Thread: Question from a TV show

Hello, my name is Jon and I'm a researcher for several television shows including CSI, CSI: Miami, Bones and a show on TNT called Rizzoli & Isles. It's my job to make sure our show is as accurate as possible. Contributors to this forum have been kind enough to help us in the past and I'm helping you can do so again.

While we often have a ballistics expert help determine the path a bullet took. we want to bring in a mathematician in this scene to use some formula to make the same determination. We have the bullet impact on a statue so we know the angle of impact. What other factors would you need and what formula would you use to help determine where the shooter was located?

Perhaps this isn't possible as there are many variables, but anything you can tell us would truly be appreciated.

Many thanks

Jon

Originally Posted by jonfromtv

Hello, my name is Jon and I'm a researcher for several television shows including CSI, CSI: Miami, Bones and a show on TNT called Rizzoli & Isles. It's my job to make sure our show is as accurate as possible. Contributors to this forum have been kind enough to help us in the past and I'm helping you can do so again.

While we often have a ballistics expert help determine the path a bullet took. we want to bring in a mathematician in this scene to use some formula to make the same determination. We have the bullet impact on a statue so we know the angle of impact. What other factors would you need and what formula would you use to help determine where the shooter was located?

Perhaps this isn't possible as there are many variables, but anything you can tell us would truly be appreciated.

Many thanks

Jon

I am a mathematician. I am also an avid shooter.

There is no "formula". What one does have are ballistic modeling programs that solve differential equations that include the mass, initial velocity of the bullet, the ballistic coefficient of the bullet and air drag to calculate a nominal trajectory. The actual trajectory will deviate slightly from the nominal due to a variety of factors related to the gun, the ammunition and atmospheric conditions.

Ballistics programs are readily available to the public. Some are better than others in that they actually solve the differential equations. RCBS has a good one, written by a physicist who is also a shooter.

I would suggest that in your TV show you bring in a physicist or a ballistics engineer rather than a mathematician. If the show is Rizzoli & Isles, based in Boston, I would have them go to the Army experts at Aberdeen Proving Grounds in Maryland.

A mark on a statue is not the ideal way to calculate a trajectory. You need at least two points of reference to determine the actual angle. In other words, an entrance and an exit would allow you to determine the exact path. Any surface that the bullet would ricochet off of does not give that to you. You could examine the mark left and attempt to determine angle based on where the bullet went afterwards, but surface irregularities, and a host of other factors could make that impossible.

In any case, once you determine the angle that the bullet was coming in at, either through a path or 'ricochet analysis', from there it would just be wind factors, air resistance, and gravity. Theoretically, if you know the type of bullet, the density and hardness of the statue, you could calculate the velocity of the bullet. From there, you would adjust for air resistance, gravities effect, and you could calculate with a reasonable degree of certainty from where the bullet was fired, but you would have to make a lot of assumptions.

If you are doing it for a TV show, you can probably eliminate a lot of the assumptions. If you give the statue a hat, clothing, thin metal, or something that can be shot through, you can arrive at an accurate angle. If you recover the firearm, then you can know for sure the mass and aerodynamics of the bullet. From there, all you have to do is take into account wind and gravity.

In any case, without an accurate angle and velocity, there will always be at least two possible shooting positions. One, shooting up and the bullet falling, and two shooting down with the bullet falling. Both could produce almost identical trajectories, although shooting down would obviously be an easier shot.

This is great info! Thank you so much. So if we recover the bullet, know where it ricocheted, we can determine the angle it took. Is there ANY formula you can give us for our scientist to put into his calculator that wouldn't make you laugh at it's ridiculousness? Something like, "We know the mass of the bullet. If we adjust for gravity, we could determine the shooter was located about BLANK yards away by BLANKING BLANK with BLANK."

You see what we're going for? Ahh, TV. Taking the complex and reducing it to one sentence. Can you think of anyway to say this on TV in a realistic manner? We really appreciate it and hope we're not cheating too much here.

Best

Jon

Originally Posted by whoisjohngalt

In any case, without an accurate angle and velocity, there will always be at least two possible shooting positions. One, shooting up and the bullet falling, and two shooting down with the bullet falling. Both could produce almost identical trajectories, although shooting down would obviously be an easier shot.

If you consider the bullet as a spinning rigid body (which it is) rather than as an idealized point mass, the impact angle of the axis of the projectile will be quite different for those two trajectories. The bullet axis is not in general tangent to the trajectory for a spin-stabilized projectile and the trajectories that you suggest require long range where this effect becomes important.

