You fire a bullet and drop one -- which host the ground first

Cecil's Columns/Staff Reports
21

Okay, let’s forget the silly vacuum. Since you cannot by any stretch of imagination either discharge a firearm or observe a bullet in flight in a vacuum, this premise is moot, please let it die.
Now then: Everyone seems to have forgotten that there is an entire

scientific discipline devoted to this type of inquiry called “ballistics”

Basically, there are 3 major physical forces at work in the forced propulsion of a projecile,

whether it’s an arrow, a bullet, or a 50lb Naval artillery shell: inertia, drag, and gravity.

All three forces are working on the fired projectile, and they each enjoy a brief moment of

superiority over the other two physical forces.
Inertia is the first question to address in this hypothetical question. It makes a

very big difference indeed, if the bullet in question was fired from a wimpy .22 short

cartridge, with little gunpowder (and therefore, less compressed gas to propel the bullet out

of the barrel and into flight,) or if it was a high-powered rifle with 8x the propellant

force. The more propellant force behind the bullet when launched (fired,) the greater the

inertia, and therefore the farther AND FOR A LONGER DURATION it will remain in the air,

before drag and gravity eventually pull it down.
Drag is the second issue: is it a blunt-nosed “wadcutter” style bullet which suffers

a great deal of drag, and will therefore be pulled to earth from its flight rather quickly,

or a tapered, more aerodynamic full-metal jacketed bullet, which will fly as far as a bullet

can, before gravity can do its work?
The last and most obvious question is, how tall is the person dropping the bullet? I

imagine a few feet would make a big difference in such an equation.
As for the other factors of the “curved earth” and “bullets flying into space”, these

are too silly for ballistics, no projectile weapon made has ever been able to fling something

that far. These are only pertinent to cruise missiles or other long-distance self-propelled

devices. BTW - the cause of “arced” trajectories in ballistics is not the bullet itself arcing upwards, yet simply because the operator of the firearm, assuming he knows how to make accurate shots, HAS ALREADY POINTED THE MUZZLE OF THE FIREARM ABOVE A LEVEL PLANE BEFORE FIRING, either by crudely aiming higher than the target, or by setting his sights to cause the same effect. It is the position of the firearm upon firing, i.e., the angle of the barrel, that causes the bullet to begin its upward arc away from the level plane.

22

You mean momentum, which is still wrong. We’re talking about a vacuum, so drag has nothing to do with anything. And gravity is blind to momentum–the force still acts on the body in question, no matter how fast or slow its going. Sure if the bullet is going eight cajillion miles per hour it my not actually fall onto the surface, but the gravitational force acting on it at any given instant is the same as a bullet just loafing at the same height.

And can we just cut the crap on this earth’s curvature stuff? Let’s say you fire a bullet at 3,000 feet per second and drop a bullet from the same height above the perfectly round and utterly air-less surface of that earth-sized world, Britneyspearoid. From a height of five feet, how long would a bullet take to hit the ground, dropped? One second? Anyone care to tell me how much Britney’s surface has curved in 3,000 feet? One half of one smidge of one microdash. For all practical purposes, it hasn’t curved at all.

Finally, I’m no physicist, as you’ve already recognized, but isn’t all this talk of straight lines and curved surfaces silly anyway? Space is curved by definition. For local effects like a bullet dropping near the surface of Britneyspearoid, I would think that space is curved enough to make the curvature of the surface a perfectly flat plane in the context of our problem.

Next we’ll be talking about oribital velocities and what happens when you fire a gun with a muzzle velocity of 350,000 feet per second or whatever and the bullet goes into orbit five feet off the ground.

23

craxonius: Yes, indeed, there is a whole science called ballistics. A few of the notions on which it’s based:
Inertia is not a force, although you might say that it’s what the forces are acting on.
In general, more than one force can act on a body at the same time.
Gravity always acts on a body.
So, for instance, when a gun is fired in atmosphere, we can consider two stages: In the first, the bullet is still inside the barrel. There’s a downward force from gravity, and an upward force, equal in magnitude, from the bottom of the barrel, canceling the gravitational force. There’s also a drag force, pushing backwards, and a stronger force forward from te explosion of the propellant which causes the bullet to accelerate forward.
Now, the moment that the bullet leaves the barrel, the gravitational force is still acting on it, but the force from the bottom of the barrel isn’t, so it starts accelerating downward immediately. There’s also a component of the drag force now acting upward, but that’s not enough to stop it from falling. There’s also no longer a force from the propellant, so the air resistance is the only horizontal force acting on it, so it also immediately starts slowing down. With the exception of turbulent effects, the horizontal and vertical motions of the bullet are completely independent-- the time before it hits the ground depends only on the vertical forces, namely gravity and the vertical component of drag, which is itself dependent on the vertical componenet of the speed.

