Not homework. (I’m way to old for that.) General interest. And no, I don’t need answer fast.
So I’m standing at the corner of 33rd and 5th Avenue, at the base of the Empire State Building. I take out my Glock 9 mm (no, I don’t really own one), aim it toward the Hudson River, place it at a 45 degree angle to the sidewalk. (Up, that is.)
And fire. Let’s say the muzzle velocity is 375 m/sec. Let’s assume there’s no encumbrances along the street that would get in the way of the bullet
Then, because no one bats an eye at a gun being fired in mid-town (this is classic, 1970s era NYC), I take the elevator up to the observation deck on the 86th floor of the ESB (320 m). Take out my Glock, aim it parallel to the street toward the Hudson and fire.
If you ignore the curvature of the earth and the rotation of the earth and the difference in elevation between the street level in Manhattan and ground level in New Jersey (fairly minor corrections in this example) and atmospheric effects (not at all minor in the real world), the street-level bullet will travel a horizontal distance of 14.3 km. The maximum height of the trajectory is 3.6 km. If you shoot along 33d St, it would end up somewhere near the corner of Post Ave. and Green Ave. in Lyndhurst, New Jersey.
Neglecting atmospheric effects, the bullet fired horizontally from the open-air observation deck (maybe around 320 m above street level and 335 m above sea level) will travel a horizontal distance of only 3.1 km., falling just short of the New Jersey bank of the Hudson but close enough to hit the Chart House restaurant, which sits at the end of a pier.
Again, in the real world you wouldn’t be able to shoot nearly that far because of the atmosphere.
The optimum angle of fire does decrease as your firing position goes up, though (albeit never to quite exactly horizontal), and your maximum range likewise increases. In fact, your maximum range increases in a very simple way: Your total three-dimensional range is increased by an amount exactly equal to your height at firing. Thus, if your maximum range from ground level is 14,300 m, and you go 300 m up a skyscraper, then your new maximum range will be 14,600 m.
Like with bibliophage’s calculations, this is of course assuming a flat plane, uniform gravity, no aerodynamic effects, and an inertial reference frame.
I can’t / won’t argue with the math but of what good is this information?
For as little as $10, I will stand, in still air & with any adjustments or 9mm pistol you might wish to use at 14,000 meters.
So I get a penny for every miss in distance only, the ability to reach that far. I will have over $10 before you even get close.
The point is, was this a math question or a request for information as close as could be gotten in the real world of NYC?
Then numbers would be wildly different with just the different ammo choices available, etc…
The OP did not state, ‘no air,’ the responders did.
If the OP is satisfied with the math answers in a non existent world and is happy to just learn a bit about what angle of fire does from about 4 feet from the ground to the top of a familiar building he knows, great, that has been shown convincingly, I even learn a bit I did not know.
I just question, from the OP as posted, if the answers given were what he was wanting.
Can I get any light on this from the posters? Or the OP?
I have an excel spreadsheet at home that models projectile flights, and incorporates atmospheric drag. I’ll mess around with it tonight and report back.
And if you fire a bullet parallel to the ground 38 stories up and also drop a bullet at the same instant, they will both hit the ground at the same time.
Same thing if you’re holding the rifle 5 feet above the ground.
A bullet dropped from 321.489 meters will strike the ground in 8.1 seconds. (9.8 meters/seconds^2).
A bullet fired horizontally from the same height will also strike the ground in 8.1 sec, but it will land much farther away. In order to tell how far away, you’ll need to know that particular bullet’s Ballistic Coefficient. The higher the BC, the farther the bullet will travel over a given time.
A headwind will slightly (relatively-speaking) reduce the distance a bullet will travel and a tailwind will slightly increase it but crosswinds will have the greatest effect on where the bullet lands. The stronger the crosswind and the longer (time-of-flight) the bullet is subjected to a crosswind, the more it will drift off-target.
But this is another situation where the atmosphere adds significant complication. For starters, it should be plausible that the dropped bullet does not maintain the same orientation as the fired one, and thus experiences different aerodynamic drag as it falls.
No. The result I posted assumes that all shots, at all heights, are fired at the optimum angle for that height. Flat horizontal is not the optimum angle for any height, and a shot that’s fired horizontally starting at ground level will travel a distance of 0 before hitting the ground.
In fact, even if both projectiles are spherical (so orientation doesn’t matter), the one that’s fired will still take longer to land than the dropped one. This is due to the fact that air resistance is proportional to the square of the speed.
So for a 9mm bullet weighing 4.1 grams, with a muzzle velocity of 375 m/s, I calculated the range to be 1420 meters, considering aerodynamic drag. Without drag, the range would be about 11,100 meters.
With drag, 1354 meters. Without drag, 3034 meters.
*Episode 125 – “Knock Your Socks Off”
Original air date: October 7, 2009
Bullet Fired vs. Bullet Dropped
Myth statement -
A bullet fired horizontally and one dropped from the same height will hit the ground simultaneously.
Status - Confirmed
Notes - Adam and Jamie first carried out two small-scale experiments, one using ball bearings (dropped vs. shot from a spring-loaded launcher), the other using paintballs (dropped vs. fired from a paintball gun). While the first experiment seemed to bear out the myth, the second one contradicted it; Adam attributed this result to imperfections in the paintball’s surfaces that caused them to veer slightly off course.
For full-scale testing, they started at a firing range and used a .45 caliber pistol to measure the distance a bullet would travel before hitting the ground. Since the ground there was not level, they set up a second test at Fort Mason. Once they had properly fine-tuned their mechanism to fire and drop the bullets at the same time, they found that the two bullets landed within 39.6 milliseconds of each other. Commenting that this difference was less than the duration of one film frame (shot at 24 frames per second), and thus short enough for the human eye not to notice, they declared the myth confirmed.*
A bullet’s design doesn’t provide for lift. Gravity will pull both bullets downward at the same accelerating rate until they can’t fall any further.
Air resistance may be proportional to the square of the speed but artillery and firearm experts rely on a projectile’s ballistic coefficient ratio, wind speed and direction, plus muzzle velocity, direction and elevation to determine where a projectile will strike a target or hit the ground.
Three bullets of different calibers and weights, but with the same BC, will travel in the same arc to the target, provided they all left the muzzle at the same muzzle velocity.
Three 115gr 9mm bullets, a flat-nose soft point, a hollow point, and a round nose, will have different BC’s and will not travel in the same arc to target (again assuming equal muzzle velocities).
The higher a bullet’s BC, the less drag it has, and the farther it will travel in the same amount of time.
I don’t see what the “but” is doing in there. All of those things are relevant, and don’t change the fact that air resistance is proportional to the square of the speed.
Air resistance and bullet velocity would begin to decrease as soon as an unpowered bullet exits the muzzle. That doesn’t provide a shooter with any real world information as to how much a barrel should be elevated in order to hit the center of a target. How quickly a bullet’s velocity will decrease is best judged by the bullet’s BC and muzzle velocity.