7000 = 8.22 TKF

The Taylor Knockout Factor, which is essentially momentum adjusted with caliber diameter, favors bigger, slower bullets. Most modern students of the gun disregard TKF entirely as it disregards factors such as sectional density which factor into how well a bullet penetrates.

Sectional Density is the weight of bullet in grains divided by 7000 (number of grains in a pound) divided by the diameter of the bullet squared. So, my favorite .45ACP has a sectional density of: 230 / 7000 / .452 x .452 = 0.161. Let’s say that .357 at 1300 we discussed earlier was a 125 grain bullet. It would have an SD of 0.140. Again, why should you care? In short, long skinny bullets tend to penetrate better than short, fat bullets; however, short, fat and sufficiently heavy bullets can penetrate as well or better than lighter, longer, skinnier bullets. More or less. Your mileage may vary. Don’t piss off grizzlies with anything short of crew served weapons. Just saying.

Last subject of the night before I succumb to the siren song of my mattress: Recoil energy. The numbers in the table from part 1 come from a Chuck Hawks article on the subject, but you can follow this link to a free online calculator ( so you can tweak figures endlessly to your heart's content if you know the parameters involved. You will need to Google Fu for your load’s charge weight (that's the amount of gun powder stuffed into the casing) or dig out a reloading manual. If you are really into the math and physics of recoil, look up recoil on Wikipedia for more details.

Have fun. Ta ta for now. I hope you enjoyed this stimulating exercise in math and science. Don’t blame me, blame GunDiva. It’s her fault.  She asked for it.