If I fired an arrow into the sky.....

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moredes

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Given this hypothetical data, how far could this bullet travel if it didn't hit anything til it hit the ground?

.45 ACP
850fps
185 grain bullet
assume a BC of .135
barometric pressure- 29.70
300' above sea level
temp 75 degrees, humidity 60%

I'm told the optimum angle is ~30 degrees; let's say I fired it 'eyes-high', say 5' 7" (doubt whether the launch height contributes much to the results, though, wouldn't you?)

I'm just looking for a ballpark number--are we talking about 1400 yards, 1800 yards, 2400yds, 2 miles? I don't care about eeking out the last +-50 - 100yd, just a rough guess.

Thank you.
 
what is the angle? how far above the ground is it fired from?
to be close we also need barametric pressure, temperature and humididty data.
btw, drag coefficient also changes with velocity.
 
I read somewhere that a bullet fired at a 45 degree angle will be traveling faster when it reaches earth than it was when it was fired. Who knows if thats true or not.
 
Dude, thats some fun physics and calc right there!

Since I've not gotten to the physics yet, I'll wait until some other answers popup. But, I think I got a good guess ;)
 
My guess is the answer's probably closer to 12-1400yd than any other distance I listed.
 
I come up with 4.25 miles in a vacuum. Not the question asked, but unfortunately, I've forgotten enough physics that I can't figure out the calculations for drag anymore. I'm embarrassed. :(

-Ogre
 
Its an apples to oranges comparison, but when I was a kid, me and hte neighbor kid used to fire our compound bows into the air as we had about a mile of hay/corn fields behind our houses. The arrows probably weighed 400 gr, and probably at about 250 fps. they typically travelled 400-500 yards. Just a WASG, I would think the bullet would land around 1600 yards.



We had an old Navy manual dated 1946 or so that belonged to my great uncle in our house. IIRC it had a mention of a .45 fired at a 45 degree angle had a range around 1200 yards, but that was a long time ago that I read that, and I was probably only 10 year sold.
 
I came up with 3.75 mi in a vacumn :uhoh:

I'll post my work when I get a chance to look over it rested, and make sure I didn't make one of my signature mathematical screw-ups (ie, 2*3 = 5).

Maybe we just need a mathematician or physicysts (sp?) in here.
 
45 degrees will give you the most horizontal distance travelled for any given projectile at any given speed.

.45 ACP
850fps
185 grain bullet
assume a BC of .135
barometric pressure- 29.70
300' above sea level
temp 75 degrees, humidity 60%
Click to expand...

the bullet calliber, barometric pressure, temp, etc don't really matter unless you want to take wind resistance into account.

Usually calculations such as this neglect wind resistance, and only factor in the initial velocity of the projectile (which is dependent on charge, weight, and barrel length) and the negative acceleration due to gravity.

at least this is what was taught to me in my college physics a few years ago.:D

vertical distance - Dv
horizontal distance - Dh
Vertical Velocity= Vv
Initial velocity - Vi
Acceleration due to gravity - Ag
Initial height = Hi
Total time in flight - T
time - t

first find Vv=0 (appex of flight)
Vv = 0 = Vi (sin(angle))-Ag*t
therefore at t = Vi(sin(angle))/Ag seconds into flight, the Vv is 0.

At this point I think you can assume that T is 2* the time to appex of flight (in this case T=2*(Vi(sin(angle))/Ag)

then to find the horizontal distance travelled Dh
Dh = Vi*cos(angle) *T

I know you said...
I'm just looking for a ballpark number
Click to expand...

but I couldn't resist trying to recall this from memory. I hope someone else can confirm that I'm not too off base here.

This is probably more information than anyone wanted.:eek:
 
I read somewhere that a bullet fired at a 45 degree angle will be traveling faster when it reaches earth than it was when it was fired. Who knows if thats true or not.
Click to expand...
Not true. Think about it - when going up, the bullet's velocity is reduced by both gravity (the vertical component) and air resistance (both vertical and translational components.) When coming down, gravity accelerates it downward, but drag forces continue to oppose both translational and vertical motion. Terminal velocity will necessarily be less than initial velocity.
45 degrees will give you the most horizontal distance travelled for any given projectile at any given speed.
Click to expand...
This is the case in a vacuum. Bullets generally will achieve maximum range in air if fired at an inclination somewhere between 30 and 35 degrees - this varies by bullet, velocity, ballistic coefficient, etc. Ballistic calculations in undergraduate college courses invariably neglect drag forces as they're not readily and tidily computed from first principles . . . physics professors don't LIKE inelegant things like numerical integration and "fudge factors" like ballistic coefficients.

If one is talking about long range artillery, things like the Earth's curvature and rotation also have a noticeable effect . . . IIRC this was first noticed in a serious way during the long range naval duels between the English navy and the German High Seas Fleet at the Battle of Jutland during WWI.
 
At this point I think you can assume that T is 2* the time to appex of flight (in this case T=2*(Vi(sin(angle))/Ag)
Click to expand...
True in a vacuum but not true on a planet with an atmosphere.

One must take into account the effect that drag has. At apex the projectile has lost some horizontal velocity and the relative effect of the gravity component over the velocity component is increased - which is why the impact angle is considerably greater than the launch angle. In addition due to drag the projectile will not regain any horizontal velocity on the way down.