Meanwhile, if you insist that a gun can’t fire in vacuum (maybe most normal firearms can’t, but that’s a technical problem. It’d be fairly easy to make one that could), then just say it’s a crossbow, or a slingshot, instead. And why, pray tell, could we not observe a bullet in vacuum?

24

I believe that this is a little incorrect; the inertia of the bullet is the inertia of the bullet, no matter what the charge is. Of course, the effect of firing the bullet depends on both the inertia of the bullet and the charge.

True in practice; but angling the muzzle upwards destroys the correspondence between the two situations and the important physics lesson contained in the comparison.

Chemical-fuel projectile guns have been made for firing something into space (typically defined as beginning 50 miles up); see Gun Launch for Orbital Vehicles. Bull achieved altitudes of 100 miles. Of course, there’s been little or no practical application. Railguns (not chemically powered) are still being investigated (see The Electromagnetic Propulsion (EMP) Homepage).

Curvature of the Earth is a factor in firing battleship guns, and, of course, in the aiming of the Paris gun.

25

Chronos’ observation that there are two stages to the firing of a bullet (inside the barrel and outside) has an interesting implication…

In our hypothetical scenario, I think it’s most natural to assume that by “simultaneously” we mean that the explosive charge on the gun is ignited at the moment the other bullet is dropped.

This means that the dropped bullet gets a head start, since the fired bullet won’t start the downward component of its motion until after it’s been accelerated out of the barrel.

This would say that the dropped bullet always hits first, regardless of the effects of the earth’s curvature.

26

Guns fire at extreme altitudes quite reliably; a dearth of atmospheric oxygen sufficient to kill humans doesn’t phase Brownings at all. Guns fire underwater quite reliably (unless you leave poorly sealed cartridges submerged for a long time); it’s not a recommended practice because case pressures get freakishly high when you fire into a water-filled barrel. One the tests Ingram carried out on his MAC-10 prototype was firing it underwater in a swimming pool. Smokeless powder (like all similar propellants) requires oxygen to ignite, it does not require atmospheric oxygen. Although I concede that ordinary lubes might not be optimal for vacuum, as dtilque says, vacuum is the ideal theoretical condition for shooting modern firearms (I don’t know about black powder), spacesuit purchase price notwithstanding.

It would be much easier to do the problem if you assume a surface that is flat (I mean flat in the Euclidean sense, not flat as in “following the curvature of the Earth for a uniform altitude at all points”) for the maximum range of a bullet fired from an unelevated gun, say, a large, supernaturally flat parking lot. Also assume the vacuum, which eliminated air friction, rifling and other annoyances to the equation. Finally, assume that we drop the unfired bullet at the same instant the fired bullet leaves the muzzle. Both bullets hit the ground at the same time.

As Cecil concludes, if you don’t assume the “supernatural flatness” condition, there is no guarantee that the fired bullet ever hits the ground at all.

27

Wow, everyone is in such a big rush to prove Cecil wrong, they don’t bother to actually read his column. I was going to blanket post lots of detailed responses, but why bother? Most of the claims of Cecil being wrong boil down to two situations: (1) failing to read what Cecil actually said, and thus stating something he already addressed (i.e. curvature of the earth); (2) nit-picking the assumptions of the problem to show off how “smart” you are in finding all the little details that Cecil ignored. For example, all the arguments over the assumption of a vacuum - will the gun fire (oxidizer, lube), or does the falling bullet get a head start for the length of the barrel?

Sure, Cecil simplified the problem a little by ignoring the atmosphere - removing drag, turbulence, wind, etc. All those things affect the real-world case answer, but this is not really a real-world case problem. This is like the logic puzzles in those other threads - it’s a simplified case to demonstrate one principle of physics, not a complex mix of three or four chapters of information. You’d never learn physics if you had to address turbulence, wind direction, the local deviation in gravity field due to location on the Earth, etc, all on the first homework assignment. Similarly, Cecil ignored those irrelevant details. He could have discussed the timing of “firing the gun” and dropping the bullet, and whether they were simultaneous on trigger pull or exit of the barrel. Or he could have discussed the reflexes of the bullet dropper vs. the trigger puller, and the variations in timing of the two events by the same person (coordination and such). But those are all irrelevant to the question.