Note that it was explicitly stated that bullet impact angle is determinable from the impact -- though admittedly this requires a fortunate choice of materials and geometry. (It would be good if the statue were wood or rather soft and thin metal.)

Dr. Rocket, since you are an avid shooter, as am I, you know that many times, even at a shooting range, you have to aim up to hit the target. I am unaware, both mathematically, and from a practical shooting matter, how you could tell the difference between a bullet that was fired downwards, from say a tower, or a bullet that was fired from ground level and was on it's way back down, assuming both bullets traveled the same distance. Both would still maintain stability until the point where air resistance caused the bullet to stop spiraling. Unless the bullet was fired straight up, that is almost a negligible consideration, especially considering how many more important factors we are leaving out for the sake of simplicity. Namely wind resistance and wind speed.

As far as equations, there are a ton of results on google to calculate a bullet trajectory, as well as a ton of programs like Dr. Rocket mentioned. Most of which make a lot of assumptions and I doubt the reasoning behind any of them can be condensed into one sentence.

The best you can hope for, realistically and short, would be a piece of clothing or thin material to give you an accurate angle, and then use some computer generated graphics to simulate a parabolic path, arriving, or course at the place where it makes sense to put it for the script. This would leave it to the viewer to assume that some math went into the specific path.

Originally Posted by whoisjohngalt

Dr. Rocket, since you are an avid shooter, as am I, you know that many times, even at a shooting range, you have to aim up to hit the target. I am unaware, both mathematically, and from a practical shooting matter, how you could tell the difference between a bullet that was fired downwards, from say a tower, or a bullet that was fired from ground level and was on it's way back down, assuming both bullets traveled the same distance. Both would still maintain stability until the point where air resistance caused the bullet to stop spiraling. Unless the bullet was fired straight up, that is almost a negligible consideration, especially considering how many more important factors we are leaving out for the sake of simplicity. Namely wind resistance and wind speed.

That is the whole point -- the angle of the axis of the bullet to a horizontal line will be relatively constant over the trajectory so long as the gyroscopic effect is stabilizing the bullet, and that distinguishes the two trajectories that you suggest. If the trajectories are nearly flat the effect is very small, but in that case the trajectories are not very different anyway. this is the difference between a spin-stabilized projectile like a bullet and an aerodynamically stabilized projectile like an arrow.

Originally Posted by whoisjohngalt

The best you can hope for, realistically and short, would be a piece of clothing or thin material to give you an accurate angle, and then use some computer generated graphics to simulate a parabolic path, arriving, or course at the place where it makes sense to put it for the script. This would leave it to the viewer to assume that some math went into the specific path.

A bullet trajectory is NOT parabolic except in a vacuum. That is precisely why you need to solve the differential equations that include air resistance. If you are close enough for a parabolic approximation to suffice, you are probably close enough for a straight line to be adequate as well -- under 100 yds. Remember that while missing a target by a foot is poor marksmanship, locating the shooting point within a foot is quite adequate.

I didn't say a bullet's path was parabolic. I said the easiest way to make it close to accurate and fast was to simulate a parabolic path. I also specifically mentioned a script and making the explanation as quick as possible.

It seems like we are arguing the same argument. The difference is the OP needs to present the info in under 15 seconds, and we have days to debate whether or not the trajectory is influenced by whether the bullet was hollow point or full metal jacket. I could argue that using a custom loaded cartridge, leaving the weapon to be found, and with a magazine full of ordinary cartridges, I could actually make it look like a bullet was fired from a 5 story window instead of a 1 story window by using a cartridge with less gunpowder, therefore less velocity, and aiming up, thereby changing the angle of impact, but still keeping the same distance; just not the height...but that might be too much of a tv script...

At this point, we are arguing semantics and points way to minute for the OP to go into during a 15 second explanation on a TV show.

The discussion has been entertaining, though, and I do agree with everything you have written. I just believe you have a higher standard of accuracy than is possible, on TV or reality. All variables just simply cannot be taken into account in anything other than a science experiment. In reality, you are trying to reconstruct data after you knew that data was important. There is no way the correct wind speed is going to have been recorded. No way to know if the bullet was factory or custom loaded. No way to know what angle the shooter aimed at. No way to know if he had modified the bullet. Simply put, CSI and the investigations that they do are fake. The calculations are, of course, possible, but we rarely have the data that we would need.

