How Wrong is "Wrong"?

WARNING: Here there be PHYSICS!

(And math too, of course).

So here's an absurd scenario that is nevertheless "simple" in all the relevant aspects.  Despite the recent sequestration, NASA has decided it's high time they hired and trained some new astronauts for all those manned missions they're not planning.  Thinking you have nothing better to do in todays dead horse of an economy, you decide to apply.  And lucky you, you're hired!

Three day's later, you find yourself in NASA's zero-G vacuum chamber, engaging in a very stenuous space station repair training exercise.  Unfortunately, budget cuts mean that the anti-gravity device was poorly developed and even less well-tested.  It suddenly cuts out in the middle of your training excercise, and you find yourself hanging from an artificial space station one hundred meeters above the ground.

Your half-cost supervisor assures you that the super-foam balls filling ball-pen floor of the chamber (budget cuts, remember) will provide more than enough to take your fall.  But you're really high up.  You remember your training manual stating that you can survive a fall into the ball-pen only if you are travelling less than fourty-five meters per second upon impact.  So what do you say?  Do you drop?

Now I know this scenario sounds pretty implausible, but it's mostly designed to make the physics simpler, so my point is not obscured by unnecessary complexity.  Now, if you know anything about physics, you'll probably know that these sorts of simple kinematic problems are typically solved using Newtonian Mechanics.  However, you may also know that when you get right down to it, Newtonian Mechanics isn't correct.  So if that's the case, why the heck to physicists still use it?  Why do they still teach it?

Well I'm glad you asked ;)  Let me show you.

Using Newtonian Mechanics, this problem can be solved very easily using the concept of energy.  When someone is sitting still at some height above the earth, we say they have an amount of "potential energy" equal to their mass "m" times their height "h" times the earth's acceleration due to gravity "g."  In math-speak:

U = mgh

Now as you fall, you lose potential energy.  But energy is still conserved, so where does it go?  Well in a vacuum, it all goes into kinetic energy "T", which is related to your velocity "v" as:

T = (1/2)mv^2

So at the top of the chamber, all of your energy is potential energy.  At the bottom, all of it is kinetic.  And both energies are equal.  So this tells us:

U = T     ->     mgh = (1/2)mv^2

From this point we execute some simple algebra and find v = sqrt(2gh).  Plugging in the standard 9.8m/s^2 for g (this is something you can look up in pretty much any textbook) and the 100m you're sitting at, and we have v = 44.271887242 m/s.  Whew, you're safe!

But wait, isn't this all wrong?  After all, it's Newtownian Mechanics.  Wouldn't we do better to use Reletivity?  Well, yes... but also no.  While it's true that this result is not perfectly correct, it is correct enough for virtually any application you may ever find yourself designing.  But let's take a look at what relativity has to say on the matter.

Reletivistically, all mass contains energy.  This is the famous E = mc^2, which is only true for thins at rest (i.e. not moving).  This is in addition to any potential energy you may have.  So when you're hanging from the artificial space station, your energy is:

Hanging Energy = mc^2 + mgh

Now when you reach the bottom, your energy isn't all in the form of kinetic energy.  There's still come energy stored up in your mass.  The equation for your kinetic energy when you have used up all your potential energy can be found in any simple textbook on relativity, or even wikipedia.  It is:

Fallen Energy = mc*sqrt(c^2 + v^2)

Where here "c" is speed of light in a vacuum, which is roughly three hundred million meters per second.  Of course, these energies must be equal.  So we do some algebra.  Terms and factors that "cancel" are highlighted:

mc^2 + mgh = mc*sqrt(c^2 + v^2)

c^4 + 2ghc^2 + g^2 h^2 = c^4 + c^2 v^2

2gh + (g^2 h^2)/c^2 = v^2

v = sqrt(2gh)*sqrt[1 + g^2 h^2 / 2c^2]

We see here that Special Relativity tells us the velocity should be different. We have that same sqrt(2gh) factor, but now it's multiplied by something that is clearly larger than one.  So wait, would you really hit the ball-pen floor at less than fourty-five meters per second?  Well, if you plug in the numbers you get v = 44.2718872428 m/s

So I guess technically you can say that our first calculation was wrong.  But it's important to say just how wrong.  And the answer is barely.  In fact, the wrongness isn't all that much bigger than what one might get from rounding errors in your standard six-function pocket calculator.  It's WAY less wrong than the wrongness that comes when you use 3.14 as pi, and it's even WAY less wrong than the wrongness that both answers have by using g = 9.8 m/s^2 instead of slightly-more-accurate g = 9.81 m/s^2.  In fact, the answer that Newtownian Mechanics gave us differes from the answer that Special Relativity gave us by less than 1% of 1% of 1% of 1% of 1%.  And for trading this tiny, tiny bit of accuracy, we gain the ease of not having to deal with that messy mc^2 stuff, or any of the other mathematical complications.  We can make our prediction faster, and lose only the tiniest bit of accuracy.

THAT is why we still teach Newtonian Mechanics.  For the VAST majority of macroscopic but earthbound applications, Newtonian Mechanics is more than accurate enough.  You can use it to build roads, rails, houses, and all manner of useful things.  So yeah, you can argue all you want that Newtonian Mechancis is "wrong," but that doesn't change the fact that it is extraordinarily accurate for a wide variety of applications.  And if you want to argue that Newotonian Mechanics is the same kind of "wrong" as phlogiston, crystal healing, or the Myan apocalypse, well then you're the one who's wrong.


P.S.  This is what I mean when I say that science is backwards-compatible.  Special Relativity doesn't just correct the errors that Newtonian Mechanics makes with large quantities, it also explains why Newtonian Mechanics doesn't appear to make mistakes with small quantities.  And since here "small" means "smaller than virtually everything observable before the twentieth century," it's no wonder that Newtonian Mechanics looked so right for so wrong.  It was.

Author's Notes
The astute among you may have noticed that I didn't use General Relativity here, even though I'm dealing with gravity.  Basically, I didn't want to make things any more mathematical than absolutely necessary, so I handled the gravity Newtonianly and handled everything else relativistically.  So I'll just say here that the corrections you'd get from using General Relativity instead of Special Relativity are nowhere near significant to this problem.

Also, just figured I'd point out that all the algebra here was performed on one side of a spare envelope I had lying around... (though the numerics required the assistance of my six-function pocket calculator).