What he did address were (1) the person who thinks that the dropped bullet falls faster, (2) the person who knows they fall at the same acceleration, (3) the person who brings up the curvature, (4) if the bullet speed is the escape velocity. I think that’s a pretty thorough answer.

Now, something I can’t ignore.

craxonius said:

While what you say is essentially true, I believe the way you are saying it is misleading. I’m not sure if you are aware of why, so I’ll explain. Yes, the amount of propellent (combined with bullet size) can effect the length of flight time, but this assumes a ballistic path with an upward arc. How it affects flight time is how much upward component of push is given. As long as we go with the assumed position of level initial path, then the amount of propellent will NOT affect the amount of time of the fall (again ignoring earth curvature). If you aim the barrel upward at, say, 30 degs angle, you are giving the bullet an upward push of sin 30 of the thrust of the flat path. That gives an initial upward force that gravity has to counteract. If you used an angle of 60 deg, you get a larger upward force. You also get a shorter range, because less of the push is straight out. (Note: some initial angle gives more range than zero angle precisely because of gravity. Initial velocity, controlled by propellent, is critical in determining range. Actually, range increases, then decreases, and two angles will give you the same range but one has a flatter path than the other. I forget the exact transition values. 60 deg above may not be right - that may match the 30 deg range.)

And again on the propellent statement, if you have enough propellent, your velocity exceeds the escape velocity, and you get an orbital path - until you find the first building, or tree, or person in the way.

Again, gravity starts pulling down right away. Drag only affects how much range the bullet has, not how long it takes to fall.

Not really. As long as the bullet is dropped from the same height that the other one is fired, it doesn’t matter if it’s Hakeem Alajowan, or Mini-Me.

28

Don’t blame us for having a discussion here. What, is every thread in this forum supposed to have just two posts:

  1. Someone says “Yep, what Cecil said.”

  2. Arnold replies with a link to the original column. :slight_smile:

What fun is that?

Actually, I agree with you that people shouldn’t quibble about ignoring atmospheric effects when discussing these sorts of mythical situations. If Cecil states this as an explicit assumption in his answer, that’s fair.

If someone wants to drag :wink: the atmosphere topic back into the discussion, they should do so as an extension to the original problem, rather than as an argument with it.

But I think that the barrel-length angle is just as good as the earth-curvature one. And there was no stated assumption of a zero-length barrel in the original column. It at least points out that the original question was a little lacking in that it didn’t fully define the timing of the drops.

29

One of the above post brought up an interesting question – at which angle would you point the gun to get the maximum horizontal range? I think that we proved 45* in physics class one (gravity being the only other force), but not sure…

30

In physics class, like in Cecil’s column, you ignored the atmosphere in your calculations, and yes, then you get 45[sup]o[/sup].

But because atmosphere puts drag on the projectile, the actual max range angle is somewhat less than that.

31

I have decided to do some more

Well, not really, but I felt like expropriating that quote since all I really do around here is pick over details, and I felt like making fun of myself. I tried to post this link earlier today, and my browser did a faceplant. This one backs up my and dtilque’s contention that vacuum is quite a friendly environment to nitro propellants.

32

I think we should all remember that as far as the question originally went,it was posed by the writer’s highschool physics teacher; it was apparently to demonstrate a simple Newtonian physical law, and not to take into account every “real world” variable that might possibly take effect. So from that standpoint, Cecil’s basic answer is correct. However, he himself had to start the whole ball
a-rollin’ by elaborating for another paragraph or two, so here we all are…

33

charizard said:

I have no problem with people asking a question for clarification. I have no problem with people adding additional commentary that was not addressed by Cecil. Even playing nitpicking is okay, in most regards. (Hey, I do that, too.)

What bothers me is the incessant need to declare Cecil wrong. Because in almost all cases, he is not wrong. So by doing so you are not showing yourself to be smarter than Cecil, just more arrogant. But hey, if you want to look like an arrogant dolt, go right ahead and yell “CECIL IS WRONG” at the top of your lungs, and then go on to show how Cecil was not wrong, just did not address your particular little nitpick.