Basically, it would be a lot quicker and easier to make something that looks realistic in photoshop than it would be to explain something that is actually realistic. Just my .02.

So this is the difference between one technique (seemingly obvious and accurate) and another (mathematically more complex and more accurate).

So, the bullet passes through the statue piercing first one part and then the other. You can look through the two holes, even using a rifle scope, and identify the window on the roof or maybe the umpteenth floor as the origin of the shot. But a mathematician might use more math and determine that the bullet trajectory must have arched, do some math, and determine the shot came from several stories below the level assumed above.

Or the ballistics expert may have assumed the bullet arched, but the mathematician may have also realized that the bullet arched AND slowed down during its trajectory, so it came from a lower spot.

HOWEVER, in reality, I'd think that the ballistics expert would know as much about the math of trajectories (maybe more) than what a mathematician knows. Unless, the ballistics expert is knowledgeable about short-range shots and internal ballistics (path inside the body), but not so knowledgeable as the mathematician about relatively long shots. But why would the mathematician know more? Maybe the mathematician grew up in the country and was well-versed with rifles and their trajectories.

Why not call upon a real life case.

Several decades ago (~50 years), a female driver on a coastal highway (in California or Florida) was shot dead in the back of the head by someone who was apparently in the back seat of the car. It took authorities a while to find the shooter.

The bullet came from a .303 Enfield rifle, which aren't so common. They investigated all the .303 Enfield owners in the area. They found an owner who was, on the day/time of the shooting, in a boat on the ocean, plinking at bottles and cans in the water. He happened to be aiming inland. They estimate that one of the the bullets ricocheted off the water, traveled about 800 yards (~½ mile) over water and land, at no greater than several yards above the surface, and entered the car's left rear window, which was rolled down, and fatally struck the driver in the back of the head.

Totally freak incident, but reckless, and (I think) they convicted the shooter of involuntary manslaughter.

Originally Posted by jrmonroe

So this is the difference between one technique (seemingly obvious and accurate) and another (mathematically more complex and more accurate).

So, the bullet passes through the statue piercing first one part and then the other. You can look through the two holes, even using a rifle scope, and identify the window on the roof or maybe the umpteenth floor as the origin of the shot. But a mathematician might use more math and determine that the bullet trajectory must have arched, do some math, and determine the shot came from several stories below the level assumed above.

Or the ballistics expert may have assumed the bullet arched, but the mathematician may have also realized that the bullet arched AND slowed down during its trajectory, so it came from a lower spot.

HOWEVER, in reality, I'd think that the ballistics expert would know as much about the math of trajectories (maybe more) than what a mathematician knows. Unless, the ballistics expert is knowledgeable about short-range shots and internal ballistics (path inside the body), but not so knowledgeable as the mathematician about relatively long shots. But why would the mathematician know more? Maybe the mathematician grew up in the country and was well-versed with rifles and their trajectories.

Why not call upon a real life case.

Several decades ago (~50 years), a female driver on a coastal highway (in California or Florida) was shot dead in the back of the head by someone who was apparently in the back seat of the car. It took authorities a while to find the shooter.

The bullet came from a .303 Enfield rifle, which aren't so common. They investigated all the .303 Enfield owners in the area. They found an owner who was, on the day/time of the shooting, in a boat on the ocean, plinking at bottles and cans in the water. He happened to be aiming inland. They estimate that one of the the bullets ricocheted off the water, traveled about 800 yards (~½ mile) over water and land, at no greater than several yards above the surface, and entered the car's left rear window, which was rolled down, and fatally struck the driver in the back of the head.

Totally freak incident, but reckless, and (I think) they convicted the shooter of involuntary manslaughter.

Interesting but
1. The .303 was the standard British cartridge for years and 50 years ago .303 Enfields were sold by the car load very cheaply -- maybe $10-$25 each.
2. Those Enfields were sold without records, often by mail, sometimes from barrels full of them in surplus stores. There is no way the authorities could investigate all Enfield owners or even identify them
3. A ricocheting bullet will almost always tumble and take a very erratic path, not, "no greater than several yards above the surface" for 600 yards.

This has all the earmarks of an urban legend.

Thank you all again for the information you provided. Hope I wasn't asking too ridiculous a question. I also hope I can come back here again when science questions for our TV shows come up. You've been a big help and we really do appreciate it. We have professionals in criminology, law, medicine, etc., but I often find the best info can be found on forums like this. Glad you proved me right again!

Cheers

jon

This is a really good post. I have nothing to add to the technical argument above...

