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).