Do you ever have one of those days where you start wondering about some esoteric little problem and can’t let go of it until you find the answer? It can be anything. How many angels can dance on the head of a pin? How many chips are in a bag of Fritos? You know…stupid stuff that won’t matter in the immediate future much less a year or a decade from now. I had one of those days last Sunday.

I had to make a quick trip down to Centerville, Texas to meet a guy who was coming up from Houston for a little business we had to conduct together face to face. It’s a 2 hour drive from Castle Erickson to Woody’s Smokehouse in Centerville. I was a little tired from staying up too late on Saturday night, and I needed a distraction to keep my mind active while on the drive.

So, what little esoteric nugget do I come up with? Something profound? No. Something amusing? Well, that depends on whether or not simple math and basic physics are your idea of a knee slapper.

It all started with a MythBusters episode I saw a little while back where Adam and Jamie were trying to determine whether a bullet fired straight up (90 degrees perpendicular to the ground) could cause a fatal injury when falling back to earth. Now the MythBusters conclusively proved that a bullet fired straight up at 90 degrees would fly straight up and fall back down at a terminal velocity of something like 100 miles per hour. Since the atmosphere is not a perfect, frictionless environment, the projectile would eventually stop spinning allowing the bullet to destabilize and start stumbling.

Add to that little memory nugget the fact that I’ve been trying to educate myself about all aspects of the shooting sports…including ballistics. So, I decided to see if I could figure out how high a 230 grain bullet fired straight up at 1000 feet per second would theoretically travel before coming back home to papa.

Now, before I go on, allow me to say that I haven’t had a math or physics class in about 18 or 19 years after getting a C in Calculus I on the second try and an A in Analytical Geometry on the third try (yeah, I’m stubborn that way sometimes). That right there pretty much killed any desire or hopes I had for pursuing a degree in engineering.

Anyway, the math necessary to figure out the answer to my problem is deceptively simple once you understand the physics part of the equation. Some people might think that there is not enough information provided above to figure out the answer, and they would be partially correct. You do have to know what the force of gravity is to get the answer. Other than that, it boils down to straight forward multiplication, division, addition and subtraction.

I’ll help you out here and clue you in that the force of gravity is roughly 32 feet per second squared (32.163 ft/sec sq. to be exact…remember that figure…it’s necessary if you want to figure out muzzle energy on your own without a ballistics table).

So, we point our theoretical pistol straight up so the barrel is 90 degrees relative to the ground and pull the trigger. Our theoretical 230 grain bullet leaves the muzzle at an initial velocity of 1000 feet per second. As a quick aside here, there are 7000 grains in a pound making a 230 grain bullet 0.5257 ounces or 0.03285714 pounds. Not considering friction (since I didn’t have a reliable source handy from which to obtain figures), the only force acting on the bullet once it leaves the barrel is gravity. Gravity is Decelerating the bullet at a constant rate of 32 feet per second squared. In other words, every second the bullet is in flight, it slows down by 32 feet per second.

Now, with this bit of information, we can arrive at the first interim step along the way to answering our original question: how much time will the bullet be in flight before running out of steam. Basic long division is our friend here. 1000 feet per second divided by 32 feet per second squared leaves us with 31.25 seconds of travel time.

That answer allows us to take the next step in answering the problem. We now know that we need to knock 32 feet per second off the initial velocity and each subsequent velocity 31.25 times. Each of those 31.25 speeds represents the distance that the bullet would have traveled in each of those 31.25 seconds. This is where we break out elementary addition and subtraction. 1000 feet per second minus 32 feet per second equals 968 feet per second. 968 minus 32 equals 936. And so on and so forth. Eventually, 31.25 times later we get to the last quarter second of the bullet’s flight where it expends the last 8 feet per second of its velocity (notice the symmetry that 8 is ¼ of 32) to ever so momentarily hover in midair at a relative speed of exactly zero feet per second just before starting its return trip to terra firma. If you add up each of those recorded velocities, you come up with a total distance traveled in 31.25 seconds of 16,128 feet. Dividing that figure by 5280 feet per mile, we learn that the maximum potential altitude for a 230 grain bullet (any weight bullet for that matter) fired at 1000 feet per second is 3.05454… miles.

