Was watching youtube and as usual, my feed had a random mythbusters scene in it. Naturally I started searching mythbusters and more stuff came into the feed.

But. With the good comes the bad and there were at least 3 videos stating how dangerous celebratory gunfire was. In fact one video's info-graphic depiction of a bullet falling after it hit its apex, was that of the bullet stabilizing itself and falling point down.

Now there is only so much BS I can take in a day but a though occurred to me.

Yes, celebratory gun fire can be lethal if the bullet retains its energy and stabilized trajectory IF fired at an angle.

Now for the question. (and I know there will be many people to correct me)
:Can a particular angle "spread" be found by calculating the BC of a bullet and will tell me at what angle that bullet will slow enough below terminal velocity and therefore could be considered a non lethal threat?

IE: projectile with a .477 BC will destabilize and slow within 90°-80° angle of fire but will retain trajectory at 79° onward making it possibly lethal.

There is a book called Hatcher's Notebook written after WW2 that covers shooting 30-06's that way. If I recall correctly the came back down butt first if they were fired at 90 degrees. At 78 degrees they would flip over and come back nose first. The return velocity wasn't that fast...280 fps maybe? I'm not sure what effect different bullet configurations would have.

A bullet will not restablilize and drop point first if the spin has stopped or reduced below the required rpm. That is my guess, the center of gravity is to the rear so I  am sure it will fall base first or i suppose tumble
A modern rifle bullet has its  center of pressure far in front of the center of gravity, it want to flip around, the spin is the only thing that prevents it.

As far as angle of fire, a bullet will destabilize and tumble at less than 1 degree if has flown far enough.

At 75 degrees it will be unstable and tumbling hundreds of feet up, and would be free falling basically, probably  quite dangerous though.

Once the bullets loses stability it will no longer have the .477 BC. So that number isn't useful.

I looked up Hatcher's Notebook. Pretty interesting read. It does touch on angle of departure but only for computing ordinates to trajectory.

The BC I guess is only as good as the speed it's traveling at so I guess the formula to find the "angle of failure" would have to include that variable.

There are times I wish there was software for this.

But in the end I'd like to come up with some simple formula to support a box O truth type theory that I have.
Where in the end I can tell people without hesitation that if the angle of fire was more than X° and if the weight was below a certain mark, that they could take their generized theory of shooting straight up and shove it.

Look at it this way what's required to make a bullet fired from say 80-110* lethal on return?

Speed.

I guess I can grab a ballistic calculator and make charts of increasing angle of departure until I see a degree in which the FPS dropped below a certain threshold.

But I'd like to make this as math related as I can.

What variables would you guys include into this formula?

Are you somehow trying to justify shooting into the air? It sounds as though you are.    We had a young boy killed here a few years ago at a July 4th celebration from celebratory gunfire from I believe a 10mm. No one was ever found or charged. The police distributed a map with a circle showing where it likely came from but I don't recall how many miles were encompassed in the circle.

He's asking for a mathematical formula that would show @ what angles a given bullet would be lethal, & which would not.  Which would actually aid police in looking for idiots who fire into the air.

OP, your top expected altitude is relatively simple; spin rates decrease relatively slowly over the time of a bullet's flight.  The hard part will be taking into account tumbling on the return fall which will further slow the bullet & delay a return to stability.

Only if you are going to ignore the wildly varying drag that you have no BC for.

He can approximate that as the velocity slows using calculus.  How, I don't remember, that was a long time ago.  

He can also do practical experiments.  If his formula predicts the bullet will run out of steam @ 3200 feet, fly a drone @ 3300 feet.  Lower by 10 feet @ a time until the drone is hit.  That will confirm or debunk his estimates.  

Best results might be to try to replicate the Mythbusters episode where they were testing this.

best I recall for a 9x19mm bullet of whatever weight (can't recall) they figured out how far it would penetrate into the local ground at  a few velocities and then started firing the pistol into the air of. a stand straight up ato 90* and waiting to hear a thud of it returning and find the hole, measure and have an estimate of impact velocity....they waited under something I think btw.

best I recall they concluded so far as the bullet went up straight enough to actually stop, then fall at terminal velocity it was non lethal, but if it had any arc on it that it was still quite plausible to be lethal.


To bad no one has tried it with a couple hundred square feet of ballistics gel lol. getting this down to a formula isn't realistic. Without very expensive fluid dynamic and physics simulation software I don't see it happening. Crude experiments are probably the best we could muster realistically.

Maybe you're overthinking the problem.  If you calculate/measure how much energy the bullet has when it hits the ground after being fired straight up, that's your minimum energy.  Your maximum energy is firing at 0 degree directly in to target, known lethal.  Shouldn't the equation be:  (maximum-minimum)*cos(angle)+minimum = expected impact energy?  Then you have to agree on what is a lethal amount of energy?

Seems like I'm going to spend some time on this page.

Lots of math goin on here.

I will return either with results or more questions.

Not really.

You need BC values valid over the range of atmospheric conditions.

With a multi degree of freedom model you can start to get some approximations but it is a lot more than just calculus as the velocity slows.

Supersonic through transonic to subsonic is still a real PITA.

The models undergo dramatic changes.

I think you have this backwards.  A very low drag bullet has its center of pressure behind its center of mass, like an arrow.

Please refer to figure 10.1, page 150, in the Book of Litz.