Do you get my point?

Oh, and if we start every thread off with the title “Cecil is Wrong!”, we’ll never keep straight which thread is which.

34
  1. assume we stand on a flat plane, with uniform gravity
  2. i don’t see that it matters, but assume a vacuum
  3. bullets can be fired in space or underwater. really.
  4. assume a firearm without an “arcing” aiming scheme
  5. modern firearms use rifling to give the bullet “spin”

i think the rifling is a more important element than
the curvature thing (important tho that may be).

the idea behind rifling is to stabilize the projectile
by giving it a sort of gyroscopic stay-on-courseness.
this helps the bullet resist some atmospheric effects,
but since we’re in a vacuum, the effect it resists
would be gravity.

wouldn’t the gyroscopic effect of the spinning bullet
make it fall slightly more slowly?

35

I just answered this in the second thread on this column, but here it is again.

Gyroscopic effects act to keep the axis of spin pointed a particular direction. They do not act to prevent motion.

Take a toy gyroscope and spin it up. You’ll find you have no trouble moving it up, down, or sideways. You only feel resistance when you try to twist its axis to a different alignment.

36

Atmosphere has two effects: The first is simple drag; anything moving through air has a force on it opposing the motion. This can be resolved into independent components, so the vertical component of drag on both bullets is the same: the force upward depends only on the characteristics of the object and the downward component of its velocity. The other effect is turbulance: The air pushed out of the way by the bullet will swirl around in unpredidictable ways, perhaps bumping into the sides of the bullet one or more times in the process, before making its way behind the bullet to fill thespace formerly occupied by the bullet. This can, in principle, produce different vertical forces on the fired and dropped bullets, causing one to hit before the other.
Actually, I think that the only reason that Cecil assumed vacuum in the first place is to get an orbit out of it.

37

Don’t forget relativistic effects! Since the bullet fired from the zero-length barrel vacuum-gun (held at exact level, although since the barrel has no length, I’m not sure how you measure that), over the supernaturally expansive flat parking lot is traveling faster than the bullet dropped from the same height, time moves slower for it. If a tiny person riding on each bullet timed their trips with stopwatches, wouldn’t the person on the fired bullet get a shorter time due to relativistic effects?

Naturally, to make this easier to observe, you’d fire the bullet pretty fast, say 0.9c or so.

I’m not even going to mention the problem of simultaneity- if two events are supposed to be truly simultaneous to all observers, don’t they have to occupy the same point in space? This would require the two bullets to either overlap in space, or have zero size. Oops, I guess I did mention it after all :slight_smile:

I think it’s clear how inadequate Cecil’s column was in addressing this issue! :slight_smile:

Arjuna34

38 
 Actually, the reason for rifling isn't to increase range, it's to increase accuracy. Sure, you get more range out of a spinning bullet, but you gave up range to get that spin. To get maximum range, you fire from an unrifled barrel and then deploy some sort of fins to spin it after it clears the barrel. Obviously not economic for ordinary bullets, though! (It *HAS* been done, though--subcaliber rounds fired from a 16" gun into space. They didn't attain orbit, though.)
39

Bullets don’t float up when fired. In all modern military rifles (and most sporting arms as well) the sights are not aligned parallel to the barrel. The front sight is minutely lower so that the arc of the bullet passes the sighting line at two different points. The first point is relatively close to the muzzle and is of little importance. The second intersection is the point at which the sights are “zeroed”. The Armalite AR-15 (aka M16) has two rear sight rings, a short range aperture and a long range aperture in an L-shaped arrangement. By flipping up the desired ring, the user can select the range at which the fired bullet will cross the sight line. This arrangement reduces the need for “Kentucky windage” indetermining the necessary elevation of the sights in relation to the target.

40

Good one, Arjuna, you got me thinking for a few minutes. Of course, the observer we’re interested in isn’t riding the bullets, he’s standing on the ground. You can define simultenaity for spatially separated events provided that you specify the reference frame: In this case, we’re restricted in our choice of frame by the fact that gravity must be uniform in our frame. In that ground frame, then, (again assuming the supernatural parking lot, etc.), even if the bullet is fired at relativistic speed, they’ll both still hit the ground at the same time.