Updated: October 26, 2012
In the words of Jon Tickle, a co-presenter on Brainiac and a physics guy himself, bless him, here's one for you: Say you have a gun and you fire it, parallel to the ground. Say the spent bullet casing comes out of the firing chamber flying and smoking and all that at the very same angle, parallel to the ground. Say your bullet hits nothing and just lands somewhere in the field, far away. Which one hits the ground first, the bullet or the casing?
This is probably a trivial question, but I'd like to explore it a little more. In other words, let me tell you how gravity treats rounds and casings, what elevation and super-elevation have got to do with it, why air resistance and firing angle are important, and what happens when merry people celebrating a wedding pop one or two into the air above the revelers. Follow me.
Gravity respects no one
Now, when you fire your gun, provided you shoot at 0-degree angle, and the empty casing comes out the same way and does not rebound against any nearby objects, the moment you've heard the casing plink, it also means your bullet's touched the ground. It's so simple.
What you need here is the most basic equation for calculating the time it takes to a free-falling object to travel the distance of about 1.5 meters, which is the average shoulder height of a typical male shooter. With the gravity pulling at around 9.8 m/sec2 and no air resistance, the object will finish its drop in:
t = sqrt(2X/a) = sqrt (3.0/9.8) = sqrt (0.31) = 0.55 sec
In this time frame, the horizontally flying bullet - and slightly dropping in a parabolic trajectory, will travel the distance of 0.55 sec times its muzzle velocity. Neglecting the air resistance, the only other parameter missing is the muzzle velocity. For typical guns, it's around 300-400 m/sec. For assault rifles, around 1,000 m/sec.
This means your typical assault rifle can lob a bullet 550 meters away, where it will nick someone's shoe. If you wish to hit at the chest height, you will need to elevate slightly and compensate for the bullet drop. Not only will this correct your aim, it will also increase your range. In theory, at the precise 45-degree angle, your shot will reach the farthest. With air resistance playing a major factor, the angle is more like 55 degrees.
So are weddings dangerous?
P.S. For those living in the Western world, let me explain the wedding culture to you. Once a couple is merrily joined in a union that will last until one or the other dies, it is customary to celebrate most loudly. Shouting works, but nothing beats a volley from an automatic rifle. Hence, it is even more customary to take your family-owned AK-47 and fire a few salvos into the air. Much rejoicing.
Let's discuss vertical and terminal ballistics. Let's see what happens when you go wild with the math, as well as the application of ballistics to common life situations, like the wedding shootout. What happens if you fire at the 90-degrees angle? Is there any chance anyone might get hurt?
Here, we need the time it takes the bullet to decelerate to 0. Then, multiply that by 2, assuming the bullet can fall freely and will not reach a terminal velocity barrier that will slow it down, which it will, coming down base first, tumbling and whatnot. However, if you shoot with a bit of deflection, the bullet might retain its spin and remain deadly enough on impact. Maybe. Anyhow, ze math:
t = V/a = 1000/9.8 = 102 sec
For the assault rifle, the vertical bullet will fly for almost two minutes before reversing direction and as long to come down. Enter terminal velocity, and the bullet comes down at a much reduced speed, so it could spend many minutes descending. Still, it could be quite deadly. How much? We will discover soon. Meanwhile, in this period, the bullet will travel a magnificent theoretical distance:
X = 0.5 x a x t2 = 0.5 x 9.8 x (102)2 = 50,900 m
It would reach a very nasty altitude, yup. That's a little drastic in fact. If it were so easy to lob bullets so high, you would not need expensive surface-to-air missiles for interceptions. You would just have to angle your el-cheapo rifle skywards and fire about 1 minute ahead of the scheduled arrival of enemy planes into their flight path. Better yet, with only a slightly higher muzzle velocity, it would be possible to shoot down satellites.
This is not what happens in reality though. In reality, there's drag, and it depends on many factors, resulting in a ballistic coefficient (BC) that tells how much of the initial velocity of the bullet is lost to air resistance. Here's quite a bit more on external ballistics on Wikipedia.
Here's an awesome video showing a scientific experiment firing a 9mm PARA bullet with the muzzle velocity of 1,500 kph (417 m/sec), landing at 350 kph (97 m/sec) and penetrating a block of ballistic gel with rather deadly results. Almost like the medieval arquebuss of sorts. The video is in German, but the message is universal:
At a full vertical angle, such a bullet would reach the theoretical height of 8.6 km, however it tapers off at 1.15 km. For such an altitude, the muzzle velocity would have to be around 120 m/sec, just a little more what it has on impact. The overall flight time is then 12 seconds to max. altitude or about 25 seconds overall, as seen in the experiment. It shows the BC to be around 0.28.
Like we mentioned earlier, the velocity will depend on many factors, including the cross-section, the length to diameter ratio, the center of pressure, the center of gravity, spin, mass, and others. Now, let's go back to our small-arms rifle bullet. According to this article right here, a 0.3-cal round can reach the altitude of only 2.8 km, which allows for a corrected muzzle velocity of about 235 m/sec and the climb time of 24 seconds. That's approx. only 25-30% of the initial momentum, all of it lost to drag, with an approx. coefficient of 0.23.
Indeed, drag is the drag [sic] here. However, the same article gives a tank 120mm SABOT round the max. range of 102 km at 55 degrees. Blimey! That's really awesome. Now, there's Abrams firing, at a more realistic angle. But if you had one of those for your nuptials, well.
And some more fun reading:
So yes, we can conclude that weddings can be dangerous. And the bigger the gun, the more fun. Now let's try to answer some of the questions raised earlier. We know all about velocity and drag. We know about the gravitational pull and how you must elevate ever so slightly to hit above the shoe level at some distance.
We're left with answering the elevation angle and the casings ground plink. At straight angles, it's a no brainer. But what if you raised your barrel ever so slightly and the casing comes out tumbling at a weird angle. What then?
So, if you wanted, for purely mathematical study purposes, to have your bullet remain at the same height at the target as it was when fired, you need to take into account the distance and the muzzle velocity. The rest is more trivial, for now. Therefore, with 0.55 sec flight time to target, your ballistic trajectory must reach the apex at half that, or 0.275 sec, I hope you're with me. This translates into only 0.4 m at half-distance. In other words, you need an angle of only 0.1 degrees or 6 mils to compensate. For all practical purposes, the trajectory is virtually flat. But it is not negligible, as you have those range marks etched into the back sights.
Now, at a much greater range, say 1,400 meters and the muzzle velocity of 905 m/sec, you get the half-distance super-elevation of 2.9 meters, which translates into 0.24 degrees or roughly 14 mils. And looking at the Wikipedia source, it turns out I'm pretty close to the calculated value of 13.2 mils. Damn.
As to the bullet casing plinking on the ground, who cares? It's just gravity.
There we go. You've sort of learned a little more about life through the wondrous information mechanisms of the hicks physics, not to be confused with Higgs. Some people, especially those who have not been tortured in high school or the university, might somehow assume that the spent, smoking casing is inferior to the mighty round zipping away, but in fact, they both suffer from the same effect of gravity.
With elevated angles, you get more fun. At low angles, you should use the word mils to show off. At 55 degrees, you get the best distance. At 90 degrees, it becomes super fun. And a bit more dangerous than you might expect. True, the bullets do not rain with the same ferocity as when fired, they veer and tumble and are carried away by wind, and there's terminal velocity to spoil the fun. But all in all, you still get a pretty decent punch of energy that can cause intellectual damage. Most gloriously demonstrated in The Mexican.
Bottom line, weddings are dangerous - avoid them if you can!
P.S. The tank and the artillery images are in public domain.