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
time
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
Therefore,
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.
Or
KE = mass x velocity x
velocity
2
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 (http://www.shooterscalculator.com/recoil-calculator.php)
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.
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