A simple question: if I fire a bullet from a gun, and drop a bullet from my other hand (which is held at exactly the same height) at the same time, which bullet will hit the ground first? This questioned aired on an episode of Qi last night here, and got me thinking.
According to Stephen Fry, host of Qi, the fact that both bullets fall the same vertical distance, and must therefore both arrive on the ground at the same time, is “counter-intuitive”.
The trouble is, Stephen, it’s counter-intuitive because it’s wrong, and your post-programme justification is just as wrong. It’s got sod-all to do with air resistance and experimental error, either. Allow me to explain…
With apologies for my atrocious inability to draw (and thus my excellent ability to borrow bits of clipart from around the globe), the scenario being posited is as follows:
From which it is clear, I hope, that both bullets traverse the same vertical distance, from shoulder to ground, regardless of the horizontal distance traveled. It is therefore (I would have thought) quite obvious, not counter-intuitive at all, that both bullets must arrive at the ground at the same time, given that both are falling the same distance in the same gravitational field.
Except, of course, that the real situation is this:
That is, the Earth’s surface is not flat and therefore the bullet which moves horizontally finds that the Earth’s surface is dropping away from it, a little bit. Putting in a couple of guidelines to make it a touch clearer:
Not only must the fired bullet fall the same vertical distance, x, as the bullet dropped from the hand, but it must then additionally fall a further vertical distance, y, resulting from the curvature of the Earth. The fired bullet must therefore fall very slightly more than the dropped bullet… and therefore will arrive on the ground slightly after the one dropped from the hand.
Now, you may argue that the extra falling distance is trivial, but it isn’t. Different firearms have different ranges, of course, but a reasonable generalisation, if a bit on the conservative side, is that an unimpeded bullet can fly 1600m (1 mile, in old money). Over 1600m, the Earth’s surface curves away from the tangent (the ‘true horizontal’) by about 20cm (the old “8 inches per mile” idea, mentioned here and governed by the formula Δ=√(R²+L²)-R, where R is the radius of the Earth, L is the horizontal distance traveled and Δ is the extra vertical distance caused by the curvature of the Earth).
So, the fired bullet has to fall 20cm further than the dropped bullet. Given a gravitational constant of 9.8m/s², and assuming a standing start, that extra 20cm will take about 0.2 seconds to fall… which isn’t much, but it’s easily within accurate measurement possibilities and certainly well outside “experimental error”.
If we generalise, therefore, any object which travels a certain distance horizontally away from an origin will, on the Earth, end up having to fall further to reach the Earth’s surface than if it had been dropped exactly at that point of origin. The extra fall might only amount to a fraction of a millimetre, but it’s there and will affect ‘drop time’. Anyone suggesting otherwise must be a Flat-Earther.
Unfortunately, in various parts of the Internet, lots of people get this point gloriously confused with the practicalities of aerodynamics, wind resistance, rifling and whether the gun’s ‘kick’ when fired affects one’s ability to fire perfectly horizontally, and much else besides! Forget all that irrelevant stuff: the problem, as described, is merely one of idealised falling bodies in a gravitational field. If you idealise a flat Earth while you’re at it, sure: you’ll get Stephen Fry’s asserted result. Unfortunately, the one thing you can guarantee in a gravitational field is that your surface won’t be flat… and that makes all the difference.
Dropped bullets really do arrive at the ground earlier than fired ones, assuming only that it’s a curvaceous world lacking an atmosphere… so Stephen Fry is accordingly wrong. (But Qi is still an entertaining programme!)
Oh… and Mythbusters measured a difference but then declared it was insignificant and that the two bullets arrived simultaneously after all. They got it wrong, too (too many factors at work to detail here, but little things like air resistance, their choice of drop mechanism, their firing mechanism and so on… all mean their results are irrelevant to the hypothetical case).
You r teh smart
Just for fun:
The trajectory of the fired bullet is not truly straight; it is, like the Earth it is flying above, curved. It never finds the Earth falling away, else it’d move upwards in relation to the Earth’s surface.
So, the bullet/Earth physics is, really, as if the Earth was flat.
Hi Richard:
Yup, the trajectory is curved (parabolic, I’d say). But that doesn’t mean it “doesn’t find” the Earth falling away from it. Getting something into orbit is all about “it finding” that the Earth is falling away from it at the same rate (or faster) as it itself is falling. Not getting into orbit (like a bullet) simply means the Earth is falling away from the object slower than it itself is falling. The rate of the Earth’s ‘falling away’ relative to the rate at which the object itself is falling is important in determining the persistence (or not) of an orbit… but in all cases, the Earth is indeed falling away.
There’s a picture of the concept in Newton’s original Principia.
I even gave you the mathematics of the rate of ‘falling away’ (i.e., the curvature of the Earth). I think you’ll find 20cm over 1600m is a significant amount of ‘falling away’, even to a bullet.
Howard, to be fair to Stephen Fry, the explanation that both hit the ground at the same time, was how I was taught in physics at school. Now admittedly, that was back in 1974-75 (second year Physics) and things may have changed by now.
Having said that, your explanation make perfect sense.
Cheers,
Norm.
I may be being silly, excuse me.
If I drop a bullet out of my hand from say 10 m up and fire a bullet from exactly the same point, my reasoning is this.
The dropped bullet has gravity pushing it down and will hit the ground in for instance 2-3 seconds
The bullet fired from a gun has force behind it and will travel for up to mile before hitting the ground.
Am I missing the point here, Im not suggesting that Phyisics are wrong, Im just confused.
Hi Dominic. You have to do what is customary in these thought experiments: do them in a vacuum, with a perfectly spherical Earth. In those conditions, a fired bullet has a horizontal motion imparted by its charge (which, obviously) fires perfectly from a perfect gun and doesn’t therefore make the bullet travel slightly upward or downward.
The only thing which makes the bullet arc downwards is the vertical motion, directed towards the centre of the Earth, due to gravity.. which will always be a constant 9.8 metres per second per second.
Your dropped bullet is also experiencing that same vertical motion. Exactly the same vertical motion as the fired bullet. It’s just that it has no horizontal motion vector to worry about.
In other words, the fired bullet does experience two forces:
|——————————————> (propulsion by propellant)
|
| (downwards by gravity)
|
/
The graceful parabolic curved trajectory that the fired bullet follows is merely what you get when you combine those two vectors at every point along the path. And the key thing is that the size of the downward force and its affect on the vertical motion of the bullet isn’t affected by the horizontal one at all. It’s entirely determined by the mass of the Earth, in fact. And it’s exactly the same force as a feather would experience; or a hammer; or indeed two cannon balls dropped from the top of the Leaning Tower of Pisa. Or, more to the point, your dropped bullet as well as your fired one.
So, in a nutshell, the fired bullet’s horizontal motion doesn’t affect its vertical motion at all (again, if we were in an atmosphere, it is quite likely that the horizontal motion would generate some lift, which definitely would counteract the downward force of gravity somewhat, so we have to idealize that away).
Hence, it is true, that in an idealised world with a completely flat surface, the two bullets would both land at the same time. It’s only the extra vertical distance that has to be covered by the fired bullet that makes a difference to the outcome. It’s its displacement across a horizontal that curves that’s the problem, not merely the fact that it’s displaced.