...but it is fascinating how a scenrio that appears simplistic on the surface is so complex when regarded in detail.

This was a bona fide news story. Perhaps .303s were a dime a dozen in many areas, but maybe not where this death occurred. The authorities were otherwise clueless, and apparently checked out the local owners that they were aware of. Erratic tumbling only affects accuracy in regards to aim and, because there was no aiming involved per se, it wouldn't matter much.

Remember, reality is not always probable, or likely, and some real-life situations are counter-intuitive. Take for example, bullets that have less penetration power closer to the muzzle and more penetration power farther away. This happened years ago when bullet manufacturing was not perfected, and this could still happen today for shooters who pour their own. What happened was impurities would get into the poured lead and cause the center of mass to misalign with the bullet's geometric center. As the bullet traveled down the barrel, it would rotate about its geometric center. Upon leaving the barrel, the bullet's center of mass would disrupt this spinning and cause gyrations. Penetration at this point in the trajectory would amount to partial keyholing. Allowed enough revolutions, the bullet's spin would restabilize about the center of mass, and the bullet longitudinal axis would better align to the bullet's trajectory, allowing greater penetration, even though its speed had decreased somewhat.

Yeah, unfortunately the math needed for this isn't the kind of thing most people, even mathematicians, would do on a calculator. That said, it might be possible to use the parabolic trajectory as a first approximation, and that could be done on a calculator. As others have pointed out, you'd need impact angle and velocity, which is a separate issue, and it wouldn't be accurate, but it could narrow down the area to search enough to be useful to the plot.

Originally Posted by jrmonroe

This was a bona fide news story. Perhaps .303s were a dime a dozen in many areas, but maybe not where this death occurred. The authorities were otherwise clueless, and apparently checked out the local owners that they were aware of. Erratic tumbling only affects accuracy in regards to aim and, because there was no aiming involved per se, it wouldn't matter much.

Erratic tumbling affects range, penetration, and lethality. The trajectory of a tmbling projectile bears no resemblance to hat of a stabilized projectile.

Do you have a reliable reference for this story ? I would believe it was a story far more readily than I would believe it is true.

Originally Posted by jrmonroe

Remember, reality is not always probable, or likely, and some real-life situations are counter-intuitive. Take for example, bullets that have less penetration power closer to the muzzle and more penetration power farther away. This happened years ago when bullet manufacturing was not perfected, and this could still happen today for shooters who pour their own. What happened was impurities would get into the poured lead and cause the center of mass to misalign with the bullet's geometric center. As the bullet traveled down the barrel, it would rotate about its geometric center. Upon leaving the barrel, the bullet's center of mass would disrupt this spinning and cause gyrations. Penetration at this point in the trajectory would amount to partial keyholing. Allowed enough revolutions, the bullet's spin would restabilize about the center of mass, and the bullet longitudinal axis would better align to the bullet's trajectory, allowing greater penetration, even though its speed had decreased somewhat.

I have seen unstable projectiles key-hole at 25 yards. The deviation from normal at that distance was measured in feet (could not reliably hit a target 2 feet on a side). At 800 yards, if it got that far, a bullet could be anywhere. The one thing that could be pretty well guaranteed is that it would not travel that distance within only a few yards of the ground.

Originally Posted by DrRocket

Do you have a reliable reference for this story ?

It was in a Reader's Digest maybe the late 1960s. A few other people on the internet remember it too. Apparently the article's title was "A shot in the dark".

http://www.thehulltruth.com/boating-...e-gunfire.html

http://msgboard.snopes.com/cgi-bin/u...2;t=000302;p=1

We're remembering the same specific details. 303 Enfield, left rear window, woman driver, etc.

Originally Posted by jrmonroe

Take for example, bullets that have less penetration power closer to the muzzle and more penetration power farther away. This happened years ago when bullet manufacturing was not perfected, and this could still happen today for shooters who pour their own. What happened was impurities would get into the poured lead and cause the center of mass to misalign with the bullet's geometric center. As the bullet traveled down the barrel, it would rotate about its geometric center. Upon leaving the barrel, the bullet's center of mass would disrupt this spinning and cause gyrations. Penetration at this point in the trajectory would amount to partial keyholing. Allowed enough revolutions, the bullet's spin would restabilize about the center of mass, and the bullet longitudinal axis would better align to the bullet's trajectory, allowing greater penetration, even though its speed had decreased somewhat.

This is a different reference from a book about 100 years old. I cannot recall title, author, etc.