I know that some more seriously minded math and science types might quibble that there are “simpler” and more precise ways to calculate the answer to that question. Possibly. A quick Google search from the comfort of home using my DSL connected laptop instead of my 3G iPhone while traveling 70 miles per hour on I-45 suggest that I use the formula “distance = initial velocity plus ½ the rate of acceleration times the square of the amount of time involved.” That would look like “d = 1000 ft/sec + (1/2 * 32 ft/sec * 31.25 sec * 31.25 sec)” or 16,625 feet (3.148674 miles).

So, all things considered, I don’t think I did too badly for a tired guy driving a car and fiddling with a phone and a pad of paper. A difference of 500 feet (or just a touch less than 1/10 of a mile) is a pretty good swag if I do say so myself.

Now, for the bonus question: if the theoretical 230 grain bullet reached its maximum altitude and is struck by a Boeing 737 traveling at an airspeed of 500 miles per hour at the very instant that the bullet’s velocity reached zero, how much kinetic energy imparted to the airplane by the impact with the bullet?

I’ll let you all noodle that one for a bit. If you’re nice, I might even post the answer.

I didn't understand one word of that. I am now, however, having a horrible flashback of making a mousetrap car in some high school science class that snapped shut on my finger and broke my nail. Awesome.

ReplyDeleteCandance, if you didn't understand the words comin' outta my mouth (fingers actually), I really have to question your choice of vocation. Just kidding. Math and science isn't for everyone...as evidenced by the fact that I make a living handling insurance claims. I still have a closet engineering/science geek hiding in my brain though.

ReplyDeleteI'm afraid your google search answer is incorrect by exactly 1000 feet. You only add the initial velocity when the object is accelerating. Think of it in reverse: the bullet is falling from an initial velocity of 0 to a final velocity of 1000 fps at impact (since we're not considering air resistance). Using the formula from my beginning physics class d = 0.5 * a * t^2 the height is 15,625 feet.

ReplyDeleteJoe, thanks for the correction. That makes sense...I think...the lab assistant in my physics I class was Chinese and barely spoke English and the prof was an old guy who was more interested in his nuclear experiments than freshmen. So, you have to pardon me for getting some of the minor details wrong.

ReplyDeleteStill...I'm impressed. Lots of stuff in there that gave me a headache (but then so did Physics in college). I drive around wondering if people can see farts outdoors when it's -2*F, not brainy stuff like you do. You'll definitely get into law school with a brain like yours :)

ReplyDeleteGunDiva, thanks, but sometimes I wonder. The Queen and I were driving home from church one time when I got a spaced out look on my face. She asked me what I'm thinking. My answer: "Tools."

ReplyDeleteAs an Engineering friend of mine would say, "now that is Math not Arithmetic"...

ReplyDeletejust saying....

and kinda like a slinky, not really much good for anything but fun to kick and watch it flop down the stairs....LOL

As a youth, I invested half an hour to climb to the top of the Foshay Tower in MPLS (tallest building in the midwest at the time) and look out over the city through the screened viewing windows.

ReplyDeleteAs kids are wont to do, the urge to toss a coin off the building was immediately stifled by a cousin who pointed out a sign posted at each viewing area: "Do not throw anything from this window... a penny dropped from this height will penetrate three inches into the concrete sidewalk below."

Years later, Jim Carmichael and I had a discussion as to how much damage a .22 bullet would cause a person if it hit them after being fired into the air. Jim's position was "not much unless it hit them directly in the eye." Mine took a different tack, and I still believe I am correct, despite having no science or math background to verify or test the assumption: "I think the bullet will go through your skull and out your chin."