The loss of horizontal velocity would - all other things being equal - imply a longer downward time. However, due to the loss of horizontal velocity and the increased effect of the gravity component intuitively one would be led to believe that post apex flight time would be shorter than pre-apex flight time. Since the impact angle is greater than the launch angle intuitively it would seem that the time of travel post apex would be less than the time of travel pre apex.

The shape of the trajectory curve due to air resistance is not the shape of a pure parabola. The shape would be skewed thus the standard equation for a parabolic curve cannot be used to calcultate distance or flight times.

Drawing the hypothetical trajectory curve with the skewed shape would also tend to indicate that the downward time would be less than the upward time though that's just an intuitive assumption. I'm not sure but I think that solving for time would require the integration of the trajectory curve and then comparing the area under the curve less than apex distance to the area under the curve greater than apex distance. The real problem is getting the formula for the trajectory curve (however I suspect that since artillery can be fired with such great precision that tables exist somewhere that make knowing the forumla for the curve for the bullet in question would not actually be necessary).

It's been way too long since I had to use mathematics capable of proving the above but it seems intuitive to me that total flight time would be less than time to apex times two.

Should be an easy exercise though for those with both physics and advanced mathematics training.
 
The shape of the trajectory curve due to air resistance is not the shape of a pure parabola.
Click to expand...

Yes, notice the firing angle was 32 deg. and the terminal angle was 62 deg. not even close to a true parabola. Almost twice as steep on the return to Earth. Also the terminal velocity of only ~250 fps. This severe velocity decay due to gravity and drag is the reason for the trajectory not being symmetrical. At 250fps it certainly did not accelerate on the steep (62 deg) angle down either. This figure would ovbiously be higher for a low drag bullet...but still way lower than the start velocity.
 
I do remember reading an article about idiots who shoot bullets into the air for celebration. The author's calculations showed that almost any bullet (rifle or pistol) will reach the ground at roughly 300fps. Air resistance was the key. The other interesting point was that the bullets go up with their point to the sky and come down with their point to the sky.
 
A draggy bullet-type trajectory in an atmosphere is sometimes called an "impetus trajectory," IIRC, to differentiate it from a pure ballistic trajectory. Impetus trajectories give greatest range at low elevations. An extreme example is the trajectory of a golf ball when hit by a driver off the tee; the ball visibly "runs out of steam" in midair and falls much more steeply than the initial trajectory. A driver with a 45-degree face angle wouldn't hit the ball very far at all. (Golf isn't an exact analogy due to the role of aerodynamic Magnus-effect lift in the early part of the ball's flight, but it's the same idea.)

At the other end of the spectrum, really heavy artillery (e.g., railroad guns) produce maximum range at elevations of 60 to 70 degrees, IIRC, to get the projectile out of the dense lower atmosphere as quickly as possible, thereby greatly reducing the kinetic energy lost to the atmosphere.
 
Usually calculations such as this neglect wind resistance, and only factor in the initial velocity of the projectile (which is dependent on charge, weight, and barrel length) and the negative acceleration due to gravity.
Click to expand...
at least this is what was taught to me in my college physics a few years ago
Click to expand...

We tell you that when we teach college physics because the calculations to solve it with air resistance is a PITA. there is actually no good way to do it analyticaly (with an equation that you simply solve). You have to set up an iterative process and genearlly use a computer code.
The vacuum assumption is valid for heavy slow moving objects, but is way off base when dealing with small arms trajectory.

Best bet is to use the online calculator that someone posted the link to above.
 
The author's calculations showed that almost any bullet (rifle or pistol) will reach the ground at roughly 300fps.
Click to expand...
True if you're shooting more or less straight up. In that case, all of the velocity is bled off by air resistance and gravity and the terminal velocity is a result of the balance between air resistance and the pull of gravity.

If you're shooting at a significant angle off of vertical, then you have to consider more than terminal velocity. The DOWNWARD velocity component (pointing straight into the ground) won't be more than 300fps (whatever terminal velocity works out to) but if the angle wasn't very steep and the bullet is very aerodynamic, it may retain a significant amount of FORWARD velocity (pointing on the horizontal in the direction of bullet travel) that could only be bled off by air resistance.
 
At the other end of the spectrum, really heavy artillery (e.g., railroad guns) produce maximum range at elevations of 60 to 70 degrees, IIRC, to get the projectile out of the dense lower atmosphere as quickly as possible, thereby greatly reducing the kinetic energy lost to the atmosphere.
Click to expand...

This is what I understood also. I would think that an angle above 45 degrees would also favor bullets as well. Due to the thinner air at the higher elevation.
 
This is what I understood also. I would think that an angle above 45 degrees would also favor bullets as well. Due to the thinner air at the higher elevation.
Click to expand...

No, rifle bullets go about 11,000 feet or so straight up (obviously depends on caliber~10K for .223/~ 12K for .300WM) that isn't nearly high enough to realize any major air thinning effects, and that is straight up...so 45 deg. would = 5,500 feet maximum ordinate roughly.
http://www.eskimo.com/~jbm/ballistics/maxdist/maxdist.html this link provides straight up distance as well and you can check the rest of the sight for lots of other equations.

"Trajectories Basic" is the most usefull for making custom ballistic tables. I have found them to be very accurate for my rifle and handloads.
 
in a vacuum (i.e. no drag), with a perfectly spehrical earth, a bullet fired at a 45 degree angle, hypotheically, would land going faster than it was shot.

why? curvature of the earth. means it will have to fall just a bit farther to get to the same elevation relative to the earth's surface.
 
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