All based on that single penny tossed off the Foshay Tower.

I maintain that thought today and do all I can to instill in people the very unwise decision of firing into the air.

Shy III

Mr. Daddy, thanks for the slinky image. There is something oddly satisfy about watching something flop down the stairs.

ReplyDeleteSly Wolf, I seriously doubt that a penny dropped from tall building would penetrate solid concrete up to three inches deep based on the amount of kinetic energy involved (about 3.2 ft. lbs. if I got all the numbers right). Having said that, I really don't like having stuff drop out of the sky and hit me in the head. Also, that MythBusters episode I mentioned did show documented proof in the form of an xray for a man who was hit in the head with a .30 caliber bullet that had been fired into the air. The conclusion was that firing at an angle significantly less than 90 degrees allows the bullet to retain more of forward and rotational momentum down range increasing the kinetic energy delivered to a particular target (someone's head in this case). Since most people are not capable of shooting a firearm straight up at 90 degrees and other forces such as wind resistance/friction, the rotation of the earth, low flying aircraft, etc. can have an impact on a bullet's actual trajectory, it remains a very bad idea to discharge an unaimed weapon into the air. Not to mention it violates the Four Rules of safe gun handling.

How do you arrive at the deceleration figure of 32 feet per second? I am assuming this figure would change depending on the muzzle velocity and weight of the bullet.

ReplyDeleteAlso, someone mentioned what would happen a .22 bullet fired straight up. Would it only hurt if it hit the eye or go through your skull. Wouldn't it be the same effect as dropping a .22 bullet from the same height as its apogee from being fired straight up from the ground, since the bullet fired stops ever so briefly before returning to earth?

Thanks!

Anonymous, I arrived at the 32 ft/sec figure for deceleration as that is the accepted figure for the force of gravity. Since we are stipulating that the bullet is fired straight up at 90 degrees and we are ignoring the effects of wind/air resistance, the only force acting on the bullet after it leaves the barrel is the force of gravity whose vector is in the opposite direction of travel until it reaches apogee.

ReplyDeleteAs to your second question, I don't think I can give you a definitive answer.

Firing a bullet straight up is equivalent to dropping a bullet from the same height as the fired bullet's apogee for the reason you mention. More or less. It's highly unlikely that anyone would succeed in firing a bullet straight up and the wind conditions allowing to fall straight back down.

The Mythbusters episode referenced proved that a bullet dropped from high enough would reach a terminal velocity of around 120 miles per hour due to air resistance. They also proved that the bullets tumble as they fall instead of pointing nose down. Plugging the numbers of a 55 grain .22 bullet falling at 120 mph into the kinetic energy formula, we get an energy at impact of 7.57 ft.lbs. of energy impacting a spot just a skosh under a quarter of an inch in diameter.

I don't know how much kinetic energy it takes to penetrate the skull. The only figure I've been able to find refers to a study on impacts to femurs (which are supposed to be the strongest bones in the body). That study supposedly found that it only took 10 ft.lbs. of energy to penetrate the femur through and through.

So, a 55 grain .22 bullet falling 90 degrees straight down at terminal velocity arguably has the potential to penetrate the skull or at least make a painful dent in the skull. Whether it WILL or not depends on several variables that we can not account for here including nose down/flat impact, hair/no hair, hat/no hat, etc.

Finally, I think we can safely agree that getting hit in the eye with a .22 bullet is going to sting just a touch no matter have fast its moving.

The further the bullet travels from the Earth, the less gravity is, so remember to factor that into the calculus. Likewise, if there is a solar eclipse directly overhead, a bullet shot straight up will travel a sliver higher (helped by both the gravity of the sun and moon).

ReplyDeleteAnonymous, you just shot right past my "C' in Calculus I and whatever grade I got in Physics I thanks that the Chinese lab asst and senile professor. What you say makes sense, but I would have to lump that into the "variables I can't account for" category.

ReplyDelete